373:, epistemic uncertainties underlying the lack of knowledge about the physics of the modeled system give rise to mathematical operators associated with the computational model, which are deficient in a certain sense. Such operators lack certain properties linked to unmodeled physics. When such operators are discretized to perform computational simulations, their accuracy is limited by the missing physics. To compensate for this deficiency of the mathematical operator, it is not enough to make the model parameters random, it is necessary to consider a mathematical operator that is random and can thus generate families of computational models in the hope that one of these captures the missing physics. Random matrices have been used in this sense, with applications in vibroacoustics, wave propagations, materials science, fluid mechanics, heat transfer, etc.
1627:
13611:
398:
324:, which guarantees the theoretical high and low limits of the eigenvalues associated with a random variable covariance matrix. This matrix calculated in this way becomes the null hypothesis that allows one to find the eigenvalues (and their eigenvectors) that deviate from the theoretical random range. The components thus excluded become the reduced dimensional space (see examples in fMRI ).
4825:
309:
the synaptic weights crosses a critical value, at the limit of infinite system size. Results on random matrices have also shown that the dynamics of random-matrix models are insensitive to mean connection strength. Instead, the stability of fluctuations depends on connection strength variation and time to synchrony depends on network topology.
1453:
8951:
6645:, the conditional distribution of the spectrum inside the disks also converges to a uniform distribution. That is, if we cut the shrinking disks along with the spectrum falling inside the disks, then scale the disks up to unit area, we would see the spectra converging to a flat distribution in the disks.
5575:
4590:
1902:
344:
control variables. With linear evolution, matrices of coefficients appear in the state equation (equation of evolution). In some problems the values of the parameters in these matrices are not known with certainty, in which case there are random matrices in the state equation and the problem is known
299:
with random matrices is a key reason for the wide use of free probability in other subjects. Voiculescu introduced the concept of freeness around 1983 in an operator algebraic context; at the beginning there was no relation at all with random matrices. This connection was only revealed later in 1991
308:
In the field of computational neuroscience, random matrices are increasingly used to model the network of synaptic connections between neurons in the brain. Dynamical models of neuronal networks with random connectivity matrix were shown to exhibit a phase transition to chaos when the variance of
7878:
3424:
9142:
6122:
6905:
8686:
5243:
320:, it is important to be able to select the number of significant components. The criteria for selecting components can be multiple (based on explained variance, Kaiser's method, eigenvalue, etc.). Random matrix theory in this content has its representative the
4574:
9648:
5883:
1275:
8715:
5386:
1749:
9297:
361:(that is, with only additive uncertainty) the optimal policy with a quadratic loss function coincides with what would be decided if the uncertainty were ignored, the optimal policy may differ if the state equation contains random coefficients.
6245:
818:
4820:{\displaystyle \mathrm {d} \mu _{N}(\mu )={\frac {1}{{\widetilde {Z}}_{N}}}e^{-H_{N}(\lambda )}\mathrm {d} \lambda ,\qquad H_{N}(\lambda )=-\sum \limits _{j\neq k}\ln |\lambda _{j}-\lambda _{k}|+N\sum \limits _{j=1}^{N}Q(\lambda _{j}),}
8325:
7655:
5397:
2759:
2613:
5766:
946:
623:
7754:
3306:
2267:
7749:
9969:
2406:
10118:
8960:
3568:
9835:
2489:
1202:
6779:
2342:
3173:
8503:
3704:
7373:
4462:
4004:
3841:
2161:
7130:
3258:
2959:
300:
by
Voiculescu; he was motivated by the fact that the limit distribution which he found in his free central limit theorem had appeared before in Wigner's semi-circle law in the random matrix context.
9472:
1579:
2888:
135:, the BohigasâGiannoniâSchmit (BGS) conjecture asserts that the spectral statistics of quantum systems whose classical counterparts exhibit chaotic behaviour are described by random matrix theory.
8398:
7508:
4189:
8095:
8137:
8001:
10026:
4292:
4107:
8212:
5128:
4374:
9195:
5959:
8048:
5117:
3093:
6643:
2003:
8472:
3926:
3767:
1043:
7583:
5967:
5772:
1699:
2052:
722:
717:
10996:
Rychkov VS, Borlenghi S, Jaffres H, Fert A, Waintal X (August 2009). "Spin torque and waviness in magnetic multilayers: a bridge between Valet-Fert theory and quantum approaches".
9170:
11768:
Burda, Z; Kornelsen, J; Nowak, MA; Porebski, B; Sboto-Frankenstein, U; Tomanek, B; Tyburczyk, J (2013). "Collective
Correlations of Brodmann Areas fMRI Study with RMT-Denoising".
8217:
7290:
7588:
7177:
7038:
6971:
6457:
5293:
2644:
2520:
6512:
4916:
1126:
863:
536:
1612:
232:, with recent work utilizing random matrices to show that hyper-parameter tunings can be cheaply transferred between large neural networks without the need for re-training.
9467:
9403:
9345:
2824:
146:
model). Moreover, such random unitary transformations can be directly implemented in an optical circuit, by mapping their parameters to optical circuit components (that is
9696:
8437:
7931:
7439:
7230:
7203:
6998:
6932:
6734:
6348:
2997:
868:
545:
112:
to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the spacings between the
6771:
6562:
5282:
2188:
381:
Random matrix theory can be applied to the electrical and communications engineering research efforts to study, model and develop
Massive Multiple-Input Multiple-Output (
1448:{\displaystyle \langle H_{ij}H_{mn}^{*}\rangle =\langle H_{ij}H_{nm}\rangle ={\frac {1}{n}}\delta _{im}\delta _{jn}+{\frac {2-\beta }{n\beta }}\delta _{in}\delta _{jm},}
8498:
8171:
1089:
649:
8946:{\displaystyle R_{n,V}^{(k)}(x_{1},\dots ,x_{k})={\frac {n!}{(n-k)!}}\int _{\mathbf {R} }dx_{k+1}\cdots \int _{\mathbb {R} }dx_{n}\,p_{n,V}(x_{1},x_{2},\dots ,x_{n}),}
6486:
6161:
6153:
5632:
3453:
2785:
2639:
2515:
11966:
Chiani M (2014). "Distribution of the largest eigenvalue for real
Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution".
1963:
9860:
7905:
7545:
7403:
5605:
4216:
3599:
6684:
5570:{\displaystyle E_{Q}=\inf \limits _{\nu \in M_{1}(\mathbb {R} )}-\int \int _{x\neq y}\ln |x-y|\mathrm {d} \nu (x)\mathrm {d} \nu (y)+\int Q(x)\mathrm {d} \nu (x).}
4936:
4877:
4857:
10031:
6689:
Typically this means the study of spacings between eigenvalues, and, more generally, in the joint distribution of eigenvalues in an interval of length of order 1/
1897:{\displaystyle {\frac {1}{Z_{\beta ,n}}}\prod _{k=1}^{n}e^{-{\frac {\beta }{4}}\lambda _{k}^{2}}\prod _{i<j}\left|\lambda _{j}-\lambda _{i}\right|^{\beta }~,}
11164:
ZumbĂŒhl DM, Miller JB, Marcus CM, Campman K, Gossard AC (December 2002). "Spin-orbit coupling, antilocalization, and parallel magnetic fields in quantum dots".
9855:
9716:
9423:
9371:
9190:
8706:
6585:
6532:
6368:
5656:
4024:
3861:
3619:
1063:
9301:
The following result expresses these correlation functions as determinants of the matrices formed from evaluating the appropriate integral kernel at the pairs
6321:
3497:
3269:
The
Gaussian ensembles are the only common special cases of these two classes of random matrices. This is a consequence of a theorem by Porter and Rosenzweig.
2094:
9721:
2411:
1131:
216:-type inequalities can typically be strengthened when applied to the maximal eigenvalue (i.e. the eigenvalue of largest magnitude) of a finite sum of random
5663:
2201:
220:. Random matrix theory is used to study the spectral properties of random matricesâsuch as sample covariance matricesâwhich is of particular interest in
142:, transformations described by random unitary matrices are crucial for demonstrating the advantage of quantum over classical computation (see, e.g., the
6541:, one is interested in the limit distribution of eigenvalues in a set that shrinks to zero, but slow enough, such that the number of eigenvalues inside
11866:
985:. Its distribution is invariant under orthogonal conjugation, and it models Hamiltonians with time-reversal symmetry. Equivalently, it is generated by
3098:
13269:
7662:
4298:
Note that generic matrix ensembles do not allow us to grow, but most of the common ones, such as the three
Gaussian ensembles, do allow us to grow.
3624:
2347:
3772:
7047:
2893:
10686:
Bohigas, O.; Giannoni, M.J.; Schmit, Schmit (1984). "Characterization of
Chaotic Quantum Spectra and Universality of Level Fluctuation Laws".
1474:
7873:{\displaystyle Y_{n}:={\sqrt {4n\gamma _{n}}}\left({\frac {1}{\sqrt {n}}}\rho \left(G_{n}\right)-1-{\sqrt {\frac {\gamma _{n}}{4n}}}\right),}
7253:
As another example, consider the
Ginibre ensemble. It can be real or complex. The real Ginibre ensemble has i.i.d. standard Gaussian entries
3419:{\displaystyle \mu _{H}(A)={\frac {1}{n}}\,\#\left\{{\text{eigenvalues of }}H{\text{ in }}A\right\}=N_{1_{A},H},\quad A\subset \mathbb {R} .}
2829:
10458:
8330:
7444:
6370:, the number of dimensions of the gaussian ensemble increases, the proportion of the eigenvalues falling within the interval converges to
3181:
are random
Hermitian matrices with density on the space of real symmetric/Hermitian/quaternionic Hermitian matrices, which is of the form
8050:
almost surely falls uniformly on the unit disc. and the edge statistics theorem states that the radius of the almost-unit-disk is about
209:
4398:
The limit of the empirical spectral measure of invariant matrix ensembles is described by a certain integral equation which arises from
2275:
10768:
Russell, Nicholas; Chakhmakhchyan, Levon; O'Brien, Jeremy; Laing, Anthony (2017). "Direct dialling of Haar random unitary matrices".
9137:{\displaystyle R_{n,V}^{(1)}(x_{1})=n\int _{\mathbb {R} }dx_{2}\cdots \int _{\mathbf {R} }dx_{n}\,p_{n,V}(x_{1},x_{2},\dots ,x_{n}).}
255:
associated with random matrix distributions implies that random matrices will not test large portions of an algorithm's input space.
13483:
12702:
7295:
6657:, one is interested in the limit distribution of eigenvalues in a set that shrinks so fast that the number of eigenvalues remains
3935:
2099:
13574:
3277:
The spectral theory of random matrices studies the distribution of the eigenvalues as the size of the matrix goes to infinity.
3184:
1714:
12063:
11944:
10320:
10293:
10274:
6900:{\displaystyle \Xi (\lambda _{0})=\sum _{j}\delta {\Big (}{\cdot }-n\rho (\lambda _{0})(\lambda _{j}-\lambda _{0}){\Big )}~,}
428:
Random matrix theory first gained attention beyond mathematics literature in the context of nuclear physics. Experiments by
13647:
13493:
13259:
10874:
Franchini F, Kravtsov VE (October 2009). "Horizon in random matrix theory, the
Hawking radiation, and flow of cold atoms".
7205:
should depend only on the symmetry class of the random matrix (and neither on the specific model of random matrices nor on
80:
4115:
8053:
8100:
12277:
8681:{\displaystyle p_{n,V}(x_{1},\dots ,x_{n})={\frac {1}{Z_{n,V}}}\prod _{i<j}(x_{i}-x_{j})^{2}e^{-\sum _{i}V(x_{i})}.}
7936:
9974:
5238:{\displaystyle \mathrm {d} \mu _{N}(\mu )={\frac {1}{{\widetilde {Z}}_{N}}}e^{-N^{2}I_{Q}(\nu )}\mathrm {d} \lambda ,}
824:
refers to the fact that the distribution is invariant under unitary conjugation. The Gaussian unitary ensemble models
11822:
11433:
10255:
10198:
4569:{\displaystyle {\frac {N_{f,H}-\int f(\lambda )\,dN(\lambda )}{\sigma _{f,n}}}{\overset {D}{\longrightarrow }}N(0,1)}
4392:
201:
4221:
4036:
316:, random matrix theory has been applied in order to perform dimension reduction. When applying an algorithm such as
11565:
Wainrib, Gilles; Touboul, Jonathan (March 2013). "Topological and Dynamical Complexity of Random Neural Networks".
11461:
Voiculescu, Dan (1991): "Limit laws for random matrices and free products". Inventiones mathematicae 104.1: 201-220
8176:
4313:
3460:
275:) is modeled by the distribution of eigenvalues of certain random matrices. The connection was first discovered by
5891:
13294:
12050:, SpringerBriefs in Mathematical Physics, vol. 26, Cham: Springer International Publishing, pp. 15â21,
11342:
Pennington, Jeffrey; Bahri, Yasaman (2017). "Geometry of Neural Network Loss Surfaces via Random Matrix Theory".
9643:{\displaystyle R_{n,V}^{(k)}(x_{1},x_{2},\dots ,x_{k})=\det _{1\leq i,j\leq k}\left(K_{n,V}(x_{i},x_{j})\right),}
8014:
4944:
825:
453:
354:
276:
121:
12042:
Livan, Giacomo; Novaes, Marcel; Vivo, Pierpaolo (2018), Livan, Giacomo; Novaes, Marcel; Vivo, Pierpaolo (eds.),
6117:{\displaystyle q(x)=-\left({\frac {Q'(x)}{2}}\right)^{2}+\int {\frac {Q'(x)-Q'(y)}{x-y}}\mathrm {d} \nu _{Q}(y)}
5878:{\displaystyle 2\int _{\mathbb {R} }\log |x-y|\mathrm {d} \nu (y)-Q(x)\leq l,\quad x\in \mathbb {R} \setminus J}
3013:
284:
12841:
6590:
4384:
2055:
1968:
1702:
170:
11626:
Timme, Marc; Wolf, Fred; Geisel, Theo (February 2004). "Topological Speed Limits to Network Synchronization".
10028:
is a complete sequence of monic polynomials, of the degrees indicated, satisfying the orthogonilty conditions
8953:
which are skew symmetric functions of their variables. In particular, the one-point correlation function, or
8442:
3872:
3713:
988:
13657:
7550:
1633:
321:
11369:
Yang, Greg (2022). "Tensor Programs V: Tuning Large Neural Networks via Zero-Shot Hyperparameter Transfer".
11103:
Janssen M, Pracz K (June 2000). "Correlated random band matrices: localization-delocalization transitions".
2023:
656:
13058:
12695:
12159:(1995). "On the Statistical Mechanics Approach in the Random Matrix Theory: Integrated Density of States".
9147:
7247:
7241:
5381:{\displaystyle M_{1}(\mathbb {R} )=\left\{\nu :\nu \geq 0,\ \int _{\mathbb {R} }\mathrm {d} \nu =1\right\}}
3175:
above the main diagonal are independent random variables with zero mean and have identical second moments.
2009:
2008:
The distribution of the largest eigenvalue for GOE, and GUE, are explicitly solvable. They converge to the
1710:
317:
68:
12408:"Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles"
13662:
13133:
7256:
7041:
1942:
221:
197:
11923:
Bohigas, Oriol; Weidenmuller, Hans (2015). Akemann, Gernot; Baik, Jinho; Di Francesco, Philippe (eds.).
11522:
Rajan, Kanaka; Abbott, L. (November 2006). "Eigenvalue Spectra of Random Matrices for Neural Networks".
7146:
7007:
6940:
6373:
6255:
The typical statement of the Wigner semicircular law is equivalent to the following statement: For each
1626:
51:
is the study of properties of random matrices, often as they become large. RMT provides techniques like
13289:
12811:
6567:
For example, the Ginibre ensemble has a mesoscopic law: For any sequence of shrinking disks with areas
2270:
1468:
10935:
SĂĄnchez D, BĂŒttiker M (September 2004). "Magnetic-field asymmetry of nonlinear mesoscopic transport".
13393:
13264:
13178:
11689:"Eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks"
10654:
Conference on Neutron Physics by Time-Of-Flight Held at Gatlinburg, Tennessee, November 1 and 2, 1956
9292:{\displaystyle \int _{B}R_{n,V}^{(1)}(x)dx=\mathbf {E} \left(\#\{{\text{eigenvalues in }}B\}\right).}
6494:
4885:
1100:
837:
510:
213:
11738:
1584:
409:
13652:
13498:
13388:
13096:
12776:
12109:
MarÄenko, V A; Pastur, L A (1967). "Distribution of eigenvalues for some sets of random matrices".
9428:
9376:
9304:
7232:). Rigorous proofs of universality are known for invariant matrix ensembles and Wigner matrices.
6741:
2793:
477:
461:
350:
252:
125:
44:
9653:
8403:
7914:
7408:
7208:
7182:
6976:
6910:
6712:
6326:
2964:
820:
is a normalization constant, chosen so that the integral of the density is equal to one. The term
13533:
13462:
13344:
13204:
12801:
12688:
12588:
11924:
11688:
6747:
6544:
6534:
increases, then we obtain strictly stronger theorems, named "local laws" or "mesoscopic regime".
6240:{\displaystyle {\frac {\mathrm {d} \nu _{Q}(x)}{\mathrm {d} x}}={\frac {1}{\pi }}{\sqrt {q(x)}}.}
5251:
2166:
370:
193:
116:
of a random matrix, and should depend only on the symmetry class of the underlying evolution. In
2190:
is the variance of off-diagonal entries. The variance of the on-diagonal entries do not matter.
13403:
12986:
12791:
8477:
8150:
5635:
1068:
813:{\displaystyle Z_{{\text{GUE}}(n)}=2^{n/2}\left({\frac {\pi }{n}}\right)^{{\frac {1}{2}}n^{2}}}
628:
358:
154:
64:
10652:
Block, R. C.; Good, W. M.; Harvey, J. A.; Schmitt, H. W.; Trammell, G. T., eds. (1957-07-01).
6462:
6131:
5610:
3431:
2764:
2618:
2494:
13349:
13086:
12936:
12931:
12766:
12741:
12736:
268:
248:
174:
40:
8320:{\displaystyle Z_{n}=\int _{M\in \mathbf {H} ^{n\times n}}d\mu _{0}(M)e^{{\text{tr}}(V(M))}}
1948:
496: = 4 for GSE. This index counts the number of real components per matrix element.
13543:
12901:
12731:
12711:
12650:
12600:
12553:
12419:
12316:
12168:
12118:
11936:
11881:
11787:
11703:
11645:
11584:
11531:
11480:
11471:
Sompolinsky, H.; Crisanti, A.; Sommers, H. (July 1988). "Chaos in Random Neural Networks".
11244:
11225:
Bahcall SR (December 1996). "Random Matrix Model for Superconductors in a Magnetic Field".
11183:
11122:
11069:
11015:
10954:
10893:
10840:
10787:
10695:
10509:
Wigner, E. (1955). "Characteristic vectors of bordered matrices with infinite dimensions".
10479:
10441:
10383:
10346:
10133:
7883:
7650:{\displaystyle \lim _{n\rightarrow \infty }{\frac {1}{\sqrt {n}}}\rho \left(G_{n}\right)=1}
7523:
7381:
5583:
4194:
3584:
1456:
465:
166:
32:
24:
6660:
4921:
4862:
4833:
2754:{\displaystyle p_{4}(s)={\frac {2^{18}}{3^{6}\pi ^{3}}}s^{4}e^{-{\frac {64}{9\pi }}s^{2}}}
2608:{\displaystyle p_{2}(s)={\frac {32}{\pi ^{2}}}s^{2}\mathrm {e} ^{-{\frac {4}{\pi }}s^{2}}}
340:
state variables through time depends at any time on their own values and on the values of
8:
13564:
13538:
13116:
12921:
12911:
12490:; RamĂrez, J.A.; Schlein, B.; Yau, H.T. (2010). "Bulk universality for Wigner matrices".
12487:
10187:
6737:
4029:
For example, we can "grow" a sequence of matrices from the Gaussian ensemble as follows:
2408:
is the mean spacing. The probability distribution of spacings is approximately given by,
1213:
117:
84:
12654:
12604:
12557:
12423:
12320:
12172:
12122:
11885:
11791:
11743:
Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization
11707:
11649:
11588:
11535:
11484:
11248:
11187:
11126:
11073:
11019:
10958:
10897:
10844:
10791:
10699:
10387:
10350:
13615:
13569:
13559:
13513:
13508:
13437:
13373:
13239:
12976:
12971:
12906:
12896:
12761:
12569:
12543:
12534:(2010). "Random matrices: universality of local eigenvalue statistics up to the edge".
12499:
12435:
12366:
12306:
12250:
12184:
12085:
11993:
11975:
11905:
11847:
11777:
11669:
11635:
11608:
11574:
11504:
11449:
11425:
11370:
11351:
11324:
11306:
11268:
11234:
11207:
11173:
11146:
11112:
11057:
11039:
11005:
10978:
10944:
10917:
10883:
10856:
10830:
10803:
10777:
10631:
10526:
10488:
10399:
10362:
10218:
10202:
9840:
9701:
9408:
9356:
9175:
8691:
6570:
6517:
6353:
5641:
4009:
3929:
3846:
3604:
3475:). If the integrated density of states is differentiable, its derivative is called the
1048:
651:
457:
346:
236:
20:
12612:
6262:
5761:{\displaystyle 2\int _{\mathbb {R} }\log |x-y|\mathrm {d} \nu (y)-Q(x)=l,\quad x\in J}
2061:
13626:
13610:
13413:
13408:
13398:
13378:
13339:
13334:
13163:
13158:
13143:
13138:
13129:
13124:
13071:
12966:
12916:
12861:
12831:
12826:
12806:
12796:
12756:
12616:
12403:
12384:
12332:
12294:
12273:
12254:
12202:
Johansson, K. (1998). "On fluctuations of eigenvalues of random Hermitian matrices".
12188:
12156:
12059:
12024:
11940:
11897:
11818:
11719:
11661:
11600:
11547:
11496:
11429:
11272:
11260:
11199:
11138:
11085:
11060:(April 1991). "Random matrices, fractional statistics, and the quantum Hall effect".
11031:
10970:
10909:
10860:
10821:
Verbaarschot JJ, Wettig T (2000). "Random Matrix Theory and Chiral Symmetry in QCD".
10665:
10623:
10467:
10403:
10316:
10305:
A First Course in Random Matrix Theory: for Physicists, Engineers and Data Scientists
10289:
10270:
10251:
3476:
3464:
1255:, and it models Hamiltonians with time-reversal symmetry but no rotational symmetry.
941:{\displaystyle {\frac {1}{Z_{{\text{GOE}}(n)}}}e^{-{\frac {n}{4}}\mathrm {tr} H^{2}}}
618:{\displaystyle {\frac {1}{Z_{{\text{GUE}}(n)}}}e^{-{\frac {n}{2}}\mathrm {tr} H^{2}}}
52:
12573:
12455:"Asymptotics for polynomials orthogonal with respect to varying exponential weights"
12439:
12230:
12130:
11997:
11909:
11838:
Turnovsky, Stephen (1974). "The stability properties of optimal economic policies".
11508:
11355:
11328:
11150:
10982:
10921:
10807:
10577:
10560:
10395:
10366:
2262:{\displaystyle \lambda _{1}<\ldots <\lambda _{n}<\lambda _{n+1}<\ldots }
13621:
13589:
13518:
13457:
13452:
13432:
13368:
13274:
13244:
13229:
13209:
13148:
13101:
13076:
13066:
13037:
12956:
12951:
12926:
12856:
12836:
12746:
12726:
12658:
12608:
12561:
12509:
12466:
12453:
Deift, P.; Kriecherbauer, T.; McLaughlin, K.T.-R.; Venakides, S.; Zhou, X. (1997).
12427:
12376:
12324:
12242:
12211:
12176:
12126:
12051:
11985:
11932:
11889:
11795:
11750:
11711:
11673:
11653:
11612:
11596:
11592:
11539:
11488:
11421:
11316:
11252:
11211:
11191:
11130:
11077:
11027:
11023:
10962:
10905:
10901:
10848:
10795:
10748:
10703:
10657:
10615:
10572:
10547:
10518:
10427:
10391:
10354:
10308:
6697:, pertaining to intervals inside the support of the limiting spectral measure, and
4399:
1934:
1263:
The ensembles as defined here have Gaussian distributed matrix elements with mean âš
1252:
539:
449:
441:
296:
244:
240:
217:
72:
13214:
12215:
11754:
11657:
11543:
11195:
11043:
10966:
10852:
10487:
Eynard, Bertrand; Kimura, Taro; Ribault, Sylvain (2015-10-15). "Random matrices".
10432:
10415:
13319:
13254:
13234:
13219:
13199:
13183:
13081:
13012:
13002:
12961:
12846:
12816:
12673:
12231:"A simple approach to the global regime of Gaussian ensembles of random matrices"
10475:
10437:
4379:
The limit of the empirical spectral measure for Wigner matrices was described by
333:
182:
158:
153:
Random matrix theory has also found applications to the chiral Dirac operator in
105:
36:
12055:
11256:
10551:
10194:
are identically distributed Gaussian random variables (either real or complex).
7292:, and the complex Ginibre ensemble has i.i.d. standard complex Gaussian entries
5391:
be the space of one-dimensional probability measures and consider the minimizer
251:. Although random entries are traditional "generic" inputs to an algorithm, the
71:, or scalar products between eigenvectors. Many physical phenomena, such as the
13579:
13523:
13503:
13488:
13447:
13324:
13284:
13249:
13173:
13112:
13091:
13032:
13022:
13007:
12941:
12886:
12876:
12871:
12781:
11989:
11715:
11492:
10799:
10449:
10411:
7744:{\displaystyle \gamma _{n}=\log \left({\frac {n}{2\pi }}\right)-2\log(\log(n))}
6537:
The mesoscopic regime is intermediate between the local and the global. In the
4388:
3456:
456:
could be modeled as a random matrix. For larger atoms, the distribution of the
225:
205:
143:
139:
91:, can be modeled mathematically as problems concerning large, random matrices.
76:
60:
12663:
12638:
12565:
12471:
12454:
12380:
12328:
12246:
12043:
11893:
11799:
11320:
10753:
10736:
10358:
9964:{\displaystyle \psi _{k}(x)={1 \over {\sqrt {h_{k}}}}\,p_{k}(z)\,e^{-V(z)/2},}
480:
are the Gaussian ensembles: GOE, GUE and GSE. They are often denoted by their
13641:
13584:
13442:
13383:
13304:
13299:
13224:
13153:
13027:
13017:
12946:
12866:
12851:
12786:
12620:
12388:
12336:
12295:"Local Semicircle Law and Complete Delocalization for Wigner Random Matrices"
11901:
11344:
ICML'17: Proceedings of the 34th International Conference on Machine Learning
11134:
11081:
10707:
10627:
10471:
10206:
6774:
4380:
2401:{\displaystyle \langle s\rangle =\langle \lambda _{n+1}-\lambda _{n}\rangle }
481:
445:
280:
264:
229:
147:
132:
109:
88:
56:
12354:
11867:"Random matrix theory for modeling uncertainties in computational mechanics"
10113:{\displaystyle \int _{\mathbf {R} }\psi _{j}(x)\psi _{k}(x)dx=\delta _{jk}.}
4459:). For many classes of random matrices, a central limit theorem of the form
13467:
13424:
13329:
13042:
12981:
12891:
12771:
11737:
Vergani, Alberto A.; Martinelli, Samuele; Binaghi, Elisabetta (July 2019).
11723:
11665:
11604:
11551:
11500:
11297:
Tropp, J. (2011). "User-Friendly Tail Bounds for Sums of Random Matrices".
11264:
11203:
11142:
11035:
10974:
10913:
10229:
8008:
429:
178:
11089:
10538:
Wishart, J. (1928). "Generalized product moment distribution in samples".
10312:
13309:
13279:
13047:
12881:
12751:
12527:
11640:
11239:
11178:
11117:
10949:
4310:, one is interested in the distribution of linear statistics of the form
4033:
Sample an infinite doubly infinite sequence of standard random variables
3563:{\displaystyle \mu _{H}(A)={\frac {1}{n}}\sum _{i}\delta _{\lambda _{i}}}
1615:
444:. Because there was no knowledge of direct nucleon-nucleon interactions,
113:
9830:{\displaystyle K_{n,V}(x,y):=\sum _{k=0}^{n-1}\psi _{k}(x)\psi _{k}(y),}
2484:{\displaystyle p_{1}(s)={\frac {\pi }{2}}s\,e^{-{\frac {\pi }{4}}s^{2}}}
1197:{\displaystyle {\frac {1}{Z_{{\text{GSE}}(n)}}}e^{-n\mathrm {tr} H^{2}}}
397:
13360:
12821:
12589:"A limit theorem at the edge of a non-Hermitian random matrix ensemble"
12431:
12180:
11851:
11412:
Keating, Jon (1993). "The Riemann zeta-function and quantum chaology".
10835:
10635:
10603:
10530:
7908:
1217:
437:
272:
162:
12513:
12084:
Meckes, Elizabeth (2021-01-08). "The Eigenvalues of Random Matrices".
12028:
12012:
10669:
10653:
10604:"Characteristic Vectors of Bordered Matrices With Infinite Dimensions"
10453:
10190:. In the important special case considered by Wishart, the entries of
6973:
captures the statistical properties of eigenvalues in the vicinity of
13594:
13168:
12531:
12452:
11375:
10767:
10493:
11739:"Resting state fMRI analysis using unsupervised learning algorithms"
10661:
10619:
10522:
4579:
3577:
Given a matrix ensemble, we say that its spectral measures converge
1945:. Eigenvalues repel as the joint probability density has a zero (of
13528:
12090:
10782:
12548:
12504:
12371:
12311:
11980:
11782:
11579:
11470:
11311:
11010:
10888:
10656:(Report ORNL-2309). Oak Ridge, Tennessee: Oak Ridge National Lab.
2015:
1933:
is a normalization constant which can be explicitly computed, see
12680:
10374:
Pastur, L.A. (1973). "Spectra of random self-adjoint operators".
2337:{\displaystyle s=(\lambda _{n+1}-\lambda _{n})/\langle s\rangle }
433:
187:
39:—that is, a matrix in which some or all of its entries are
12293:
ErdĆs, LĂĄszlĂł; Schlein, Benjamin; Yau, Horng-Tzer (April 2009).
10416:"Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture"
3168:{\displaystyle \left\{H_{n}(i,j)~,\,1\leq i\leq j\leq n\right\}}
1701:. They are normalized so that the distributions converge to the
1091:
matrix with IID samples from the standard normal distribution.
3699:{\displaystyle \lim _{n\to \infty }\mathbb {E} _{H}=\rho (A)}
460:
of the Hamiltonian could be computed in order to approximate
7368:{\displaystyle {\mathcal {N}}(0,1/2)+i{\mathcal {N}}(0,1/2)}
6701:, pertaining to intervals near the boundary of the support.
3999:{\displaystyle \lim _{n\to \infty }\mu _{H_{n}}(A)=\rho (A)}
3836:{\displaystyle \lim _{n\to \infty }\mu _{H_{n}}(A)=\rho (A)}
2156:{\displaystyle \rho (x)={\frac {1}{2\pi }}{\sqrt {4-x^{2}}}}
12407:
11767:
11625:
10995:
10199:
limit of the empirical spectral measure of Wishart matrices
382:
313:
11452:. Fields Institute Monographs, Vol. 35, Springer, New York
11163:
7125:{\displaystyle K(x,y)={\frac {\sin \pi (x-y)}{\pi (x-y)}}}
4443:, one is also interested in the fluctuations about â«
3253:{\textstyle {\frac {1}{Z_{n}}}e^{-nV(\mathrm {tr} (H))}~,}
2954:{\displaystyle \int _{0}^{\infty }ds\,s\,p_{\beta }(s)=1,}
120:, random matrices model the behaviour of large disordered
3769:
independently from the ensemble, then with probability 1,
3455:
is a deterministic measure; this is a particular case of
1251:. Its distribution is invariant under conjugation by the
12485:
12353:
Bourgade, Paul; Yau, Horng-Tzer; Yin, Jun (2014-08-01).
12270:
Random Matrices, Random Processes and Integrable Systems
11736:
11564:
10264:
7510:
We have the following theorem for the edge statistics:
4391:. As far as sample covariance matrices are concerned, a
1574:{\displaystyle E=e^{{\frac {1}{4N}}\|V+V^{T}\|_{F}^{2}}}
10283:
8011:. In words, the circular law says that the spectrum of
12013:"STATISTICAL PROPERTIES OF ATOMIC AND NUCLEAR SPECTRA"
10651:
10337:
Edelman, A.; Rao, N.R (2005). "Random matrix theory".
9172:
gives the expected number of eigenvalues contained in
7405:
be sampled from the real or complex ensemble, and let
3187:
3016:
2883:{\displaystyle \int _{0}^{\infty }ds\,p_{\beta }(s)=1}
436:
cannot be approximated to move independently, leading
11874:
Computer Methods in Applied Mechanics and Engineering
10685:
10303:
Potters, Marc; Bouchaud, Jean-Philippe (2020-11-30).
10034:
9977:
9863:
9843:
9724:
9704:
9656:
9475:
9431:
9411:
9379:
9359:
9307:
9198:
9178:
9150:
8963:
8718:
8694:
8506:
8480:
8445:
8406:
8393:{\displaystyle V(x):=\sum _{j=1}^{\infty }v_{j}x^{j}}
8333:
8220:
8179:
8153:
8103:
8056:
8017:
7939:
7917:
7886:
7757:
7665:
7591:
7553:
7526:
7503:{\displaystyle \rho (G_{n}):=\max _{j}|\lambda _{j}|}
7447:
7411:
7384:
7298:
7259:
7211:
7185:
7149:
7050:
7010:
6979:
6943:
6913:
6782:
6750:
6715:
6663:
6593:
6573:
6547:
6520:
6497:
6465:
6376:
6356:
6329:
6265:
6164:
6134:
5970:
5894:
5775:
5666:
5644:
5613:
5586:
5400:
5296:
5284:
is the above functional inside the squared brackets.
5254:
5131:
4947:
4924:
4888:
4865:
4836:
4593:
4465:
4316:
4224:
4197:
4118:
4039:
4012:
3938:
3875:
3849:
3775:
3716:
3627:
3607:
3587:
3500:
3434:
3309:
3272:
3101:
2967:
2896:
2832:
2796:
2767:
2647:
2621:
2523:
2497:
2414:
2350:
2278:
2204:
2169:
2102:
2064:
2026:
1971:
1951:
1752:
1636:
1587:
1477:
1278:
1134:
1103:
1071:
1051:
991:
871:
840:
725:
659:
631:
548:
513:
247:
to describe computation errors in operations such as
12048:
Introduction to Random Matrices: Theory and Practice
11389:
10558:
8147:
The joint probability density of the eigenvalues of
4184:{\displaystyle H_{n}=(G_{n}+G_{n}^{T})/{\sqrt {2n}}}
3869:, weak almost sure convergence means that we sample
11922:
10486:
10265:Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010).
10122:
8090:{\displaystyle 1-{\sqrt {\frac {\gamma _{n}}{4n}}}}
239:, random matrices have been used since the work of
10820:
10112:
10020:
9963:
9849:
9829:
9710:
9690:
9642:
9461:
9417:
9397:
9365:
9339:
9291:
9184:
9164:
9136:
8945:
8700:
8680:
8492:
8466:
8431:
8392:
8319:
8206:
8165:
8132:{\displaystyle {\frac {1}{\sqrt {4n\gamma _{n}}}}}
8131:
8089:
8042:
7995:
7925:
7899:
7872:
7743:
7649:
7577:
7539:
7502:
7433:
7397:
7367:
7284:
7224:
7197:
7171:
7124:
7032:
6992:
6965:
6926:
6899:
6765:
6728:
6678:
6637:
6579:
6556:
6526:
6506:
6480:
6451:
6362:
6342:
6315:
6239:
6147:
6116:
5953:
5877:
5760:
5650:
5626:
5599:
5569:
5380:
5276:
5237:
5111:
4930:
4910:
4871:
4851:
4819:
4568:
4368:
4286:
4210:
4183:
4101:
4018:
3998:
3920:
3855:
3835:
3761:
3698:
3613:
3593:
3562:
3447:
3418:
3252:
3167:
3087:
2991:
2953:
2882:
2818:
2779:
2753:
2633:
2607:
2509:
2483:
2400:
2336:
2261:
2182:
2155:
2088:
2046:
1997:
1957:
1896:
1693:
1606:
1573:
1447:
1196:
1128:is described by the Gaussian measure with density
1120:
1083:
1057:
1037:
940:
865:is described by the Gaussian measure with density
857:
812:
711:
643:
617:
530:
10873:
7996:{\displaystyle F_{\mathrm {Gum} }(x)=e^{-e^{-x}}}
6886:
6820:
6488:is the density of the semicircular distribution.
4580:The variational problem for the unitary ensembles
2193:
432:and others demonstrated evidence that individual
13639:
12401:
12154:
12010:
11815:Analysis and Control of Dynamic Economic Systems
11686:
10734:
10284:Akemann, G.; Baik, J.; Di Francesco, P. (2011).
10223:
10021:{\displaystyle \{p_{k}(x)\}_{k\in \mathbf {N} }}
9555:
7593:
7471:
7441:be the absolute value of its maximal eigenvalue:
3940:
3777:
3629:
224:. Random matrix theory also saw applications in
12292:
12041:
11521:
11341:
10934:
10737:"The computational complexity of linear optics"
10561:"Numerical inverting of matrices of high order"
10302:
2016:Convergence to Wigner semicircular distribution
12593:Journal of Physics A: Mathematical and General
12492:Communications on Pure and Applied Mathematics
12352:
12108:
8439:is the standard Lebesgue measure on the space
6587:inside the unite disk, if the disks have area
4287:{\displaystyle \{G_{i,j}\}_{i,j=1,2,\dots ,n}}
4102:{\displaystyle \{G_{i,j}\}_{i,j=1,2,3,\dots }}
1462:
1258:
188:Mathematical statistics and numerical analysis
12696:
10420:Bulletin of the American Mathematical Society
8207:{\displaystyle M\in \mathbf {H} ^{n\times n}}
4369:{\displaystyle N_{f,H}=n^{-1}{\text{tr}}f(H)}
3280:
1941: = 2), the formula (1) describes a
1455:from which all higher correlations follow by
1216:, e.g. symmetric square matrices composed of
303:
12011:Porter, C. E.; Rosenzweig, N. (1960-01-01).
11414:Proc. Internat. School of Phys. Enrico Fermi
11102:
11056:
10459:Notices of the American Mathematical Society
10001:
9978:
9857:, written in terms of the quasipolynomials
9278:
9267:
5954:{\displaystyle J=\bigcup \limits _{j=1}^{q}}
4245:
4225:
4060:
4040:
2395:
2363:
2357:
2351:
2331:
2325:
1595:
1588:
1555:
1535:
1348:
1319:
1313:
1279:
10286:The Oxford Handbook of Random Matrix Theory
9347:of points appearing within the correlator.
8043:{\displaystyle {\frac {1}{\sqrt {n}}}G_{n}}
5122:the probability measure is now of the form
5112:{\displaystyle H_{N}(\lambda )=N^{2}\left,}
4393:theory was developed by MarÄenko and Pastur
3088:{\textstyle H_{n}=(H_{n}(i,j))_{i,j=1}^{n}}
1272:â© = 0, and two-point correlations given by
13270:Fundamental (linear differential equation)
12703:
12689:
12459:International Mathematics Research Notices
11687:Muir, Dylan; Mrsic-Flogel, Thomas (2015).
11448:Mingo, James A.; Speicher, Roland (2017):
11400:
11224:
10336:
10212:
6934:are the eigenvalues of the random matrix.
5607:there exists a unique equilibrium measure
3490:
2012:after shifting and scaling appropriately.
364:
12662:
12547:
12503:
12470:
12370:
12310:
12201:
12089:
11979:
11837:
11781:
11639:
11578:
11374:
11310:
11238:
11177:
11116:
11009:
10948:
10887:
10834:
10781:
10752:
10576:
10559:von Neumann, J.; Goldstine, H.H. (1947).
10492:
10431:
10269:. Cambridge: Cambridge University Press.
9927:
9907:
9066:
9019:
8875:
8856:
7919:
6638:{\displaystyle A_{n}=O(n^{-1+\epsilon })}
5961:is the support of the measure and define
5865:
5785:
5676:
5439:
5353:
5311:
4859:is the potential of the ensemble and let
4503:
3928:, not independently, but by "growing" (a
3646:
3409:
3342:
3138:
2922:
2918:
2854:
2450:
2198:From the ordered sequence of eigenvalues
1998:{\displaystyle \lambda _{j}=\lambda _{i}}
12636:
12355:"Local circular law for random matrices"
12137:
11965:
11299:Foundations of Computational Mathematics
10735:Aaronson, Scott; Arkhipov, Alex (2013).
10448:
10410:
10234:
8467:{\displaystyle \mathbf {H} ^{n\times n}}
8142:
6155:has the following RadonâNikodym density
3921:{\displaystyle H_{1},H_{2},H_{3},\dots }
3762:{\displaystyle H_{1},H_{2},H_{3},\dots }
2641:, and
1625:
1038:{\displaystyle H=(G+G^{T})/{\sqrt {2n}}}
476:The most-commonly studied random matrix
357:does not apply: while in the absence of
312:In the analysis of massive data such as
13575:Matrix representation of conic sections
12261:
11411:
11285:
11279:
10719:
10717:
10537:
8214:, with partition functions of the form
7578:{\displaystyle \rho \left(G_{n}\right)}
7515:Edge statistics of the Ginibre ensemble
7143:principle postulates that the limit of
7001:
3572:
1694:{\displaystyle N=2^{0},2^{1},...,2^{5}}
13640:
12536:Communications in Mathematical Physics
12299:Communications in Mathematical Physics
12267:
12228:
12143:
12083:
10601:
10508:
10373:
10250:. Amsterdam: Elsevier/Academic Press.
7933:with cumulative distribution function
7246:One example of edge statistics is the
3463:of the limiting measure is called the
2790:The numerical constants are such that
2047:{\displaystyle {\sqrt {N\sigma ^{2}}}}
1705:. The number of "humps" is equal to N.
712:{\displaystyle H=(H_{ij})_{i,j=1}^{n}}
12684:
12671:
12586:
12526:
12359:Probability Theory and Related Fields
12348:
12346:
12103:
12101:
11864:
11296:
10723:
10245:
9165:{\displaystyle B\subset \mathbf {R} }
5636:Euler-Lagrange variational conditions
1965:th order) for coinciding eigenvalues
471:
353:with stochastic matrices is that the
196:, random matrices were introduced by
108:, random matrices were introduced by
11937:10.1093/oxfordhb/9780198744191.013.2
11812:
11749:(3). Taylor&Francis: 2168â1171.
11450:Free Probability and Random Matrices
11368:
10714:
10681:
10679:
10647:
10645:
10597:
10595:
8009:circular law of the Ginibre ensemble
6250:
1743:
1630:Spectral density of GOE/GUE/GSE, as
392:
12017:Ann. Acad. Sci. Fennicae. Ser. A VI
11383:
10288:. Oxford: Oxford University Press.
10127:
7911:, i.e., the probability measure on
7285:{\displaystyle {\mathcal {N}}(0,1)}
5902:
4879:be the empirical spectral measure.
4775:
4714:
2054:, converges in distribution to the
1621:
290:
267:, the distribution of zeros of the
13:
12710:
12343:
12098:
11426:10.1016/b978-0-444-81588-0.50008-0
11394:
10330:
10267:An introduction to random matrices
9264:
8365:
7952:
7949:
7946:
7603:
7337:
7301:
7262:
7235:
7192:
7172:{\displaystyle \Xi (\lambda _{0})}
7150:
7033:{\displaystyle \Xi (\lambda _{0})}
7011:
6966:{\displaystyle \Xi (\lambda _{0})}
6944:
6783:
6704:
6551:
6498:
6452:{\displaystyle \int _{}\rho (t)dt}
6420:
6398:
6304:
6282:
6195:
6169:
6091:
5817:
5708:
5548:
5513:
5496:
5360:
5225:
5133:
5085:
5050:
5033:
4677:
4595:
3950:
3787:
3639:
3343:
3273:Spectral theory of random matrices
3226:
3223:
3002:
2907:
2843:
2575:
1178:
1175:
922:
919:
599:
596:
327:
14:
13674:
12630:
10676:
10642:
10592:
10502:
8708:-point correlation functions (or
7907:converges in distribution to the
5869:
3621:, the ensemble-average converges:
13609:
11968:Journal of Multivariate Analysis
11390:von Neumann & Goldstine 1947
10182:) with independent entries, and
10123:Other classes of random matrices
10041:
10012:
9469:can be written as a determinant
9255:
9158:
9047:
8822:
8448:
8247:
8188:
7585:as above, with probability one,
7040:is known; thus, for GUE it is a
4301:
3461:cumulative distribution function
2761:for the symplectic ensemble GSE
2491:for the orthogonal ensemble GOE
828:lacking time-reversal symmetry.
396:
258:
13477:Used in science and engineering
12580:
12520:
12479:
12446:
12395:
12286:
12222:
12195:
12148:
12131:10.1070/SM1967v001n04ABEH001994
12111:Mathematics of the USSR-Sbornik
12077:
12035:
12004:
11959:
11916:
11858:
11831:
11806:
11761:
11730:
11680:
11619:
11558:
11515:
11464:
11455:
11442:
11405:
11362:
11335:
11290:
11218:
11157:
11096:
11050:
10989:
10578:10.1090/S0002-9904-1947-08909-6
10396:10.1070/RM1973v028n01ABEH001396
8139:, according to the Gumbel law.
8097:, and fluctuates on a scale of
6648:
6507:{\displaystyle \Delta \lambda }
5857:
5748:
4911:{\displaystyle H_{N}(\lambda )}
4687:
4405:
3401:
1121:{\displaystyle {\text{GSE}}(n)}
858:{\displaystyle {\text{GOE}}(n)}
531:{\displaystyle {\text{GUE}}(n)}
355:certainty equivalence principle
94:
12720:Explicitly constrained entries
12412:Journal of Statistical Physics
12272:. Springer. pp. 263â266.
11597:10.1103/PhysRevLett.110.118101
11028:10.1103/PhysRevLett.103.066602
10928:
10906:10.1103/PhysRevLett.103.166401
10867:
10814:
10761:
10728:
10454:"What is ... a random matrix?"
10307:. Cambridge University Press.
10082:
10076:
10063:
10057:
9997:
9991:
9945:
9939:
9924:
9918:
9880:
9874:
9821:
9815:
9802:
9796:
9753:
9741:
9718:th Christoffel-Darboux kernel
9685:
9673:
9629:
9603:
9548:
9503:
9498:
9492:
9454:
9448:
9334:
9308:
9242:
9236:
9231:
9225:
9144:Its integral over a Borel set
9128:
9083:
9004:
8991:
8986:
8980:
8937:
8892:
8807:
8795:
8778:
8746:
8741:
8735:
8670:
8657:
8627:
8600:
8555:
8523:
8426:
8420:
8343:
8337:
8312:
8309:
8303:
8297:
8284:
8278:
7964:
7958:
7738:
7735:
7729:
7720:
7600:
7496:
7481:
7464:
7451:
7428:
7415:
7362:
7342:
7326:
7306:
7279:
7267:
7189:
7166:
7153:
7116:
7104:
7096:
7084:
7066:
7054:
7027:
7014:
6960:
6947:
6881:
6855:
6852:
6839:
6799:
6786:
6760:
6754:
6673:
6667:
6632:
6610:
6548:
6514:can be allowed to decrease as
6475:
6469:
6440:
6434:
6426:
6382:
6310:
6266:
6229:
6223:
6189:
6183:
6111:
6105:
6073:
6067:
6053:
6047:
6011:
6005:
5980:
5974:
5948:
5922:
5845:
5839:
5830:
5824:
5812:
5798:
5736:
5730:
5721:
5715:
5703:
5689:
5561:
5555:
5544:
5538:
5526:
5520:
5509:
5503:
5491:
5477:
5443:
5435:
5315:
5307:
5271:
5265:
5219:
5213:
5153:
5147:
5098:
5092:
5081:
5075:
5063:
5057:
5046:
5040:
5028:
5014:
4964:
4958:
4905:
4899:
4846:
4840:
4811:
4798:
4764:
4736:
4704:
4698:
4671:
4665:
4615:
4609:
4563:
4551:
4540:
4516:
4510:
4500:
4494:
4385:Wigner semicircle distribution
4363:
4357:
4218:is the matrix made of entries
4163:
4132:
3993:
3987:
3978:
3972:
3947:
3830:
3824:
3815:
3809:
3784:
3693:
3687:
3678:
3675:
3669:
3656:
3636:
3517:
3511:
3326:
3320:
3239:
3236:
3230:
3219:
3129:
3117:
3059:
3055:
3043:
3030:
3010:are random Hermitian matrices
2939:
2933:
2871:
2865:
2813:
2807:
2664:
2658:
2540:
2534:
2431:
2425:
2317:
2285:
2194:Distribution of level spacings
2112:
2106:
2083:
2065:
1607:{\displaystyle \|\cdot \|_{F}}
1509:
1504:
1495:
1481:
1156:
1150:
1115:
1109:
1017:
998:
893:
887:
852:
846:
742:
736:
683:
666:
570:
564:
525:
519:
499:
452:approximated that the nuclear
376:
349:. A key result in the case of
171:fractional quantum Hall effect
1:
13494:Fundamental (computer vision)
12268:Harnad, John (15 July 2013).
12216:10.1215/S0012-7094-98-09108-6
11755:10.1080/21681163.2019.1636413
11658:10.1103/PhysRevLett.92.074101
11544:10.1103/PhysRevLett.97.188104
11196:10.1103/PhysRevLett.89.276803
10967:10.1103/PhysRevLett.93.106802
10853:10.1146/annurev.nucl.50.1.343
10586:
10433:10.1090/S0273-0979-03-00975-3
10224:Non-Hermitian random matrices
9462:{\displaystyle R_{n,V}^{(k)}}
9398:{\displaystyle 1\leq k\leq n}
9340:{\displaystyle (x_{i},x_{j})}
2819:{\displaystyle p_{\beta }(s)}
2615:for the unitary ensemble GUE
2269:, one defines the normalized
322:Marchenko-Pastur distribution
16:Matrix-valued random variable
10149:random matrices of the form
9691:{\displaystyle K_{n,V}(x,y)}
9425:-point correlation function
8432:{\displaystyle d\mu _{0}(M)}
7926:{\displaystyle \mathbb {R} }
7434:{\displaystyle \rho (G_{n})}
7225:{\displaystyle \lambda _{0}}
7198:{\displaystyle n\to \infty }
6993:{\displaystyle \lambda _{0}}
6927:{\displaystyle \lambda _{j}}
6729:{\displaystyle \lambda _{0}}
6693:. One distinguishes between
6343:{\displaystyle \lambda _{0}}
3932:), then with probability 1,
3465:integrated density of states
2992:{\displaystyle \beta =1,2,4}
1096:Gaussian symplectic ensemble
833:Gaussian orthogonal ensemble
492: = 2 for GUE, and
202:estimate covariance matrices
55:, diagrammatic methods, the
7:
13648:Algebra of random variables
13260:Duplication and elimination
13059:eigenvalues or eigenvectors
12613:10.1088/0305-4470/36/12/331
12056:10.1007/978-3-319-70885-0_3
11257:10.1103/PhysRevLett.77.5276
7042:determinantal point process
6766:{\displaystyle N(\lambda )}
6557:{\displaystyle \to \infty }
5415:
5277:{\displaystyle I_{Q}(\nu )}
3601:iff for any measurable set
2183:{\displaystyle \sigma ^{2}}
1943:determinantal point process
1910:
1741:of GUE/GOE/GSE is given by
1463:Moment generating functions
1259:Point correlation functions
440:to formulate the idea of a
222:high-dimensional statistics
63:to compute quantities like
10:
13679:
13193:With specific applications
12822:Discrete Fourier Transform
11990:10.1016/j.jmva.2014.04.002
11716:10.1103/PhysRevE.91.042808
11493:10.1103/PhysRevLett.61.259
10823:Annu. Rev. Nucl. Part. Sci
10602:Wigner, Eugene P. (1955).
10227:
10216:
10131:
8173:random Hermitian matrices
7239:
4410:For the linear statistics
3287:empirical spectral measure
3281:Empirical spectral measure
3179:Invariant matrix ensembles
1469:moment generating function
388:
304:Computational neuroscience
99:
49:Random matrix theory (RMT)
13603:
13552:
13484:CabibboâKobayashiâMaskawa
13476:
13422:
13358:
13192:
13111:Satisfying conditions on
13110:
13056:
12995:
12719:
12664:10.4249/scholarpedia.9886
12566:10.1007/s00220-010-1044-5
12472:10.1155/S1073792897000500
12381:10.1007/s00440-013-0514-z
12329:10.1007/s00220-008-0636-9
12247:10.1007/s11253-005-0241-4
11894:10.1016/j.cma.2004.06.038
11813:Chow, Gregory P. (1976).
11800:10.5506/APhysPolB.44.1243
11321:10.1007/s10208-011-9099-z
10754:10.4086/toc.2013.v009a004
10552:10.1093/biomet/20a.1-2.32
10359:10.1017/S0962492904000236
8493:{\displaystyle n\times n}
8166:{\displaystyle n\times n}
8007:This theorem refines the
3266:is called the potential.
2890:and the mean spacing is,
2056:semicircular distribution
2020:The spectrum, divided by
1084:{\displaystyle n\times n}
644:{\displaystyle n\times n}
506:Gaussian unitary ensemble
462:scattering cross sections
336:theory, the evolution of
283:. It is connected to the
11865:Soize, C. (2005-04-08).
11840:American Economic Review
11135:10.1103/PhysRevE.61.6278
11082:10.1103/PhysRevB.43.8641
10800:10.1088/1367-2630/aa60ed
10708:10.1103/PhysRevLett.52.1
10239:
7248:TracyâWidom distribution
7242:TracyâWidom distribution
6481:{\displaystyle \rho (t)}
6148:{\displaystyle \nu _{Q}}
6128:The equilibrium measure
5627:{\displaystyle \nu _{Q}}
3448:{\displaystyle \mu _{H}}
2780:{\displaystyle \beta =4}
2634:{\displaystyle \beta =2}
2510:{\displaystyle \beta =1}
2010:TracyâWidom distribution
956:real symmetric matrices
488: = 1 for GOE,
351:linear-quadratic control
285:HilbertâPĂłlya conjecture
253:concentration of measure
126:mean-field approximation
45:probability distribution
12842:Generalized permutation
12587:Rider, B (2003-03-28).
11925:"History â an overview"
11770:Acta Physica Polonica B
11628:Physical Review Letters
11567:Physical Review Letters
11524:Physical Review Letters
11473:Physical Review Letters
10213:Random unitary matrices
5638:for some real constant
4006:for any measurable set
3843:for any measurable set
3491:Alternative expressions
1703:semicircle distribution
371:computational mechanics
365:Computational mechanics
194:multivariate statistics
13616:Mathematics portal
12639:"Random matrix theory"
11401:Edelman & Rao 2005
10114:
10022:
9965:
9851:
9831:
9785:
9712:
9692:
9644:
9463:
9419:
9399:
9367:
9341:
9293:
9186:
9166:
9138:
8947:
8710:marginal distributions
8702:
8682:
8500:matrices, is given by
8494:
8468:
8433:
8394:
8369:
8321:
8208:
8167:
8133:
8091:
8044:
7997:
7927:
7901:
7874:
7745:
7651:
7579:
7541:
7504:
7435:
7399:
7369:
7286:
7226:
7199:
7173:
7126:
7034:
6994:
6967:
6928:
6901:
6767:
6730:
6680:
6639:
6581:
6558:
6528:
6508:
6482:
6453:
6364:
6344:
6317:
6241:
6149:
6118:
5955:
5921:
5879:
5762:
5652:
5628:
5601:
5571:
5382:
5278:
5239:
5113:
4932:
4912:
4873:
4853:
4821:
4794:
4570:
4370:
4288:
4212:
4185:
4103:
4020:
4000:
3922:
3857:
3837:
3763:
3700:
3615:
3595:
3564:
3449:
3428:Usually, the limit of
3420:
3254:
3169:
3095:such that the entries
3089:
2993:
2955:
2884:
2820:
2781:
2755:
2635:
2609:
2511:
2485:
2402:
2338:
2263:
2184:
2157:
2090:
2048:
1999:
1959:
1958:{\displaystyle \beta }
1937:. In the case of GUE (
1898:
1796:
1706:
1695:
1608:
1575:
1449:
1198:
1122:
1085:
1059:
1039:
942:
859:
814:
713:
645:
619:
532:
359:multiplier uncertainty
155:quantum chromodynamics
87:, or the emergence of
12637:Fyodorov, Y. (2011).
12229:Pastur, L.A. (2005).
12044:"Classified Material"
10608:Annals of Mathematics
10565:Bull. Amer. Math. Soc
10511:Annals of Mathematics
10313:10.1017/9781108768900
10235:Selected bibliography
10115:
10023:
9966:
9852:
9832:
9759:
9713:
9693:
9645:
9464:
9420:
9400:
9368:
9342:
9294:
9187:
9167:
9139:
8948:
8703:
8683:
8495:
8469:
8434:
8395:
8349:
8322:
8209:
8168:
8143:Correlation functions
8134:
8092:
8045:
7998:
7928:
7902:
7900:{\displaystyle Y_{n}}
7875:
7746:
7652:
7580:
7542:
7540:{\displaystyle G_{n}}
7505:
7436:
7400:
7398:{\displaystyle G_{n}}
7370:
7287:
7227:
7200:
7174:
7127:
7035:
6995:
6968:
6929:
6902:
6768:
6731:
6681:
6640:
6582:
6559:
6529:
6509:
6483:
6454:
6365:
6345:
6318:
6242:
6150:
6119:
5956:
5901:
5880:
5763:
5653:
5629:
5602:
5600:{\displaystyle E_{Q}}
5572:
5383:
5279:
5240:
5114:
4933:
4913:
4874:
4854:
4822:
4774:
4584:Consider the measure
4571:
4371:
4289:
4213:
4211:{\displaystyle G_{n}}
4186:
4104:
4021:
4001:
3923:
3858:
3838:
3764:
3701:
3616:
3596:
3594:{\displaystyle \rho }
3565:
3450:
3421:
3255:
3170:
3090:
2994:
2956:
2885:
2821:
2782:
2756:
2636:
2610:
2512:
2486:
2403:
2339:
2264:
2185:
2158:
2091:
2049:
2000:
1960:
1899:
1776:
1696:
1629:
1609:
1576:
1450:
1214:quaternionic matrices
1199:
1123:
1086:
1060:
1040:
943:
860:
815:
714:
646:
620:
533:
269:Riemann zeta function
249:matrix multiplication
175:Anderson localization
150:and phase shifters).
13658:Mathematical physics
12676:. Wolfram MathWorld.
11880:(12â16): 1333â1366.
10246:Mehta, M.L. (2004).
10134:Wishart distribution
10032:
9975:
9861:
9841:
9722:
9702:
9654:
9473:
9429:
9409:
9377:
9357:
9305:
9272:eigenvalues in
9196:
9176:
9148:
8961:
8716:
8692:
8504:
8478:
8443:
8404:
8331:
8218:
8177:
8151:
8101:
8054:
8015:
7937:
7915:
7884:
7755:
7663:
7589:
7551:
7524:
7445:
7409:
7382:
7296:
7257:
7209:
7183:
7147:
7048:
7008:
6977:
6941:
6911:
6780:
6773:. Then consider the
6748:
6713:
6679:{\displaystyle O(1)}
6661:
6591:
6571:
6545:
6518:
6495:
6463:
6374:
6354:
6327:
6323:centered at a point
6263:
6162:
6132:
5968:
5892:
5773:
5664:
5642:
5611:
5584:
5398:
5294:
5252:
5129:
4945:
4931:{\displaystyle \nu }
4922:
4886:
4872:{\displaystyle \nu }
4863:
4852:{\displaystyle Q(M)}
4834:
4591:
4463:
4314:
4222:
4195:
4116:
4037:
4010:
3936:
3873:
3847:
3773:
3714:
3708:weakly almost surely
3625:
3605:
3585:
3573:Types of convergence
3498:
3479:and is denoted
3432:
3353:eigenvalues of
3307:
3185:
3099:
3014:
2965:
2894:
2830:
2794:
2765:
2645:
2619:
2521:
2495:
2412:
2348:
2276:
2202:
2167:
2100:
2062:
2024:
1969:
1949:
1750:
1634:
1585:
1475:
1276:
1132:
1101:
1069:
1049:
989:
869:
838:
723:
657:
629:
546:
538:is described by the
511:
466:Wishart distribution
167:spin-transfer torque
81:thermal conductivity
79:of heavy atoms, the
25:mathematical physics
13565:Linear independence
12812:Diagonally dominant
12655:2011SchpJ...6.9886F
12605:2003JPhA...36.3401R
12558:2010CMaPh.298..549T
12424:1997JSP....86..109P
12321:2009CMaPh.287..641E
12173:1995JSP....79..585D
12123:1967SbMat...1..457M
11886:2005CMAME.194.1333S
11817:. New York: Wiley.
11792:2013AcPPB..44.1243B
11708:2015PhRvE..91d2808M
11650:2004PhRvL..92g4101T
11589:2013PhRvL.110k8101W
11536:2006PhRvL..97r8104R
11485:1988PhRvL..61..259S
11249:1996PhRvL..77.5276B
11188:2002PhRvL..89A6803Z
11127:2000PhRvE..61.6278J
11111:(6 Pt A): 6278â86.
11074:1991PhRvB..43.8641C
11020:2009PhRvL.103f6602R
10959:2004PhRvL..93j6802S
10898:2009PhRvL.103p6401F
10845:2000ARNPS..50..343V
10792:2017NJPh...19c3007R
10741:Theory of Computing
10700:1984PhRvL..52....1B
10388:1973RuMaS..28....1P
10351:2005AcNum..14..233E
10188:conjugate transpose
9502:
9458:
9235:
8990:
8745:
7518: —
4162:
3260:where the function
3084:
2911:
2847:
1829:
1711:probability density
1568:
1312:
708:
204:of large samples.
161:in two dimensions,
118:solid-state physics
13663:Probability theory
13570:Matrix exponential
13560:Jordan normal form
13394:Fisher information
13265:Euclidean distance
13179:Totally unimodular
12432:10.1007/BF02180200
12181:10.1007/BF02184872
11931:. pp. 15â40.
10219:Circular ensembles
10203:Vladimir Marchenko
10110:
10018:
9961:
9847:
9827:
9708:
9688:
9640:
9581:
9476:
9459:
9432:
9415:
9395:
9363:
9337:
9289:
9209:
9182:
9162:
9134:
8964:
8943:
8719:
8712:) are defined as
8698:
8678:
8653:
8599:
8490:
8464:
8429:
8390:
8317:
8204:
8163:
8129:
8087:
8040:
7993:
7923:
7897:
7870:
7741:
7647:
7607:
7575:
7537:
7516:
7500:
7479:
7431:
7395:
7365:
7282:
7222:
7195:
7169:
7122:
7030:
7002:Gaussian ensembles
6990:
6963:
6937:The point process
6924:
6897:
6814:
6763:
6726:
6676:
6635:
6577:
6554:
6524:
6504:
6478:
6449:
6360:
6340:
6313:
6237:
6145:
6114:
5951:
5875:
5758:
5648:
5624:
5597:
5567:
5447:
5378:
5274:
5235:
5109:
4928:
4908:
4869:
4849:
4817:
4728:
4566:
4366:
4284:
4208:
4181:
4148:
4099:
4016:
3996:
3954:
3930:stochastic process
3918:
3853:
3833:
3791:
3759:
3696:
3643:
3611:
3591:
3560:
3542:
3445:
3416:
3250:
3165:
3085:
3058:
2989:
2951:
2897:
2880:
2833:
2816:
2777:
2751:
2631:
2605:
2507:
2481:
2398:
2334:
2259:
2180:
2153:
2086:
2044:
1995:
1955:
1894:
1847:
1815:
1707:
1691:
1604:
1571:
1554:
1445:
1295:
1194:
1118:
1081:
1055:
1035:
938:
855:
810:
709:
682:
652:Hermitian matrices
641:
615:
528:
472:Gaussian ensembles
458:energy eigenvalues
408:. You can help by
347:stochastic control
237:numerical analysis
218:Hermitian matrices
163:mesoscopic physics
69:spectral densities
21:probability theory
13635:
13634:
13627:Category:Matrices
13499:Fuzzy associative
13389:Doubly stochastic
13097:Positive-definite
12777:Block tridiagonal
12672:Weisstein, E. W.
12599:(12): 3401â3409.
12514:10.1002/cpa.20317
12235:Ukrainian Math. J
12065:978-3-319-70885-0
11946:978-0-19-874419-1
11233:(26): 5276â5279.
11068:(10): 8641â8643.
10571:(11): 1021â1099.
10466:(11): 1348â1349.
10322:978-1-108-76890-0
10295:978-0-19-957400-1
10276:978-0-521-19452-5
9905:
9903:
9850:{\displaystyle V}
9711:{\displaystyle n}
9554:
9418:{\displaystyle k}
9366:{\displaystyle k}
9273:
9185:{\displaystyle B}
8955:density of states
8814:
8701:{\displaystyle k}
8644:
8584:
8582:
8295:
8127:
8126:
8085:
8084:
8028:
8027:
7860:
7859:
7806:
7805:
7789:
7702:
7618:
7617:
7592:
7514:
7470:
7120:
6893:
6805:
6580:{\displaystyle u}
6539:mesoscopic regime
6527:{\displaystyle N}
6363:{\displaystyle N}
6251:Mesoscopic regime
6232:
6216:
6203:
6088:
6018:
5651:{\displaystyle l}
5414:
5346:
5183:
5174:
4713:
4645:
4636:
4546:
4536:
4352:
4179:
4019:{\displaystyle A}
3939:
3856:{\displaystyle A}
3776:
3628:
3614:{\displaystyle A}
3533:
3531:
3477:density of states
3362:
3354:
3340:
3246:
3203:
3134:
2737:
2704:
2591:
2561:
2467:
2445:
2151:
2131:
2042:
1918:
1917:
1890:
1832:
1813:
1774:
1533:
1457:Isserlis' theorem
1414:
1362:
1161:
1148:
1107:
1058:{\displaystyle G}
1033:
916:
898:
885:
844:
796:
781:
734:
593:
575:
562:
517:
426:
425:
385:) radio systems.
226:neuronal networks
53:mean-field theory
13670:
13622:List of matrices
13614:
13613:
13590:Row echelon form
13534:State transition
13463:Seidel adjacency
13345:Totally positive
13205:Alternating sign
12802:Complex Hadamard
12705:
12698:
12691:
12682:
12681:
12677:
12668:
12666:
12625:
12624:
12584:
12578:
12577:
12551:
12524:
12518:
12517:
12507:
12483:
12477:
12476:
12474:
12450:
12444:
12443:
12418:(1â2): 109â147.
12399:
12393:
12392:
12374:
12350:
12341:
12340:
12314:
12290:
12284:
12283:
12265:
12259:
12258:
12226:
12220:
12219:
12199:
12193:
12192:
12167:(3â4): 585â611.
12152:
12146:
12141:
12135:
12134:
12105:
12096:
12095:
12093:
12081:
12075:
12074:
12073:
12072:
12039:
12033:
12032:
12008:
12002:
12001:
11983:
11963:
11957:
11956:
11954:
11953:
11929:academic.oup.com
11920:
11914:
11913:
11871:
11862:
11856:
11855:
11835:
11829:
11828:
11810:
11804:
11803:
11785:
11765:
11759:
11758:
11734:
11728:
11727:
11693:
11684:
11678:
11677:
11643:
11641:cond-mat/0306512
11623:
11617:
11616:
11582:
11562:
11556:
11555:
11519:
11513:
11512:
11468:
11462:
11459:
11453:
11446:
11440:
11439:
11409:
11403:
11398:
11392:
11387:
11381:
11380:
11378:
11366:
11360:
11359:
11339:
11333:
11332:
11314:
11294:
11288:
11283:
11277:
11276:
11242:
11240:cond-mat/9611136
11222:
11216:
11215:
11181:
11179:cond-mat/0208436
11161:
11155:
11154:
11120:
11118:cond-mat/9911467
11100:
11094:
11093:
11054:
11048:
11047:
11013:
10993:
10987:
10986:
10952:
10950:cond-mat/0404387
10932:
10926:
10925:
10891:
10871:
10865:
10864:
10838:
10818:
10812:
10811:
10785:
10765:
10759:
10758:
10756:
10732:
10726:
10721:
10712:
10711:
10683:
10674:
10673:
10649:
10640:
10639:
10599:
10582:
10580:
10555:
10534:
10498:
10496:
10483:
10445:
10435:
10407:
10376:Russ. Math. Surv
10370:
10326:
10299:
10280:
10261:
10161:
10139:Wishart matrices
10128:Wishart matrices
10119:
10117:
10116:
10111:
10106:
10105:
10075:
10074:
10056:
10055:
10046:
10045:
10044:
10027:
10025:
10024:
10019:
10017:
10016:
10015:
9990:
9989:
9970:
9968:
9967:
9962:
9957:
9956:
9952:
9917:
9916:
9906:
9904:
9902:
9901:
9892:
9887:
9873:
9872:
9856:
9854:
9853:
9848:
9836:
9834:
9833:
9828:
9814:
9813:
9795:
9794:
9784:
9773:
9740:
9739:
9717:
9715:
9714:
9709:
9697:
9695:
9694:
9689:
9672:
9671:
9649:
9647:
9646:
9641:
9636:
9632:
9628:
9627:
9615:
9614:
9602:
9601:
9580:
9547:
9546:
9528:
9527:
9515:
9514:
9501:
9490:
9468:
9466:
9465:
9460:
9457:
9446:
9424:
9422:
9421:
9416:
9404:
9402:
9401:
9396:
9372:
9370:
9369:
9364:
9346:
9344:
9343:
9338:
9333:
9332:
9320:
9319:
9298:
9296:
9295:
9290:
9285:
9281:
9274:
9271:
9258:
9234:
9223:
9208:
9207:
9191:
9189:
9188:
9183:
9171:
9169:
9168:
9163:
9161:
9143:
9141:
9140:
9135:
9127:
9126:
9108:
9107:
9095:
9094:
9082:
9081:
9065:
9064:
9052:
9051:
9050:
9037:
9036:
9024:
9023:
9022:
9003:
9002:
8989:
8978:
8952:
8950:
8949:
8944:
8936:
8935:
8917:
8916:
8904:
8903:
8891:
8890:
8874:
8873:
8861:
8860:
8859:
8846:
8845:
8827:
8826:
8825:
8815:
8813:
8793:
8785:
8777:
8776:
8758:
8757:
8744:
8733:
8707:
8705:
8704:
8699:
8687:
8685:
8684:
8679:
8674:
8673:
8669:
8668:
8652:
8635:
8634:
8625:
8624:
8612:
8611:
8598:
8583:
8581:
8580:
8562:
8554:
8553:
8535:
8534:
8522:
8521:
8499:
8497:
8496:
8491:
8473:
8471:
8470:
8465:
8463:
8462:
8451:
8438:
8436:
8435:
8430:
8419:
8418:
8399:
8397:
8396:
8391:
8389:
8388:
8379:
8378:
8368:
8363:
8326:
8324:
8323:
8318:
8316:
8315:
8296:
8293:
8277:
8276:
8264:
8263:
8262:
8261:
8250:
8230:
8229:
8213:
8211:
8210:
8205:
8203:
8202:
8191:
8172:
8170:
8169:
8164:
8138:
8136:
8135:
8130:
8128:
8125:
8124:
8109:
8105:
8096:
8094:
8093:
8088:
8086:
8083:
8075:
8074:
8065:
8064:
8049:
8047:
8046:
8041:
8039:
8038:
8029:
8023:
8019:
8002:
8000:
7999:
7994:
7992:
7991:
7990:
7989:
7957:
7956:
7955:
7932:
7930:
7929:
7924:
7922:
7906:
7904:
7903:
7898:
7896:
7895:
7879:
7877:
7876:
7871:
7866:
7862:
7861:
7858:
7850:
7849:
7840:
7839:
7828:
7824:
7823:
7807:
7801:
7797:
7790:
7788:
7787:
7772:
7767:
7766:
7750:
7748:
7747:
7742:
7707:
7703:
7701:
7690:
7675:
7674:
7656:
7654:
7653:
7648:
7640:
7636:
7635:
7619:
7613:
7609:
7606:
7584:
7582:
7581:
7576:
7574:
7570:
7569:
7546:
7544:
7543:
7538:
7536:
7535:
7519:
7509:
7507:
7506:
7501:
7499:
7494:
7493:
7484:
7478:
7463:
7462:
7440:
7438:
7437:
7432:
7427:
7426:
7404:
7402:
7401:
7396:
7394:
7393:
7374:
7372:
7371:
7366:
7358:
7341:
7340:
7322:
7305:
7304:
7291:
7289:
7288:
7283:
7266:
7265:
7231:
7229:
7228:
7223:
7221:
7220:
7204:
7202:
7201:
7196:
7178:
7176:
7175:
7170:
7165:
7164:
7131:
7129:
7128:
7123:
7121:
7119:
7099:
7073:
7044:with the kernel
7039:
7037:
7036:
7031:
7026:
7025:
6999:
6997:
6996:
6991:
6989:
6988:
6972:
6970:
6969:
6964:
6959:
6958:
6933:
6931:
6930:
6925:
6923:
6922:
6906:
6904:
6903:
6898:
6891:
6890:
6889:
6880:
6879:
6867:
6866:
6851:
6850:
6829:
6824:
6823:
6813:
6798:
6797:
6772:
6770:
6769:
6764:
6735:
6733:
6732:
6727:
6725:
6724:
6685:
6683:
6682:
6677:
6644:
6642:
6641:
6636:
6631:
6630:
6603:
6602:
6586:
6584:
6583:
6578:
6563:
6561:
6560:
6555:
6533:
6531:
6530:
6525:
6513:
6511:
6510:
6505:
6487:
6485:
6484:
6479:
6458:
6456:
6455:
6450:
6430:
6429:
6416:
6415:
6394:
6393:
6369:
6367:
6366:
6361:
6349:
6347:
6346:
6341:
6339:
6338:
6322:
6320:
6319:
6316:{\displaystyle }
6314:
6300:
6299:
6278:
6277:
6246:
6244:
6243:
6238:
6233:
6219:
6217:
6209:
6204:
6202:
6198:
6192:
6182:
6181:
6172:
6166:
6154:
6152:
6151:
6146:
6144:
6143:
6123:
6121:
6120:
6115:
6104:
6103:
6094:
6089:
6087:
6076:
6066:
6046:
6037:
6029:
6028:
6023:
6019:
6014:
6004:
5995:
5960:
5958:
5957:
5952:
5947:
5946:
5934:
5933:
5920:
5915:
5884:
5882:
5881:
5876:
5868:
5820:
5815:
5801:
5790:
5789:
5788:
5767:
5765:
5764:
5759:
5711:
5706:
5692:
5681:
5680:
5679:
5657:
5655:
5654:
5649:
5633:
5631:
5630:
5625:
5623:
5622:
5606:
5604:
5603:
5598:
5596:
5595:
5576:
5574:
5573:
5568:
5551:
5516:
5499:
5494:
5480:
5469:
5468:
5446:
5442:
5434:
5433:
5410:
5409:
5387:
5385:
5384:
5379:
5377:
5373:
5363:
5358:
5357:
5356:
5344:
5314:
5306:
5305:
5283:
5281:
5280:
5275:
5264:
5263:
5244:
5242:
5241:
5236:
5228:
5223:
5222:
5212:
5211:
5202:
5201:
5184:
5182:
5181:
5176:
5175:
5167:
5160:
5146:
5145:
5136:
5118:
5116:
5115:
5110:
5105:
5101:
5088:
5053:
5036:
5031:
5017:
5006:
5005:
4979:
4978:
4957:
4956:
4937:
4935:
4934:
4929:
4917:
4915:
4914:
4909:
4898:
4897:
4878:
4876:
4875:
4870:
4858:
4856:
4855:
4850:
4826:
4824:
4823:
4818:
4810:
4809:
4793:
4788:
4767:
4762:
4761:
4749:
4748:
4739:
4727:
4697:
4696:
4680:
4675:
4674:
4664:
4663:
4646:
4644:
4643:
4638:
4637:
4629:
4622:
4608:
4607:
4598:
4575:
4573:
4572:
4567:
4547:
4539:
4537:
4535:
4534:
4519:
4484:
4483:
4467:
4442:
4400:potential theory
4375:
4373:
4372:
4367:
4353:
4350:
4348:
4347:
4332:
4331:
4293:
4291:
4290:
4285:
4283:
4282:
4243:
4242:
4217:
4215:
4214:
4209:
4207:
4206:
4190:
4188:
4187:
4182:
4180:
4172:
4170:
4161:
4156:
4144:
4143:
4128:
4127:
4108:
4106:
4105:
4100:
4098:
4097:
4058:
4057:
4025:
4023:
4022:
4017:
4005:
4003:
4002:
3997:
3971:
3970:
3969:
3968:
3953:
3927:
3925:
3924:
3919:
3911:
3910:
3898:
3897:
3885:
3884:
3867:In another sense
3862:
3860:
3859:
3854:
3842:
3840:
3839:
3834:
3808:
3807:
3806:
3805:
3790:
3768:
3766:
3765:
3760:
3752:
3751:
3739:
3738:
3726:
3725:
3705:
3703:
3702:
3697:
3668:
3667:
3655:
3654:
3649:
3642:
3620:
3618:
3617:
3612:
3600:
3598:
3597:
3592:
3569:
3567:
3566:
3561:
3559:
3558:
3557:
3556:
3541:
3532:
3524:
3510:
3509:
3454:
3452:
3451:
3446:
3444:
3443:
3425:
3423:
3422:
3417:
3412:
3397:
3396:
3389:
3388:
3371:
3367:
3363:
3360:
3355:
3352:
3341:
3333:
3319:
3318:
3302:
3296:
3265:
3259:
3257:
3256:
3251:
3244:
3243:
3242:
3229:
3204:
3202:
3201:
3189:
3174:
3172:
3171:
3166:
3164:
3160:
3132:
3116:
3115:
3094:
3092:
3091:
3086:
3083:
3078:
3042:
3041:
3026:
3025:
2998:
2996:
2995:
2990:
2960:
2958:
2957:
2952:
2932:
2931:
2910:
2905:
2889:
2887:
2886:
2881:
2864:
2863:
2846:
2841:
2825:
2823:
2822:
2817:
2806:
2805:
2786:
2784:
2783:
2778:
2760:
2758:
2757:
2752:
2750:
2749:
2748:
2747:
2738:
2736:
2725:
2715:
2714:
2705:
2703:
2702:
2701:
2692:
2691:
2681:
2680:
2671:
2657:
2656:
2640:
2638:
2637:
2632:
2614:
2612:
2611:
2606:
2604:
2603:
2602:
2601:
2592:
2584:
2578:
2572:
2571:
2562:
2560:
2559:
2547:
2533:
2532:
2516:
2514:
2513:
2508:
2490:
2488:
2487:
2482:
2480:
2479:
2478:
2477:
2468:
2460:
2446:
2438:
2424:
2423:
2407:
2405:
2404:
2399:
2394:
2393:
2381:
2380:
2343:
2341:
2340:
2335:
2324:
2316:
2315:
2303:
2302:
2268:
2266:
2265:
2260:
2252:
2251:
2233:
2232:
2214:
2213:
2189:
2187:
2186:
2181:
2179:
2178:
2162:
2160:
2159:
2154:
2152:
2150:
2149:
2134:
2132:
2130:
2119:
2095:
2093:
2092:
2089:{\displaystyle }
2087:
2058:on the interval
2053:
2051:
2050:
2045:
2043:
2041:
2040:
2028:
2004:
2002:
2001:
1996:
1994:
1993:
1981:
1980:
1964:
1962:
1961:
1956:
1935:Selberg integral
1912:
1903:
1901:
1900:
1895:
1888:
1887:
1886:
1881:
1877:
1876:
1875:
1863:
1862:
1846:
1831:
1830:
1828:
1823:
1814:
1806:
1795:
1790:
1775:
1773:
1772:
1754:
1744:
1740:
1700:
1698:
1697:
1692:
1690:
1689:
1665:
1664:
1652:
1651:
1622:Spectral density
1613:
1611:
1610:
1605:
1603:
1602:
1580:
1578:
1577:
1572:
1570:
1569:
1567:
1562:
1553:
1552:
1534:
1532:
1521:
1508:
1507:
1454:
1452:
1451:
1446:
1441:
1440:
1428:
1427:
1415:
1413:
1405:
1394:
1389:
1388:
1376:
1375:
1363:
1355:
1347:
1346:
1334:
1333:
1311:
1306:
1294:
1293:
1253:symplectic group
1250:
1249:
1248:
1204:on the space of
1203:
1201:
1200:
1195:
1193:
1192:
1191:
1190:
1181:
1162:
1160:
1159:
1149:
1146:
1136:
1127:
1125:
1124:
1119:
1108:
1105:
1090:
1088:
1087:
1082:
1064:
1062:
1061:
1056:
1044:
1042:
1041:
1036:
1034:
1026:
1024:
1016:
1015:
984:
983:
948:on the space of
947:
945:
944:
939:
937:
936:
935:
934:
925:
917:
909:
899:
897:
896:
886:
883:
873:
864:
862:
861:
856:
845:
842:
819:
817:
816:
811:
809:
808:
807:
806:
797:
789:
786:
782:
774:
767:
766:
762:
746:
745:
735:
732:
718:
716:
715:
710:
707:
702:
681:
680:
650:
648:
647:
642:
625:on the space of
624:
622:
621:
616:
614:
613:
612:
611:
602:
594:
586:
576:
574:
573:
563:
560:
550:
540:Gaussian measure
537:
535:
534:
529:
518:
515:
464:by invoking the
450:Leonard Eisenbud
442:compound nucleus
421:
418:
400:
393:
297:free probability
295:The relation of
291:Free probability
245:Herman Goldstine
241:John von Neumann
200:, who sought to
43:randomly from a
13678:
13677:
13673:
13672:
13671:
13669:
13668:
13667:
13653:Random matrices
13638:
13637:
13636:
13631:
13608:
13599:
13548:
13472:
13418:
13354:
13188:
13106:
13052:
12991:
12792:Centrosymmetric
12715:
12709:
12674:"Random Matrix"
12633:
12628:
12585:
12581:
12525:
12521:
12484:
12480:
12465:(16): 759â782.
12451:
12447:
12400:
12396:
12351:
12344:
12291:
12287:
12280:
12266:
12262:
12227:
12223:
12200:
12196:
12153:
12149:
12142:
12138:
12106:
12099:
12082:
12078:
12070:
12068:
12066:
12040:
12036:
12009:
12005:
11964:
11960:
11951:
11949:
11947:
11921:
11917:
11869:
11863:
11859:
11836:
11832:
11825:
11811:
11807:
11766:
11762:
11735:
11731:
11691:
11685:
11681:
11624:
11620:
11563:
11559:
11520:
11516:
11469:
11465:
11460:
11456:
11447:
11443:
11436:
11410:
11406:
11399:
11395:
11388:
11384:
11367:
11363:
11340:
11336:
11295:
11291:
11284:
11280:
11227:Phys. Rev. Lett
11223:
11219:
11166:Phys. Rev. Lett
11162:
11158:
11101:
11097:
11055:
11051:
10998:Phys. Rev. Lett
10994:
10990:
10937:Phys. Rev. Lett
10933:
10929:
10876:Phys. Rev. Lett
10872:
10868:
10819:
10815:
10766:
10762:
10733:
10729:
10722:
10715:
10688:Phys. Rev. Lett
10684:
10677:
10662:10.2172/4319287
10650:
10643:
10620:10.2307/1970079
10600:
10593:
10589:
10523:10.2307/1970079
10505:
10450:Diaconis, Persi
10412:Diaconis, Persi
10333:
10331:Survey articles
10323:
10296:
10277:
10258:
10248:Random Matrices
10242:
10237:
10232:
10226:
10221:
10215:
10174:random matrix (
10150:
10136:
10130:
10125:
10098:
10094:
10070:
10066:
10051:
10047:
10040:
10039:
10035:
10033:
10030:
10029:
10011:
10004:
10000:
9985:
9981:
9976:
9973:
9972:
9948:
9932:
9928:
9912:
9908:
9897:
9893:
9891:
9886:
9868:
9864:
9862:
9859:
9858:
9842:
9839:
9838:
9809:
9805:
9790:
9786:
9774:
9763:
9729:
9725:
9723:
9720:
9719:
9703:
9700:
9699:
9661:
9657:
9655:
9652:
9651:
9623:
9619:
9610:
9606:
9591:
9587:
9586:
9582:
9558:
9542:
9538:
9523:
9519:
9510:
9506:
9491:
9480:
9474:
9471:
9470:
9447:
9436:
9430:
9427:
9426:
9410:
9407:
9406:
9378:
9375:
9374:
9358:
9355:
9354:
9328:
9324:
9315:
9311:
9306:
9303:
9302:
9270:
9263:
9259:
9254:
9224:
9213:
9203:
9199:
9197:
9194:
9193:
9177:
9174:
9173:
9157:
9149:
9146:
9145:
9122:
9118:
9103:
9099:
9090:
9086:
9071:
9067:
9060:
9056:
9046:
9045:
9041:
9032:
9028:
9018:
9017:
9013:
8998:
8994:
8979:
8968:
8962:
8959:
8958:
8931:
8927:
8912:
8908:
8899:
8895:
8880:
8876:
8869:
8865:
8855:
8854:
8850:
8835:
8831:
8821:
8820:
8816:
8794:
8786:
8784:
8772:
8768:
8753:
8749:
8734:
8723:
8717:
8714:
8713:
8693:
8690:
8689:
8664:
8660:
8648:
8640:
8636:
8630:
8626:
8620:
8616:
8607:
8603:
8588:
8570:
8566:
8561:
8549:
8545:
8530:
8526:
8511:
8507:
8505:
8502:
8501:
8479:
8476:
8475:
8452:
8447:
8446:
8444:
8441:
8440:
8414:
8410:
8405:
8402:
8401:
8384:
8380:
8374:
8370:
8364:
8353:
8332:
8329:
8328:
8292:
8291:
8287:
8272:
8268:
8251:
8246:
8245:
8238:
8234:
8225:
8221:
8219:
8216:
8215:
8192:
8187:
8186:
8178:
8175:
8174:
8152:
8149:
8148:
8145:
8120:
8116:
8104:
8102:
8099:
8098:
8076:
8070:
8066:
8063:
8055:
8052:
8051:
8034:
8030:
8018:
8016:
8013:
8012:
8005:
7982:
7978:
7974:
7970:
7945:
7944:
7940:
7938:
7935:
7934:
7918:
7916:
7913:
7912:
7891:
7887:
7885:
7882:
7881:
7851:
7845:
7841:
7838:
7819:
7815:
7811:
7796:
7795:
7791:
7783:
7779:
7771:
7762:
7758:
7756:
7753:
7752:
7694:
7689:
7685:
7670:
7666:
7664:
7661:
7660:
7631:
7627:
7623:
7608:
7596:
7590:
7587:
7586:
7565:
7561:
7557:
7552:
7549:
7548:
7531:
7527:
7525:
7522:
7521:
7517:
7495:
7489:
7485:
7480:
7474:
7458:
7454:
7446:
7443:
7442:
7422:
7418:
7410:
7407:
7406:
7389:
7385:
7383:
7380:
7379:
7354:
7336:
7335:
7318:
7300:
7299:
7297:
7294:
7293:
7261:
7260:
7258:
7255:
7254:
7244:
7238:
7236:Edge statistics
7216:
7212:
7210:
7207:
7206:
7184:
7181:
7180:
7160:
7156:
7148:
7145:
7144:
7100:
7074:
7072:
7049:
7046:
7045:
7021:
7017:
7009:
7006:
7005:
7004:, the limit of
6984:
6980:
6978:
6975:
6974:
6954:
6950:
6942:
6939:
6938:
6918:
6914:
6912:
6909:
6908:
6885:
6884:
6875:
6871:
6862:
6858:
6846:
6842:
6825:
6819:
6818:
6809:
6793:
6789:
6781:
6778:
6777:
6749:
6746:
6745:
6720:
6716:
6714:
6711:
6710:
6707:
6705:Bulk statistics
6699:edge statistics
6695:bulk statistics
6662:
6659:
6658:
6651:
6617:
6613:
6598:
6594:
6592:
6589:
6588:
6572:
6569:
6568:
6546:
6543:
6542:
6519:
6516:
6515:
6496:
6493:
6492:
6464:
6461:
6460:
6411:
6407:
6389:
6385:
6381:
6377:
6375:
6372:
6371:
6355:
6352:
6351:
6334:
6330:
6328:
6325:
6324:
6295:
6291:
6273:
6269:
6264:
6261:
6260:
6253:
6218:
6208:
6194:
6193:
6177:
6173:
6168:
6167:
6165:
6163:
6160:
6159:
6139:
6135:
6133:
6130:
6129:
6099:
6095:
6090:
6077:
6059:
6039:
6038:
6036:
6024:
5997:
5996:
5994:
5990:
5989:
5969:
5966:
5965:
5942:
5938:
5929:
5925:
5916:
5905:
5893:
5890:
5889:
5864:
5816:
5811:
5797:
5784:
5783:
5779:
5774:
5771:
5770:
5707:
5702:
5688:
5675:
5674:
5670:
5665:
5662:
5661:
5643:
5640:
5639:
5618:
5614:
5612:
5609:
5608:
5591:
5587:
5585:
5582:
5581:
5547:
5512:
5495:
5490:
5476:
5458:
5454:
5438:
5429:
5425:
5418:
5405:
5401:
5399:
5396:
5395:
5359:
5352:
5351:
5347:
5325:
5321:
5310:
5301:
5297:
5295:
5292:
5291:
5259:
5255:
5253:
5250:
5249:
5224:
5207:
5203:
5197:
5193:
5189:
5185:
5177:
5166:
5165:
5164:
5159:
5141:
5137:
5132:
5130:
5127:
5126:
5084:
5049:
5032:
5027:
5013:
4995:
4991:
4984:
4980:
4974:
4970:
4952:
4948:
4946:
4943:
4942:
4923:
4920:
4919:
4893:
4889:
4887:
4884:
4883:
4882:We can rewrite
4864:
4861:
4860:
4835:
4832:
4831:
4805:
4801:
4789:
4778:
4763:
4757:
4753:
4744:
4740:
4735:
4717:
4692:
4688:
4676:
4659:
4655:
4651:
4647:
4639:
4628:
4627:
4626:
4621:
4603:
4599:
4594:
4592:
4589:
4588:
4582:
4538:
4524:
4520:
4473:
4469:
4468:
4466:
4464:
4461:
4460:
4440:
4423:
4411:
4408:
4349:
4340:
4336:
4321:
4317:
4315:
4312:
4311:
4304:
4248:
4244:
4232:
4228:
4223:
4220:
4219:
4202:
4198:
4196:
4193:
4192:
4171:
4166:
4157:
4152:
4139:
4135:
4123:
4119:
4117:
4114:
4113:
4063:
4059:
4047:
4043:
4038:
4035:
4034:
4011:
4008:
4007:
3964:
3960:
3959:
3955:
3943:
3937:
3934:
3933:
3906:
3902:
3893:
3889:
3880:
3876:
3874:
3871:
3870:
3848:
3845:
3844:
3801:
3797:
3796:
3792:
3780:
3774:
3771:
3770:
3747:
3743:
3734:
3730:
3721:
3717:
3715:
3712:
3711:
3710:: If we sample
3663:
3659:
3650:
3645:
3644:
3632:
3626:
3623:
3622:
3606:
3603:
3602:
3586:
3583:
3582:
3575:
3552:
3548:
3547:
3543:
3537:
3523:
3505:
3501:
3499:
3496:
3495:
3493:
3467:and is denoted
3439:
3435:
3433:
3430:
3429:
3408:
3384:
3380:
3379:
3375:
3359:
3351:
3350:
3346:
3332:
3314:
3310:
3308:
3305:
3304:
3298:
3294:
3289:
3283:
3275:
3261:
3222:
3209:
3205:
3197:
3193:
3188:
3186:
3183:
3182:
3111:
3107:
3106:
3102:
3100:
3097:
3096:
3079:
3062:
3037:
3033:
3021:
3017:
3015:
3012:
3011:
3008:Wigner matrices
3005:
3003:Generalizations
2966:
2963:
2962:
2927:
2923:
2906:
2901:
2895:
2892:
2891:
2859:
2855:
2842:
2837:
2831:
2828:
2827:
2826:is normalized:
2801:
2797:
2795:
2792:
2791:
2766:
2763:
2762:
2743:
2739:
2729:
2724:
2720:
2716:
2710:
2706:
2697:
2693:
2687:
2683:
2682:
2676:
2672:
2670:
2652:
2648:
2646:
2643:
2642:
2620:
2617:
2616:
2597:
2593:
2583:
2579:
2574:
2573:
2567:
2563:
2555:
2551:
2546:
2528:
2524:
2522:
2519:
2518:
2496:
2493:
2492:
2473:
2469:
2459:
2455:
2451:
2437:
2419:
2415:
2413:
2410:
2409:
2389:
2385:
2370:
2366:
2349:
2346:
2345:
2320:
2311:
2307:
2292:
2288:
2277:
2274:
2273:
2241:
2237:
2228:
2224:
2209:
2205:
2203:
2200:
2199:
2196:
2174:
2170:
2168:
2165:
2164:
2145:
2141:
2133:
2123:
2118:
2101:
2098:
2097:
2063:
2060:
2059:
2036:
2032:
2027:
2025:
2022:
2021:
2018:
1989:
1985:
1976:
1972:
1970:
1967:
1966:
1950:
1947:
1946:
1932:
1882:
1871:
1867:
1858:
1854:
1853:
1849:
1848:
1836:
1824:
1819:
1805:
1801:
1797:
1791:
1780:
1762:
1758:
1753:
1751:
1748:
1747:
1739:
1730:
1723:
1717:
1685:
1681:
1660:
1656:
1647:
1643:
1635:
1632:
1631:
1624:
1598:
1594:
1586:
1583:
1582:
1563:
1558:
1548:
1544:
1525:
1520:
1519:
1515:
1488:
1484:
1476:
1473:
1472:
1465:
1433:
1429:
1420:
1416:
1406:
1395:
1393:
1381:
1377:
1368:
1364:
1354:
1339:
1335:
1326:
1322:
1307:
1299:
1286:
1282:
1277:
1274:
1273:
1271:
1261:
1247:
1237:
1236:
1235:
1233:
1221:
1186:
1182:
1174:
1167:
1163:
1145:
1144:
1140:
1135:
1133:
1130:
1129:
1104:
1102:
1099:
1098:
1070:
1067:
1066:
1050:
1047:
1046:
1025:
1020:
1011:
1007:
990:
987:
986:
982:
972:
971:
970:
968:
930:
926:
918:
908:
904:
900:
882:
881:
877:
872:
870:
867:
866:
841:
839:
836:
835:
802:
798:
788:
787:
773:
769:
768:
758:
754:
750:
731:
730:
726:
724:
721:
720:
703:
686:
673:
669:
658:
655:
654:
630:
627:
626:
607:
603:
595:
585:
581:
577:
559:
558:
554:
549:
547:
544:
543:
514:
512:
509:
508:
502:
474:
422:
416:
413:
406:needs expansion
391:
379:
367:
334:optimal control
330:
328:Optimal control
306:
293:
277:Hugh Montgomery
261:
190:
183:superconductors
159:quantum gravity
106:nuclear physics
102:
97:
37:random variable
17:
12:
11:
5:
13676:
13666:
13665:
13660:
13655:
13650:
13633:
13632:
13630:
13629:
13624:
13619:
13604:
13601:
13600:
13598:
13597:
13592:
13587:
13582:
13580:Perfect matrix
13577:
13572:
13567:
13562:
13556:
13554:
13550:
13549:
13547:
13546:
13541:
13536:
13531:
13526:
13521:
13516:
13511:
13506:
13501:
13496:
13491:
13486:
13480:
13478:
13474:
13473:
13471:
13470:
13465:
13460:
13455:
13450:
13445:
13440:
13435:
13429:
13427:
13420:
13419:
13417:
13416:
13411:
13406:
13401:
13396:
13391:
13386:
13381:
13376:
13371:
13365:
13363:
13356:
13355:
13353:
13352:
13350:Transformation
13347:
13342:
13337:
13332:
13327:
13322:
13317:
13312:
13307:
13302:
13297:
13292:
13287:
13282:
13277:
13272:
13267:
13262:
13257:
13252:
13247:
13242:
13237:
13232:
13227:
13222:
13217:
13212:
13207:
13202:
13196:
13194:
13190:
13189:
13187:
13186:
13181:
13176:
13171:
13166:
13161:
13156:
13151:
13146:
13141:
13136:
13127:
13121:
13119:
13108:
13107:
13105:
13104:
13099:
13094:
13089:
13087:Diagonalizable
13084:
13079:
13074:
13069:
13063:
13061:
13057:Conditions on
13054:
13053:
13051:
13050:
13045:
13040:
13035:
13030:
13025:
13020:
13015:
13010:
13005:
12999:
12997:
12993:
12992:
12990:
12989:
12984:
12979:
12974:
12969:
12964:
12959:
12954:
12949:
12944:
12939:
12937:Skew-symmetric
12934:
12932:Skew-Hermitian
12929:
12924:
12919:
12914:
12909:
12904:
12899:
12894:
12889:
12884:
12879:
12874:
12869:
12864:
12859:
12854:
12849:
12844:
12839:
12834:
12829:
12824:
12819:
12814:
12809:
12804:
12799:
12794:
12789:
12784:
12779:
12774:
12769:
12767:Block-diagonal
12764:
12759:
12754:
12749:
12744:
12742:Anti-symmetric
12739:
12737:Anti-Hermitian
12734:
12729:
12723:
12721:
12717:
12716:
12708:
12707:
12700:
12693:
12685:
12679:
12678:
12669:
12632:
12631:External links
12629:
12627:
12626:
12579:
12542:(2): 549â572.
12519:
12498:(7): 895â925.
12478:
12445:
12404:Shcherbina, M.
12394:
12365:(3): 545â595.
12342:
12305:(2): 641â655.
12285:
12279:978-1461428770
12278:
12260:
12241:(6): 936â966.
12221:
12210:(1): 151â204.
12194:
12157:Shcherbina, M.
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10943:(10): 106802.
10927:
10882:(16): 166401.
10866:
10836:hep-ph/0003017
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10614:(3): 548â564.
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10504:
10503:Historic works
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10426:(2): 155â178.
10422:. New Series.
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7240:Main article:
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6709:Formally, fix
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4389:Wigner surmise
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3457:self-averaging
3442:
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3415:
3411:
3407:
3404:
3400:
3395:
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1992:
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1885:
1880:
1874:
1870:
1866:
1861:
1857:
1852:
1845:
1842:
1839:
1835:
1827:
1822:
1818:
1812:
1809:
1804:
1800:
1794:
1789:
1786:
1783:
1779:
1771:
1768:
1765:
1761:
1757:
1735:
1728:
1721:
1688:
1684:
1680:
1677:
1674:
1671:
1668:
1663:
1659:
1655:
1650:
1646:
1642:
1639:
1623:
1620:
1616:Frobenius norm
1601:
1597:
1593:
1590:
1566:
1561:
1557:
1551:
1547:
1543:
1540:
1537:
1531:
1528:
1524:
1518:
1514:
1511:
1506:
1503:
1500:
1497:
1494:
1491:
1487:
1483:
1480:
1471:for the GOE is
1464:
1461:
1444:
1439:
1436:
1432:
1426:
1423:
1419:
1412:
1409:
1404:
1401:
1398:
1392:
1387:
1384:
1380:
1374:
1371:
1367:
1361:
1358:
1353:
1350:
1345:
1342:
1338:
1332:
1329:
1325:
1321:
1318:
1315:
1310:
1305:
1302:
1298:
1292:
1289:
1285:
1281:
1267:
1260:
1257:
1238:
1229:
1189:
1185:
1180:
1177:
1173:
1170:
1166:
1158:
1155:
1152:
1143:
1139:
1117:
1114:
1111:
1080:
1077:
1074:
1054:
1032:
1029:
1023:
1019:
1014:
1010:
1006:
1003:
1000:
997:
994:
973:
964:
960: = (
933:
929:
924:
921:
915:
912:
907:
903:
895:
892:
889:
880:
876:
854:
851:
848:
805:
801:
795:
792:
785:
780:
777:
772:
765:
761:
757:
753:
749:
744:
741:
738:
729:
706:
701:
698:
695:
692:
689:
685:
679:
676:
672:
668:
665:
662:
640:
637:
634:
610:
606:
601:
598:
592:
589:
584:
580:
572:
569:
566:
557:
553:
527:
524:
521:
501:
498:
473:
470:
424:
423:
403:
401:
390:
387:
378:
375:
366:
363:
329:
326:
305:
302:
292:
289:
260:
257:
189:
186:
148:beam splitters
144:boson sampling
140:quantum optics
101:
98:
96:
93:
61:replica method
15:
9:
6:
4:
3:
2:
13675:
13664:
13661:
13659:
13656:
13654:
13651:
13649:
13646:
13645:
13643:
13628:
13625:
13623:
13620:
13618:
13617:
13612:
13606:
13605:
13602:
13596:
13593:
13591:
13588:
13586:
13585:Pseudoinverse
13583:
13581:
13578:
13576:
13573:
13571:
13568:
13566:
13563:
13561:
13558:
13557:
13555:
13553:Related terms
13551:
13545:
13544:Z (chemistry)
13542:
13540:
13537:
13535:
13532:
13530:
13527:
13525:
13522:
13520:
13517:
13515:
13512:
13510:
13507:
13505:
13502:
13500:
13497:
13495:
13492:
13490:
13487:
13485:
13482:
13481:
13479:
13475:
13469:
13466:
13464:
13461:
13459:
13456:
13454:
13451:
13449:
13446:
13444:
13441:
13439:
13436:
13434:
13431:
13430:
13428:
13426:
13421:
13415:
13412:
13410:
13407:
13405:
13402:
13400:
13397:
13395:
13392:
13390:
13387:
13385:
13382:
13380:
13377:
13375:
13372:
13370:
13367:
13366:
13364:
13362:
13357:
13351:
13348:
13346:
13343:
13341:
13338:
13336:
13333:
13331:
13328:
13326:
13323:
13321:
13318:
13316:
13313:
13311:
13308:
13306:
13303:
13301:
13298:
13296:
13293:
13291:
13288:
13286:
13283:
13281:
13278:
13276:
13273:
13271:
13268:
13266:
13263:
13261:
13258:
13256:
13253:
13251:
13248:
13246:
13243:
13241:
13238:
13236:
13233:
13231:
13228:
13226:
13223:
13221:
13218:
13216:
13213:
13211:
13208:
13206:
13203:
13201:
13198:
13197:
13195:
13191:
13185:
13182:
13180:
13177:
13175:
13172:
13170:
13167:
13165:
13162:
13160:
13157:
13155:
13152:
13150:
13147:
13145:
13142:
13140:
13137:
13135:
13131:
13128:
13126:
13123:
13122:
13120:
13118:
13114:
13109:
13103:
13100:
13098:
13095:
13093:
13090:
13088:
13085:
13083:
13080:
13078:
13075:
13073:
13070:
13068:
13065:
13064:
13062:
13060:
13055:
13049:
13046:
13044:
13041:
13039:
13036:
13034:
13031:
13029:
13026:
13024:
13021:
13019:
13016:
13014:
13011:
13009:
13006:
13004:
13001:
13000:
12998:
12994:
12988:
12985:
12983:
12980:
12978:
12975:
12973:
12970:
12968:
12965:
12963:
12960:
12958:
12955:
12953:
12950:
12948:
12945:
12943:
12940:
12938:
12935:
12933:
12930:
12928:
12925:
12923:
12920:
12918:
12915:
12913:
12910:
12908:
12905:
12903:
12902:Pentadiagonal
12900:
12898:
12895:
12893:
12890:
12888:
12885:
12883:
12880:
12878:
12875:
12873:
12870:
12868:
12865:
12863:
12860:
12858:
12855:
12853:
12850:
12848:
12845:
12843:
12840:
12838:
12835:
12833:
12830:
12828:
12825:
12823:
12820:
12818:
12815:
12813:
12810:
12808:
12805:
12803:
12800:
12798:
12795:
12793:
12790:
12788:
12785:
12783:
12780:
12778:
12775:
12773:
12770:
12768:
12765:
12763:
12760:
12758:
12755:
12753:
12750:
12748:
12745:
12743:
12740:
12738:
12735:
12733:
12732:Anti-diagonal
12730:
12728:
12725:
12724:
12722:
12718:
12713:
12706:
12701:
12699:
12694:
12692:
12687:
12686:
12683:
12675:
12670:
12665:
12660:
12656:
12652:
12648:
12644:
12640:
12635:
12634:
12622:
12618:
12614:
12610:
12606:
12602:
12598:
12594:
12590:
12583:
12575:
12571:
12567:
12563:
12559:
12555:
12550:
12545:
12541:
12537:
12533:
12529:
12523:
12515:
12511:
12506:
12501:
12497:
12493:
12489:
12482:
12473:
12468:
12464:
12460:
12456:
12449:
12441:
12437:
12433:
12429:
12425:
12421:
12417:
12413:
12409:
12405:
12398:
12390:
12386:
12382:
12378:
12373:
12368:
12364:
12360:
12356:
12349:
12347:
12338:
12334:
12330:
12326:
12322:
12318:
12313:
12308:
12304:
12300:
12296:
12289:
12281:
12275:
12271:
12264:
12256:
12252:
12248:
12244:
12240:
12236:
12232:
12225:
12217:
12213:
12209:
12205:
12198:
12190:
12186:
12182:
12178:
12174:
12170:
12166:
12162:
12161:J. Stat. Phys
12158:
12151:
12145:
12140:
12132:
12128:
12124:
12120:
12116:
12112:
12104:
12102:
12092:
12087:
12080:
12067:
12061:
12057:
12053:
12049:
12045:
12038:
12030:
12026:
12022:
12018:
12014:
12007:
11999:
11995:
11991:
11987:
11982:
11977:
11973:
11969:
11962:
11948:
11942:
11938:
11934:
11930:
11926:
11919:
11911:
11907:
11903:
11899:
11895:
11891:
11887:
11883:
11879:
11875:
11868:
11861:
11853:
11849:
11845:
11841:
11834:
11826:
11824:0-471-15616-7
11820:
11816:
11809:
11801:
11797:
11793:
11789:
11784:
11779:
11775:
11771:
11764:
11756:
11752:
11748:
11744:
11740:
11733:
11725:
11721:
11717:
11713:
11709:
11705:
11702:(4): 042808.
11701:
11697:
11690:
11683:
11675:
11671:
11667:
11663:
11659:
11655:
11651:
11647:
11642:
11637:
11634:(7): 074101.
11633:
11629:
11622:
11614:
11610:
11606:
11602:
11598:
11594:
11590:
11586:
11581:
11576:
11572:
11568:
11561:
11553:
11549:
11545:
11541:
11537:
11533:
11529:
11525:
11518:
11510:
11506:
11502:
11498:
11494:
11490:
11486:
11482:
11478:
11474:
11467:
11458:
11451:
11445:
11437:
11435:9780444815880
11431:
11427:
11423:
11419:
11415:
11408:
11402:
11397:
11391:
11386:
11377:
11372:
11365:
11357:
11353:
11349:
11345:
11338:
11330:
11326:
11322:
11318:
11313:
11308:
11304:
11300:
11293:
11287:
11282:
11274:
11270:
11266:
11262:
11258:
11254:
11250:
11246:
11241:
11236:
11232:
11228:
11221:
11213:
11209:
11205:
11201:
11197:
11193:
11189:
11185:
11180:
11175:
11171:
11167:
11160:
11152:
11148:
11144:
11140:
11136:
11132:
11128:
11124:
11119:
11114:
11110:
11106:
11099:
11091:
11087:
11083:
11079:
11075:
11071:
11067:
11063:
11059:
11053:
11045:
11041:
11037:
11033:
11029:
11025:
11021:
11017:
11012:
11007:
11004:(6): 066602.
11003:
10999:
10992:
10984:
10980:
10976:
10972:
10968:
10964:
10960:
10956:
10951:
10946:
10942:
10938:
10931:
10923:
10919:
10915:
10911:
10907:
10903:
10899:
10895:
10890:
10885:
10881:
10877:
10870:
10862:
10858:
10854:
10850:
10846:
10842:
10837:
10832:
10828:
10824:
10817:
10809:
10805:
10801:
10797:
10793:
10789:
10784:
10779:
10776:(3): 033007.
10775:
10771:
10764:
10755:
10750:
10746:
10742:
10738:
10731:
10725:
10720:
10718:
10709:
10705:
10701:
10697:
10693:
10689:
10682:
10680:
10671:
10667:
10663:
10659:
10655:
10648:
10646:
10637:
10633:
10629:
10625:
10621:
10617:
10613:
10609:
10605:
10598:
10596:
10591:
10579:
10574:
10570:
10566:
10562:
10557:
10553:
10549:
10545:
10541:
10536:
10532:
10528:
10524:
10520:
10516:
10512:
10507:
10506:
10495:
10490:
10485:
10481:
10477:
10473:
10469:
10465:
10461:
10460:
10455:
10451:
10447:
10443:
10439:
10434:
10429:
10425:
10421:
10417:
10413:
10409:
10405:
10401:
10397:
10393:
10389:
10385:
10381:
10377:
10372:
10368:
10364:
10360:
10356:
10352:
10348:
10344:
10340:
10339:Acta Numerica
10335:
10334:
10324:
10318:
10314:
10310:
10306:
10301:
10297:
10291:
10287:
10282:
10278:
10272:
10268:
10263:
10259:
10257:0-12-088409-7
10253:
10249:
10244:
10243:
10231:
10220:
10210:
10208:
10207:Leonid Pastur
10204:
10201:was found by
10200:
10195:
10193:
10189:
10185:
10181:
10178: â„
10177:
10173:
10170: Ă
10169:
10165:
10160:
10157:
10153:
10148:
10145: Ă
10144:
10140:
10135:
10120:
10107:
10102:
10099:
10095:
10091:
10088:
10085:
10079:
10071:
10067:
10060:
10052:
10048:
10036:
10008:
10005:
9994:
9986:
9982:
9958:
9953:
9949:
9942:
9936:
9933:
9929:
9921:
9913:
9909:
9898:
9894:
9888:
9883:
9877:
9869:
9865:
9844:
9824:
9818:
9810:
9806:
9799:
9791:
9787:
9781:
9778:
9775:
9770:
9767:
9764:
9760:
9756:
9750:
9747:
9744:
9736:
9733:
9730:
9726:
9705:
9682:
9679:
9676:
9668:
9665:
9662:
9658:
9637:
9633:
9624:
9620:
9616:
9611:
9607:
9598:
9595:
9592:
9588:
9583:
9577:
9574:
9571:
9568:
9565:
9562:
9559:
9551:
9543:
9539:
9535:
9532:
9529:
9524:
9520:
9516:
9511:
9507:
9495:
9487:
9484:
9481:
9477:
9451:
9443:
9440:
9437:
9433:
9412:
9392:
9389:
9386:
9383:
9380:
9360:
9352:
9348:
9329:
9325:
9321:
9316:
9312:
9299:
9286:
9282:
9275:
9260:
9251:
9248:
9245:
9239:
9228:
9220:
9217:
9214:
9210:
9204:
9200:
9179:
9154:
9151:
9131:
9123:
9119:
9115:
9112:
9109:
9104:
9100:
9096:
9091:
9087:
9078:
9075:
9072:
9068:
9061:
9057:
9053:
9042:
9038:
9033:
9029:
9025:
9014:
9010:
9007:
8999:
8995:
8983:
8975:
8972:
8969:
8965:
8956:
8940:
8932:
8928:
8924:
8921:
8918:
8913:
8909:
8905:
8900:
8896:
8887:
8884:
8881:
8877:
8870:
8866:
8862:
8851:
8847:
8842:
8839:
8836:
8832:
8828:
8817:
8810:
8804:
8801:
8798:
8790:
8787:
8781:
8773:
8769:
8765:
8762:
8759:
8754:
8750:
8738:
8730:
8727:
8724:
8720:
8711:
8695:
8675:
8665:
8661:
8654:
8649:
8645:
8641:
8637:
8631:
8621:
8617:
8613:
8608:
8604:
8595:
8592:
8589:
8585:
8577:
8574:
8571:
8567:
8563:
8558:
8550:
8546:
8542:
8539:
8536:
8531:
8527:
8518:
8515:
8512:
8508:
8487:
8484:
8481:
8474:of Hermitian
8459:
8456:
8453:
8423:
8415:
8411:
8407:
8385:
8381:
8375:
8371:
8360:
8357:
8354:
8350:
8346:
8340:
8334:
8306:
8300:
8288:
8281:
8273:
8269:
8265:
8258:
8255:
8252:
8242:
8239:
8235:
8231:
8226:
8222:
8199:
8196:
8193:
8183:
8180:
8160:
8157:
8154:
8140:
8121:
8117:
8113:
8110:
8106:
8080:
8077:
8071:
8067:
8060:
8057:
8035:
8031:
8024:
8020:
8010:
8004:
7986:
7983:
7979:
7975:
7971:
7967:
7961:
7941:
7910:
7892:
7888:
7867:
7863:
7855:
7852:
7846:
7842:
7835:
7832:
7829:
7825:
7820:
7816:
7812:
7808:
7802:
7798:
7792:
7784:
7780:
7776:
7773:
7768:
7763:
7759:
7732:
7726:
7723:
7717:
7714:
7711:
7708:
7704:
7698:
7695:
7691:
7686:
7682:
7679:
7676:
7671:
7667:
7659:Moreover, if
7657:
7644:
7641:
7637:
7632:
7628:
7624:
7620:
7614:
7610:
7597:
7571:
7566:
7562:
7558:
7554:
7532:
7528:
7511:
7490:
7486:
7475:
7467:
7459:
7455:
7448:
7423:
7419:
7412:
7390:
7386:
7376:
7359:
7355:
7351:
7348:
7345:
7332:
7329:
7323:
7319:
7315:
7312:
7309:
7276:
7273:
7270:
7251:
7249:
7243:
7233:
7217:
7213:
7186:
7161:
7157:
7142:
7137:
7135:
7113:
7110:
7107:
7101:
7093:
7090:
7087:
7081:
7078:
7075:
7069:
7063:
7060:
7057:
7051:
7043:
7022:
7018:
7003:
6985:
6981:
6955:
6951:
6935:
6919:
6915:
6894:
6876:
6872:
6868:
6863:
6859:
6847:
6843:
6836:
6833:
6830:
6826:
6815:
6810:
6806:
6802:
6794:
6790:
6776:
6775:point process
6757:
6751:
6743:
6739:
6721:
6717:
6702:
6700:
6696:
6692:
6687:
6670:
6664:
6656:
6646:
6627:
6624:
6621:
6618:
6614:
6607:
6604:
6599:
6595:
6574:
6565:
6540:
6535:
6521:
6501:
6489:
6472:
6466:
6446:
6443:
6437:
6431:
6423:
6417:
6412:
6408:
6404:
6401:
6395:
6390:
6386:
6378:
6357:
6335:
6331:
6307:
6301:
6296:
6292:
6288:
6285:
6279:
6274:
6270:
6258:
6234:
6226:
6220:
6213:
6210:
6205:
6199:
6186:
6178:
6174:
6158:
6157:
6156:
6140:
6136:
6108:
6100:
6096:
6084:
6081:
6078:
6070:
6063:
6060:
6056:
6050:
6043:
6040:
6033:
6030:
6025:
6020:
6015:
6008:
6001:
5998:
5991:
5986:
5983:
5977:
5971:
5964:
5963:
5962:
5943:
5939:
5935:
5930:
5926:
5917:
5912:
5909:
5906:
5898:
5895:
5872:
5861:
5858:
5854:
5851:
5848:
5842:
5836:
5833:
5827:
5821:
5808:
5805:
5802:
5794:
5791:
5780:
5776:
5769:
5755:
5752:
5749:
5745:
5742:
5739:
5733:
5727:
5724:
5718:
5712:
5699:
5696:
5693:
5685:
5682:
5671:
5667:
5660:
5659:
5658:
5645:
5637:
5619:
5615:
5592:
5588:
5564:
5558:
5552:
5541:
5535:
5532:
5529:
5523:
5517:
5506:
5500:
5487:
5484:
5481:
5473:
5470:
5465:
5462:
5459:
5455:
5451:
5448:
5430:
5426:
5422:
5419:
5411:
5406:
5402:
5394:
5393:
5392:
5374:
5370:
5367:
5364:
5348:
5341:
5338:
5335:
5332:
5329:
5326:
5322:
5318:
5302:
5298:
5290:
5289:
5288:
5285:
5268:
5260:
5256:
5232:
5229:
5216:
5208:
5204:
5198:
5194:
5190:
5186:
5178:
5171:
5168:
5161:
5156:
5150:
5142:
5138:
5125:
5124:
5123:
5106:
5102:
5095:
5089:
5078:
5072:
5069:
5066:
5060:
5054:
5043:
5037:
5024:
5021:
5018:
5010:
5007:
5002:
4999:
4996:
4992:
4988:
4985:
4981:
4975:
4971:
4967:
4961:
4953:
4949:
4941:
4940:
4939:
4925:
4902:
4894:
4890:
4880:
4866:
4843:
4837:
4814:
4806:
4802:
4795:
4790:
4785:
4782:
4779:
4771:
4768:
4758:
4754:
4750:
4745:
4741:
4732:
4729:
4724:
4721:
4718:
4710:
4707:
4701:
4693:
4689:
4684:
4681:
4668:
4660:
4656:
4652:
4648:
4640:
4633:
4630:
4623:
4618:
4612:
4604:
4600:
4587:
4586:
4585:
4577:
4560:
4557:
4554:
4548:
4543:
4531:
4528:
4525:
4521:
4513:
4507:
4504:
4497:
4491:
4488:
4485:
4480:
4477:
4474:
4470:
4458:
4454:
4450:
4446:
4439:
4435:
4431:
4427:
4422:
4418:
4414:
4403:
4401:
4396:
4394:
4390:
4386:
4382:
4381:Eugene Wigner
4377:
4360:
4354:
4344:
4341:
4337:
4333:
4328:
4325:
4322:
4318:
4309:
4308:global regime
4302:Global regime
4299:
4279:
4276:
4273:
4270:
4267:
4264:
4261:
4258:
4255:
4252:
4249:
4239:
4236:
4233:
4229:
4203:
4199:
4176:
4173:
4167:
4158:
4153:
4149:
4145:
4140:
4136:
4129:
4124:
4120:
4111:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4054:
4051:
4048:
4044:
4032:
4031:
4030:
4027:
4013:
3990:
3984:
3981:
3975:
3965:
3961:
3956:
3944:
3931:
3915:
3912:
3907:
3903:
3899:
3894:
3890:
3886:
3881:
3877:
3868:
3864:
3850:
3827:
3821:
3818:
3812:
3802:
3798:
3793:
3781:
3756:
3753:
3748:
3744:
3740:
3735:
3731:
3727:
3722:
3718:
3709:
3690:
3684:
3681:
3672:
3664:
3660:
3651:
3633:
3608:
3588:
3580:
3570:
3553:
3549:
3544:
3538:
3534:
3528:
3525:
3520:
3514:
3506:
3502:
3488:
3486:
3482:
3478:
3474:
3470:
3466:
3462:
3458:
3440:
3436:
3426:
3413:
3405:
3402:
3398:
3393:
3390:
3385:
3381:
3376:
3372:
3368:
3364:
3356:
3347:
3337:
3334:
3329:
3323:
3315:
3311:
3303:is defined by
3301:
3295:
3288:
3278:
3270:
3267:
3264:
3247:
3233:
3216:
3213:
3210:
3206:
3198:
3194:
3190:
3180:
3176:
3161:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3135:
3126:
3123:
3120:
3112:
3108:
3103:
3080:
3075:
3072:
3069:
3066:
3063:
3052:
3049:
3046:
3038:
3034:
3027:
3022:
3018:
3009:
3000:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2948:
2945:
2942:
2936:
2928:
2924:
2919:
2915:
2912:
2902:
2898:
2877:
2874:
2868:
2860:
2856:
2851:
2848:
2838:
2834:
2810:
2802:
2798:
2788:
2774:
2771:
2768:
2744:
2740:
2733:
2730:
2726:
2721:
2717:
2711:
2707:
2698:
2694:
2688:
2684:
2677:
2673:
2667:
2661:
2653:
2649:
2628:
2625:
2622:
2598:
2594:
2588:
2585:
2580:
2568:
2564:
2556:
2552:
2548:
2543:
2537:
2529:
2525:
2504:
2501:
2498:
2474:
2470:
2464:
2461:
2456:
2452:
2447:
2442:
2439:
2434:
2428:
2420:
2416:
2390:
2386:
2382:
2377:
2374:
2371:
2367:
2360:
2354:
2328:
2321:
2312:
2308:
2304:
2299:
2296:
2293:
2289:
2282:
2279:
2272:
2256:
2253:
2248:
2245:
2242:
2238:
2234:
2229:
2225:
2221:
2218:
2215:
2210:
2206:
2191:
2175:
2171:
2146:
2142:
2138:
2135:
2127:
2124:
2120:
2115:
2109:
2103:
2080:
2077:
2074:
2071:
2068:
2057:
2037:
2033:
2029:
2013:
2011:
2006:
1990:
1986:
1982:
1977:
1973:
1952:
1944:
1940:
1936:
1931:
1927:
1923:
1914:
1907:
1905:
1891:
1883:
1878:
1872:
1868:
1864:
1859:
1855:
1850:
1843:
1840:
1837:
1833:
1825:
1820:
1816:
1810:
1807:
1802:
1798:
1792:
1787:
1784:
1781:
1777:
1769:
1766:
1763:
1759:
1755:
1746:
1745:
1742:
1738:
1734:
1727:
1720:
1716:
1712:
1704:
1686:
1682:
1678:
1675:
1672:
1669:
1666:
1661:
1657:
1653:
1648:
1644:
1640:
1637:
1628:
1619:
1617:
1599:
1591:
1564:
1559:
1549:
1545:
1541:
1538:
1529:
1526:
1522:
1516:
1512:
1501:
1498:
1492:
1489:
1485:
1478:
1470:
1460:
1458:
1442:
1437:
1434:
1430:
1424:
1421:
1417:
1410:
1407:
1402:
1399:
1396:
1390:
1385:
1382:
1378:
1372:
1369:
1365:
1359:
1356:
1351:
1343:
1340:
1336:
1330:
1327:
1323:
1316:
1308:
1303:
1300:
1296:
1290:
1287:
1283:
1270:
1266:
1256:
1254:
1245:
1241:
1232:
1228:
1224:
1219:
1215:
1211:
1208: Ă
1207:
1187:
1183:
1171:
1168:
1164:
1153:
1141:
1137:
1112:
1097:
1092:
1078:
1075:
1072:
1052:
1030:
1027:
1021:
1012:
1008:
1004:
1001:
995:
992:
980:
976:
967:
963:
959:
955:
952: Ă
951:
931:
927:
913:
910:
905:
901:
890:
878:
874:
849:
834:
829:
827:
823:
803:
799:
793:
790:
783:
778:
775:
770:
763:
759:
755:
751:
747:
739:
727:
704:
699:
696:
693:
690:
687:
677:
674:
670:
663:
660:
653:
638:
635:
632:
608:
604:
590:
587:
582:
578:
567:
555:
551:
542:with density
541:
522:
507:
497:
495:
491:
487:
483:
479:
478:distributions
469:
467:
463:
459:
455:
451:
447:
446:Eugene Wigner
443:
439:
435:
431:
420:
411:
407:
404:This section
402:
399:
395:
394:
386:
384:
374:
372:
362:
360:
356:
352:
348:
343:
339:
335:
325:
323:
319:
315:
310:
301:
298:
288:
286:
282:
281:Freeman Dyson
278:
274:
270:
266:
265:number theory
259:Number theory
256:
254:
250:
246:
242:
238:
233:
231:
230:deep learning
227:
223:
219:
215:
211:
207:
203:
199:
195:
185:
184:
180:
176:
172:
168:
164:
160:
156:
151:
149:
145:
141:
136:
134:
133:quantum chaos
129:
127:
123:
119:
115:
111:
110:Eugene Wigner
107:
92:
90:
89:quantum chaos
86:
82:
78:
74:
70:
66:
62:
58:
57:cavity method
54:
50:
46:
42:
38:
34:
30:
29:random matrix
26:
22:
13607:
13539:Substitution
13425:graph theory
13314:
12922:Quaternionic
12912:Persymmetric
12646:
12643:Scholarpedia
12642:
12596:
12592:
12582:
12539:
12535:
12528:Tao, Terence
12522:
12495:
12491:
12481:
12462:
12458:
12448:
12415:
12411:
12402:Pastur, L.;
12397:
12362:
12358:
12302:
12298:
12288:
12269:
12263:
12238:
12234:
12224:
12207:
12204:Duke Math. J
12203:
12197:
12164:
12160:
12155:Pastur, L.;
12150:
12139:
12114:
12110:
12079:
12069:, retrieved
12047:
12037:
12020:
12016:
12006:
11971:
11967:
11961:
11950:. Retrieved
11928:
11918:
11877:
11873:
11860:
11843:
11839:
11833:
11814:
11808:
11773:
11769:
11763:
11746:
11742:
11732:
11699:
11696:Phys. Rev. E
11695:
11682:
11631:
11627:
11621:
11570:
11566:
11560:
11527:
11523:
11517:
11476:
11472:
11466:
11457:
11444:
11417:
11413:
11407:
11396:
11385:
11376:2203.03466v2
11364:
11347:
11343:
11337:
11302:
11298:
11292:
11286:Wishart 1928
11281:
11230:
11226:
11220:
11169:
11165:
11159:
11108:
11105:Phys. Rev. E
11104:
11098:
11065:
11062:Phys. Rev. B
11061:
11058:Callaway DJE
11052:
11001:
10997:
10991:
10940:
10936:
10930:
10879:
10875:
10869:
10826:
10822:
10816:
10773:
10769:
10763:
10744:
10740:
10730:
10691:
10687:
10611:
10607:
10568:
10564:
10543:
10539:
10514:
10510:
10494:1510.04430v2
10463:
10457:
10423:
10419:
10379:
10375:
10342:
10338:
10304:
10285:
10266:
10247:
10230:Circular law
10196:
10191:
10183:
10179:
10175:
10171:
10167:
10163:
10158:
10155:
10151:
10146:
10142:
10138:
10137:
9350:
9349:
9300:
8954:
8709:
8146:
8006:
7658:
7513:
7377:
7252:
7245:
7141:universality
7140:
7138:
7133:
6936:
6708:
6698:
6694:
6690:
6688:
6655:local regime
6654:
6652:
6649:Local regime
6566:
6538:
6536:
6490:
6256:
6254:
6127:
5887:
5634:through the
5579:
5390:
5286:
5247:
5121:
4881:
4829:
4583:
4456:
4452:
4448:
4444:
4437:
4433:
4429:
4425:
4420:
4416:
4412:
4409:
4406:Fluctuations
4397:
4378:
4307:
4305:
4297:
4112:Define each
4028:
3866:
3865:
3707:
3706:Convergence
3578:
3576:
3494:
3484:
3480:
3472:
3468:
3427:
3299:
3290:
3286:
3284:
3276:
3268:
3262:
3178:
3177:
3007:
3006:
2789:
2197:
2019:
2007:
1938:
1929:
1925:
1921:
1919:
1908:
1736:
1732:
1725:
1718:
1708:
1466:
1268:
1264:
1262:
1243:
1239:
1230:
1226:
1222:
1209:
1205:
1095:
1093:
978:
974:
965:
961:
957:
953:
949:
832:
830:
826:Hamiltonians
821:
505:
503:
493:
489:
485:
475:
430:Enrico Fermi
427:
414:
410:adding to it
405:
380:
368:
341:
337:
331:
311:
307:
294:
262:
234:
198:John Wishart
191:
179:quantum dots
152:
137:
130:
122:Hamiltonians
103:
95:Applications
48:
28:
18:
13514:Hamiltonian
13438:Biadjacency
13374:Correlation
13290:Householder
13240:Commutation
12977:Vandermonde
12972:Tridiagonal
12907:Permutation
12897:Nonnegative
12882:Matrix unit
12762:Bisymmetric
12649:(3): 9886.
12486:ErdĆs, L.;
12144:Pastur 1973
11776:(6): 1243.
11420:: 145â185.
10829:: 343â410.
10770:New J. Phys
10747:: 143â252.
10382:(1): 1â67.
10345:: 233â297.
7134:sine kernel
1715:eigenvalues
1218:quaternions
500:Definitions
454:Hamiltonian
377:Engineering
273:L-functions
271:(and other
114:eigenvalues
13642:Categories
13414:Transition
13409:Stochastic
13379:Covariance
13361:statistics
13340:Symplectic
13335:Similarity
13164:Unimodular
13159:Orthogonal
13144:Involutory
13139:Invertible
13134:Projection
13130:Idempotent
13072:Convergent
12967:Triangular
12917:Polynomial
12862:Hessenberg
12832:Equivalent
12827:Elementary
12807:Copositive
12797:Conference
12757:Bidiagonal
12532:Vu, Van H.
12091:2101.02928
12071:2023-05-17
11952:2024-04-22
10783:1506.06220
10724:Mehta 2004
10694:(1): 1â4.
10587:References
10540:Biometrika
7909:Gumbel law
7000:. For the
4576:is known.
1709:The joint
1212:Hermitian
438:Niels Bohr
417:April 2024
345:as one of
13595:Wronskian
13519:Irregular
13509:Gell-Mann
13458:Laplacian
13453:Incidence
13433:Adjacency
13404:Precision
13369:Centering
13275:Generator
13245:Confusion
13230:Circulant
13210:Augmented
13169:Unipotent
13149:Nilpotent
13125:Congruent
13102:Stieltjes
13077:Defective
13067:Companion
13038:Redheffer
12957:Symmetric
12952:Sylvester
12927:Signature
12857:Hermitian
12837:Frobenius
12747:Arrowhead
12727:Alternant
12621:0305-4470
12549:0908.1982
12505:0905.4176
12488:Péché, S.
12389:1432-2064
12372:1206.1449
12337:0010-3616
12312:0803.0542
12255:121531907
12189:120731790
11981:1209.3394
11974:: 69â81.
11902:1879-2138
11783:1306.3825
11580:1210.5082
11312:1004.4389
11273:206326136
11011:0902.4360
10889:0905.3533
10861:119470008
10628:0003-486X
10472:0002-9920
10404:250796916
10096:δ
10068:ψ
10049:ψ
10037:∫
10009:∈
9934:−
9866:ψ
9807:ψ
9788:ψ
9779:−
9761:∑
9575:≤
9563:≤
9533:…
9390:≤
9384:≤
9353:For any
9265:#
9201:∫
9155:⊂
9113:…
9043:∫
9039:⋯
9015:∫
8922:…
8852:∫
8848:⋯
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8863:d
8857:R
8843:1
8840:+
8837:k
8833:x
8829:d
8823:R
8811:!
8808:)
8805:k
8799:n
8796:(
8791:!
8788:n
8782:=
8779:)
8774:k
8770:x
8766:,
8760:,
8755:1
8751:x
8747:(
8742:)
8739:k
8736:(
8731:V
8728:,
8725:n
8721:R
8696:k
8676:.
8671:)
8666:i
8662:x
8658:(
8655:V
8650:i
8638:e
8632:2
8628:)
8622:j
8618:x
8609:i
8605:x
8601:(
8596:j
8590:i
8578:V
8575:,
8572:n
8568:Z
8564:1
8559:=
8556:)
8551:n
8547:x
8543:,
8537:,
8532:1
8528:x
8524:(
8519:V
8516:,
8513:n
8509:p
8488:n
8482:n
8460:n
8454:n
8449:H
8427:)
8424:M
8421:(
8416:0
8408:d
8386:j
8382:x
8376:j
8372:v
8361:1
8358:=
8355:j
8344:)
8341:x
8338:(
8335:V
8313:)
8310:)
8307:M
8304:(
8301:V
8298:(
8289:e
8285:)
8282:M
8279:(
8274:0
8266:d
8259:n
8253:n
8248:H
8240:M
8232:=
8227:n
8223:Z
8200:n
8194:n
8189:H
8181:M
8161:n
8155:n
8122:n
8114:n
8111:4
8107:1
8081:n
8078:4
8072:n
8058:1
8036:n
8032:G
8025:n
8021:1
7987:x
7980:e
7972:e
7968:=
7965:)
7962:x
7959:(
7953:m
7950:u
7947:G
7942:F
7920:R
7893:n
7889:Y
7868:,
7864:)
7856:n
7853:4
7847:n
7833:1
7826:)
7821:n
7817:G
7813:(
7803:n
7799:1
7793:(
7785:n
7777:n
7774:4
7764:n
7760:Y
7739:)
7736:)
7733:n
7730:(
7721:(
7712:2
7705:)
7696:2
7692:n
7687:(
7677:=
7672:n
7645:1
7642:=
7638:)
7633:n
7629:G
7625:(
7615:n
7611:1
7598:n
7572:)
7567:n
7563:G
7559:(
7533:n
7529:G
7497:|
7491:j
7482:|
7476:j
7465:)
7460:n
7456:G
7452:(
7429:)
7424:n
7420:G
7416:(
7391:n
7387:G
7363:)
7360:2
7356:/
7352:1
7349:,
7346:0
7343:(
7338:N
7333:i
7330:+
7327:)
7324:2
7320:/
7316:1
7313:,
7310:0
7307:(
7302:N
7280:)
7277:1
7274:,
7271:0
7268:(
7263:N
7218:0
7187:n
7167:)
7162:0
7154:(
7117:)
7114:y
7108:x
7105:(
7097:)
7094:y
7088:x
7085:(
7070:=
7067:)
7064:y
7061:,
7058:x
7055:(
7052:K
7028:)
7023:0
7015:(
6986:0
6961:)
6956:0
6948:(
6920:j
6895:,
6887:)
6882:)
6877:0
6864:j
6856:(
6853:)
6848:0
6840:(
6834:n
6821:(
6811:j
6803:=
6800:)
6795:0
6787:(
6761:)
6755:(
6752:N
6722:0
6691:n
6674:)
6671:1
6668:(
6665:O
6633:)
6625:+
6622:1
6615:n
6611:(
6608:O
6605:=
6600:n
6596:A
6575:u
6522:N
6476:)
6473:t
6470:(
6447:t
6444:d
6441:)
6438:t
6435:(
6427:]
6418:+
6413:0
6405:,
6391:0
6383:[
6358:N
6336:0
6311:]
6302:+
6297:0
6289:,
6275:0
6267:[
6235:.
6230:)
6227:x
6224:(
6221:q
6211:1
6206:=
6200:x
6196:d
6190:)
6187:x
6184:(
6179:Q
6170:d
6141:Q
6124:.
6112:)
6109:y
6106:(
6101:Q
6092:d
6085:y
6079:x
6074:)
6071:y
6068:(
6061:Q
6054:)
6051:x
6048:(
6041:Q
6031:+
6026:2
6021:)
6016:2
6012:)
6009:x
6006:(
5999:Q
5992:(
5984:=
5981:)
5978:x
5975:(
5972:q
5949:]
5944:j
5940:b
5936:,
5931:j
5927:a
5923:[
5918:q
5913:1
5910:=
5907:j
5899:=
5896:J
5873:J
5866:R
5859:x
5855:,
5852:l
5846:)
5843:x
5840:(
5837:Q
5831:)
5828:y
5825:(
5818:d
5813:|
5809:y
5803:x
5799:|
5786:R
5777:2
5756:J
5750:x
5746:,
5743:l
5740:=
5737:)
5734:x
5731:(
5728:Q
5722:)
5719:y
5716:(
5709:d
5704:|
5700:y
5694:x
5690:|
5677:R
5668:2
5646:l
5620:Q
5593:Q
5589:E
5565:.
5562:)
5559:x
5556:(
5549:d
5545:)
5542:x
5539:(
5536:Q
5530:+
5527:)
5524:y
5521:(
5514:d
5510:)
5507:x
5504:(
5497:d
5492:|
5488:y
5482:x
5478:|
5466:y
5460:x
5444:)
5440:R
5436:(
5431:1
5427:M
5412:=
5407:Q
5403:E
5375:}
5371:1
5368:=
5361:d
5354:R
5342:,
5339:0
5330::
5323:{
5319:=
5316:)
5312:R
5308:(
5303:1
5299:M
5272:)
5266:(
5261:Q
5257:I
5233:,
5226:d
5220:)
5214:(
5209:Q
5205:I
5199:2
5195:N
5187:e
5179:N
5169:Z
5162:1
5157:=
5154:)
5148:(
5143:N
5134:d
5107:,
5103:]
5099:)
5096:x
5093:(
5086:d
5082:)
5079:x
5076:(
5073:Q
5067:+
5064:)
5061:y
5058:(
5051:d
5047:)
5044:x
5041:(
5034:d
5029:|
5025:y
5019:x
5015:|
5003:y
4997:x
4982:[
4976:2
4972:N
4968:=
4965:)
4959:(
4954:N
4950:H
4906:)
4900:(
4895:N
4891:H
4847:)
4844:M
4841:(
4838:Q
4815:,
4812:)
4807:j
4799:(
4796:Q
4791:N
4786:1
4783:=
4780:j
4772:N
4769:+
4765:|
4759:k
4746:j
4737:|
4725:k
4719:j
4708:=
4705:)
4699:(
4694:N
4690:H
4685:,
4678:d
4672:)
4666:(
4661:N
4657:H
4649:e
4641:N
4631:Z
4624:1
4619:=
4616:)
4610:(
4605:N
4596:d
4564:)
4561:1
4558:,
4555:0
4552:(
4549:N
4544:D
4532:n
4529:,
4526:f
4517:)
4511:(
4508:N
4505:d
4501:)
4495:(
4492:f
4481:H
4478:,
4475:f
4471:N
4457:λ
4455:(
4449:λ
4447:(
4445:f
4441:)
4438:j
4434:λ
4432:(
4430:f
4426:n
4421:H
4419:,
4417:f
4413:N
4364:)
4361:H
4358:(
4355:f
4345:1
4338:n
4334:=
4329:H
4326:,
4323:f
4319:N
4294:.
4280:n
4277:,
4271:,
4268:2
4265:,
4262:1
4259:=
4256:j
4253:,
4250:i
4246:}
4240:j
4237:,
4234:i
4230:G
4226:{
4204:n
4200:G
4177:n
4174:2
4168:/
4164:)
4159:T
4154:n
4150:G
4146:+
4141:n
4137:G
4133:(
4130:=
4125:n
4121:H
4109:.
4092:,
4089:3
4086:,
4083:2
4080:,
4077:1
4074:=
4071:j
4068:,
4065:i
4061:}
4055:j
4052:,
4049:i
4045:G
4041:{
4014:A
3994:)
3991:A
3988:(
3982:=
3979:)
3976:A
3973:(
3966:n
3962:H
3945:n
3913:,
3908:3
3904:H
3900:,
3895:2
3891:H
3887:,
3882:1
3878:H
3851:A
3831:)
3828:A
3825:(
3819:=
3816:)
3813:A
3810:(
3803:n
3799:H
3782:n
3754:,
3749:3
3745:H
3741:,
3736:2
3732:H
3728:,
3723:1
3719:H
3694:)
3691:A
3688:(
3682:=
3679:]
3676:)
3673:A
3670:(
3665:H
3657:[
3652:H
3647:E
3634:n
3609:A
3554:i
3539:i
3529:n
3526:1
3521:=
3518:)
3515:A
3512:(
3507:H
3485:λ
3483:(
3481:Ï
3473:λ
3471:(
3469:N
3441:H
3414:.
3410:R
3403:A
3399:,
3394:H
3391:,
3386:A
3382:1
3377:N
3373:=
3369:}
3365:A
3357:H
3348:{
3338:n
3335:1
3330:=
3327:)
3324:A
3321:(
3316:H
3300:H
3293:H
3291:Ό
3263:V
3248:,
3240:)
3237:)
3234:H
3231:(
3227:r
3224:t
3220:(
3217:V
3214:n
3207:e
3199:n
3195:Z
3191:1
3162:}
3158:n
3152:j
3146:i
3140:1
3136:,
3130:)
3127:j
3124:,
3121:i
3118:(
3113:n
3109:H
3104:{
3081:n
3076:1
3073:=
3070:j
3067:,
3064:i
3060:)
3056:)
3053:j
3050:,
3047:i
3044:(
3039:n
3035:H
3031:(
3028:=
3023:n
3019:H
2987:4
2984:,
2981:2
2978:,
2975:1
2972:=
2949:,
2946:1
2943:=
2940:)
2937:s
2934:(
2925:p
2920:s
2916:s
2913:d
2903:0
2878:1
2875:=
2872:)
2869:s
2866:(
2857:p
2852:s
2849:d
2839:0
2814:)
2811:s
2808:(
2799:p
2775:4
2772:=
2745:2
2741:s
2731:9
2718:e
2712:4
2708:s
2699:3
2689:6
2685:3
2674:2
2668:=
2665:)
2662:s
2659:(
2654:4
2650:p
2629:2
2626:=
2599:2
2595:s
2586:4
2576:e
2569:2
2565:s
2557:2
2544:=
2541:)
2538:s
2535:(
2530:2
2526:p
2505:1
2502:=
2475:2
2471:s
2465:4
2453:e
2448:s
2443:2
2435:=
2432:)
2429:s
2426:(
2421:1
2417:p
2391:n
2378:1
2375:+
2372:n
2361:=
2355:s
2329:s
2322:/
2318:)
2313:n
2300:1
2297:+
2294:n
2286:(
2283:=
2280:s
2249:1
2246:+
2243:n
2230:n
2211:1
2176:2
2147:2
2143:x
2136:4
2125:2
2121:1
2116:=
2113:)
2110:x
2107:(
2084:]
2081:2
2078:+
2075:,
2072:2
2066:[
2038:2
2030:N
1991:i
1983:=
1978:j
1939:ÎČ
1930:n
1928:,
1926:ÎČ
1922:Z
1913:)
1911:1
1909:(
1892:,
1879:|
1873:i
1860:j
1851:|
1844:j
1838:i
1826:2
1821:k
1811:4
1799:e
1793:n
1788:1
1785:=
1782:k
1770:n
1767:,
1760:Z
1756:1
1737:n
1733:λ
1729:2
1726:λ
1722:1
1719:λ
1687:5
1683:2
1679:,
1676:.
1673:.
1670:.
1667:,
1662:1
1658:2
1654:,
1649:0
1645:2
1641:=
1638:N
1600:F
1565:2
1560:F
1550:T
1546:V
1542:+
1539:V
1530:N
1527:4
1523:1
1517:e
1513:=
1510:]
1505:)
1502:H
1499:V
1496:(
1493:r
1490:t
1486:e
1482:[
1479:E
1443:,
1438:m
1435:j
1425:n
1422:i
1408:n
1397:2
1391:+
1386:n
1383:j
1373:m
1370:i
1360:n
1357:1
1352:=
1344:m
1341:n
1337:H
1331:j
1328:i
1324:H
1317:=
1304:n
1301:m
1297:H
1291:j
1288:i
1284:H
1265:H
1244:j
1242:,
1240:i
1234:)
1227:H
1223:H
1210:n
1206:n
1188:2
1184:H
1179:r
1176:t
1172:n
1165:e
1157:)
1154:n
1151:(
1142:Z
1138:1
1116:)
1113:n
1110:(
1079:n
1073:n
1053:G
1031:n
1028:2
1022:/
1018:)
1013:T
1009:G
1005:+
1002:G
999:(
996:=
993:H
979:j
977:,
975:i
969:)
962:H
958:H
954:n
950:n
932:2
928:H
923:r
920:t
914:4
911:n
902:e
894:)
891:n
888:(
879:Z
875:1
853:)
850:n
847:(
804:2
800:n
794:2
791:1
784:)
779:n
771:(
764:2
760:/
756:n
752:2
748:=
743:)
740:n
737:(
728:Z
705:n
700:1
697:=
694:j
691:,
688:i
684:)
678:j
675:i
671:H
667:(
664:=
661:H
639:n
633:n
609:2
605:H
600:r
597:t
591:2
588:n
579:e
571:)
568:n
565:(
556:Z
552:1
526:)
523:n
520:(
494:ÎČ
490:ÎČ
486:ÎČ
419:)
415:(
342:k
338:n
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