E. P. Wigner, Random matrices in physics, SIAM Review, 9 (1967), pp. 1-23.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... value of the inverse temperature from Ginibre s work extending Wigner s characterization of the real spectra of random matrices in terms of the thermodynamics of a Coulomb gas on the real line, to the complex spectra of random matrices in terms of the 2 D Coulomb gas [15] Wigner s construction [41] is at the center of a recent surge of research on random matrix theory, relating several diverse areas such as zeroes of Riemann zeta functions [35] 18] Riemann Hilbert problems [7] representations of the symmetric group [36] 32] non commutative probability theory [14] 40] mathematical ....

E. Wigner, Random Matrices in Physics, SIAM Review 9; 1-23, 1967. 21

....M ) Gamma(T ) Gamma(T 1 Gamma M) 2 M M 2 (2T M 1) Z d Omega e itr y Phi GammaI M ) A.8) where the integration is over the Hermitian matrix Omega . But this simply the desired result (13) B A Useful Integral Formula In the next result we generalize a formula of Wigner [11] for positive functions f( Delta) to arbitrary ones. Lemma 1 (An Integral Formula) Consider the single variable function f( Delta) Then we have the following identity for the M fold integral Z M Y m=1 dm f(m ) Y l m ( Gammai m i l ) 2 = M det F; B.1) where F is an M Theta M ....

E. Wigner, "Random matrices in physics," SIAM Review, vol. 9, pp. 1--23, 1967.

.... Random matrices, particularly their eigenvalues, have been a familiar topic for about fty years in physics [37] statistics [38,59] and numerical analysis [15] and more recently in number theory [3,33] In condensed matter physics, the subject was made famous by Wigner s semicircle law [58] and the phenomenon of Anderson localization (exponential decay of eigenvectors) 1] These and many other developments in random matrix theory have emphasized hermitian matrices, the setting for classical quantum mechanics, but random nonhermitian problems have been studied too, since the early ....

E. P. Wigner, Random matrices in physics, SIAM Review, 9 (1967), pp. 1-23.

....of the eigenvalues by integrating out the eigenvectors. This can be readily done when the distribution of the elements is invariant to unitary similarity transformations. 3. Obtain the marginal distribution of a single eigenvalue from the joint distribution using a technique of Wigner [22]. This applies to a special class of eigenvalue distributions, and the resulting marginal distribution is expressed as a Christoffel Darboux sum of orthogonal polynomials. 4. Obtain the asymptotic marginal distribution by studying the asymptotics of the Christoffel Darboux sum using a saddlepoint ....

....The marginal eigenvalue distribution To obtain the marginal distribution, we need to integrate out the variables, 2 ; M . An effective way to do this for any distribution of the form: p( 1 ; M ) K M Y m=1 f(m ) Y i j ( i Gamma j ) 2 ; A. 19) was introduced by Wigner [22]. 6 Lemma 4 (Wigner [22] Let the random variables 1 ; M be distributed according to (A.19) Then the marginal distribution for an arbitrary is given by p( f( M M Gamma1 X i=0 OE 2 i ( A.20) 6 Wigner introduced this method in his study of the so called Gaussian unitary ....

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E. P. Wigner, "Random matrices in physics", SIAM Review, vol. 9, pp. 1--23, 1967.

.... O( p p=T ) and hence that the ratio of off diagonal to on diagonal terms is O(1= p pT ) A series of numerical experiments confirms the Gaussian nature of the entries of S and also that the estimate for the ratio is sound (see Figure 11 and Table 1) According to Wigner s semi circle theorem[12, 13], a symmetric random matrix of order N, whose entries have zero mean and variance oe 2 has the eigenvalue density ae( 1 2 N oe 2 q 4N 2 Gamma 2 : 31) The theorem applies to S and since eigenvalues of M are p plus the eigenvalues of S it follows that the eigenvalues of M are such ....

E.P. Wigner. Random matrices in physics. SIAM Review, 9(1), 1967.

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E. P. Wigner, Random matrices in physics, SIAM Review, 9 (1967), pp. 1-23.

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E.P. Wigner, Random matrices in physics, S.I.A.M. Review, Vol. 9, No. 1 (1967), 1-23.

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