Universal wide correlators in non-gaussian orthogonal, unitary and symplectic random matrix ensembles

Abstract

We calculate wide distance connected correlators in non-gaussian orthogonal, unitary and symplectic random matrix ensembles by solving the loop equation in the 1/N-expansion. The multi-level correlator is shown to be universal in large N limit. We show the algorithm to obtain the connected correlator to an arbitrary order in the 1/N-expansion.1

Replica method for wide correlators in Gaussian orthogonal, unitary and symplectic random matrix ensembles

“…This result with respect to the connected part agrees with that obtained by solving functional equations [4,13] and by replica method [14].…”

“…with the one point Green function defined by eq(31). Three level correlators in GOE and GSE obtained by the diagrammatic method are identical those calculated by the replica method and the functional method [13].…”

“…For these ensembles, we have to sum over both orientable and non-orientable diagrams. The results can be checked to compare those calculated by solving functional equations [4,13] or by replica method [14], when time dependence is suppressed.…”

Spectra of Large Random Matrices: A Method of Study

“…The first group includes different approximate methods useful to explore global spectral characteristics which manifest themselves on the scale of n ≫ 1 eigenlevels. Among these methods there are (i) the mean-field approximation proposed by Dyson [4,18] which allows one to compute the density of levels in random matrix ensemble; (ii) the Schwinger-Dyson loop equations' technique [19,20,21] that had led to discovery of the phenomenon of the global spectral universality; (iii) the method of functional derivative of Beenakker [22,23], and (iv) the diagrammatic approach of Brézin and Zee [24] whose development enabled their authors to study the phenomenon of global universality [19] in more detail as well as to generalize it in the context of mesoscopic physics.…”

“…The multiple integrals in the last equation can exactly be calculated by using the representation Eq. (21). The result of the integration reads [1]…”