On the moments of the characteristic polynomial of a Ginibre random matrix

Abstract

In this article, we study the large N asymptotics of complex moments of the absolute value of the characteristic polynomial of an N×N complex Ginibre random matrix with the characteristic polynomial evaluated at a point in the unit disk. More precisely, we calculate the large N asymptotics of double-struckEfalse|trueprefixdetfalse(GN−zfalse)|γ, where GN is an N×N matrix whose entries are i.i.d. and distributed as N−1/2Z, Z being a standard complex Gaussian, Re (γ)>−2, and |z|<1. This expectation is proportional to the determinant of a complex moment matrix with a symbol which is supported in the whole complex plane and has a Fisher–Hartwig type of singularity: 0truedet(∫Cwiw¯jfalse|w−zfalse|γe−Nfalse|wfalse|2d2wfalse)i,j=0N−1. We study the asymptotics of this determinant using recent results due to Lee and Yang concerning the asymptotics of orthogonal polynomials with respect to the weight |w−z|γe−N|w|2d2w along with differential identities familiar from the study of asymptotics of Toeplitz and Hankel determinants with Fisher–Hartwig singularities. To our knowledge, even in the case of one singularity, the asymptotics of the determinant of such a moment matrix whose symbol has support in a two‐dimensional set and a Fisher–Hartwig singularity have been previously unknown.

The distribution of overlaps between eigenvectors of Ginibre matrices

“…The following crude a priori estimates are used in this paper. Note that for z strictly in the bulk of the spectrum (|z| < 1 − ε for fixed ε > 0), the first statement is a simple consequence of the main result in [55].…”

Characteristic Polynomials of Complex Random Matrices and Painlevé Transcendents

“…More recently there has been a resurgence of interest in characteristic polynomials of random matrices due to their association with log-correlated Gaussian fields [50,43,66,46]. Along these lines, quantity (1.1) can be interpteted as the multi-point Laplace transform of the random process φ N (z) = log | det(G N − z)| which is asymptotically Gaussian and logarithmically correlated, see [62,6,17,67]. This is also closely related to the recently studied question of 'moments of moments' [11,10,32] (where Painlevé V also arises), albeit in the Hermitian context.…”

Local Density for Two-Dimensional One-Component Plasma

“…[24]. In particular, convergence of the fluctuations at macroscopic scale to a Gaussian free field has been established [2,3,37,46]. Charge fluctuations have been studied [28], and heuristic arguments for very detailed behaviour are given in [47,[50][51][52].…”