Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality

Gernot Akemann,Thorsten Neuschel,Friedrich Götze

doi:10.1214/21-ecp398

Abstract

We compute the average characteristic polynomial of the hermitised product of $M$ real or complex Wigner matrices of size $N\times N$ and the average of the characteristic polynomial of a product of $M$ such Wigner matrices times the characteristic polynomial of the conjugate matrix. Surprisingly, the results agree with that of the product of $M$ real or complex Ginibre matrices at finite-$N$, which have i.i.d. Gaussian entries. For the latter the average characteristic polynomial yields the orthogonal polynomial for the singular values of the product matrix, whereas the product of the two characteristic polynomials involves the kernel of complex eigenvalues. This extends the result of Forrester and Gamburd for one characteristic polynomial of a single random matrix and only depends on the first two moments. In the limit $M\to\infty$ at fixed $N$ we determine the locations of the zeros of a single characteristic polynomial, rescaled as Lyapunov exponents by taking the logarithm of the $M$th root. The position of the $j$th zero agrees asymptotically for large-$j$ with the position of the $j$th Lyapunov exponent for products of Gaussian random matrices, hinting at the universality of the latter.

Highlights

Introduction and main resultsCharacteristic polynomials represent one of the central building blocks when studying the spectral statistics of random matrices
For the latter the average characteristic polynomial yields the orthogonal polynomial for the singular values of the product matrix, whereas the product of the two characteristic polynomials involves the kernel of complex eigenvalues
In the limit M → ∞ at fixed N we determine the locations of the zeros of a single characteristic polynomial, rescaled as Lyapunov exponents by taking the logarithm of the M th root

Summary

Introduction and main results

Characteristic polynomials represent one of the central building blocks when studying the spectral statistics of random matrices. Invariant ensembles represent determinantal point processes, and the corresponding kernel of orthogonal polynomials follows from the expectation value of two characteristic polynomials [31, 9] at finite-N This statement extends to non-Hermitian ensembles as well [6]. Due to independence of matrices and matrix elements, for both products of Ginibre and non-Hermitian Wigner matrices the expectation value of a single characteristic polynomial is trivial,. Let us emphasise, that this does not imply an identity for all k-point singular value or complex eigenvalue correlation functions at finite-N , as (non-Hermitian) Wigner ensembles do not possess any determinantal or Pfaffian structure. For the ordered zeros zj(M) of the averaged characteristic polynomials (1.3) of the product of M non-Hermitian Wigner matrices with variances σk > 0, satisfying limM→∞ τM1/M = σ > 0, it holds lim log zj(M). This strongly suggests that the jth Lyapunov exponents for products of non-Hermitian Wigner matrices become universal

Finite-N identities for non-Hermitian Wigner and Ginibre matrices

Large-M asymptotic of the zeros and Lyapunov spectrum