Universality conjecture for all Airy, sine and Bessel kernels in the complex plane

Summary

We address the question of how the celebrated universality of local correlations for the real eigenvalues of Hermitian random matrices of size N x N can be extended to complex eigenvalues in the case of random matrices without symmetry. Depending on the location in the spectrum, particular large-N limits (the so-called weakly non-Hermitian limits) lead to one-parameter deformations of the Airy, sine and Bessel kernels into the complex plane. This makes their universality highly suggestive for all symmetry classes. We compare all the known limiting real kernels and their deformations into the complex plane for all three Dyson indices β= 1, 2, 4, corresponding to real, complex and quaternion real matrix elements. This includes new results for Airy kernels in the complex plane for β= 1, 4. For the Gaussian ensembles of elliptic Ginibre and non-Hermitian Wishart matrices we give all kernels for finite N, built from orthogonal and skew-orthogonal polynomials in the complex plane. Finally we comment on how much is known to date regarding the universality of these kernels in the complex plane, and discuss some open problems.

Introduction

The topic of universality in Hermitian random matrix theory (RMT) has attracted a lot of attention in the mathematics community recently, particularly in the context of matrices with elements that are independent random variables, as reviewed in [Tao and Vu 2012; Erdʺos and Yau 2012]. The question that one tries to answer is this: under what conditions are the statistics of eigenvalues of N x N matrices with independent Gaussian variables the same (for large matrices) as for more general RMT where matrix elements may become coupled? This has been answered under very general assumptions, and we refer to some recent reviews on invariant [Kuijlaars 2011; Deift and Gioev 2009] and noninvariant [Tao and Vu 2012; Erdʺos and Yau 2012] ensembles.

In this short note we would like to advocate the idea that non-Hermitian RMT with eigenvalues in the complex plane also warrants the investigation of universality. Apart from the interest in its own right, these models have important applications in physics and other sciences (see, e.g., [Akemann et al. 2011a]). We will focus here on RMT that is close to Hermitian, a regime that is particularly important for applications in quantum chaotic scattering (see [Fyodorov and Sommers 2003] for a review) and quantum chromodynamics (QCD), for example. In the latter case, the non-Hermiticity may arise from describing the effect of quark chemical potential (as reviewed in [Verbaarschot 2011; Akemann 2007]), or from finite lattice spacing effects of theWilson–Dirac operator (see [Damgaard et al. 2010; Akemann et al. 2011b] as well as [Kieburg 2012] for the solution of this non-Hermitian RMT).