Novel Universal Correlations in Invariant Random-Matrix Models

Abstract

We show that eigenvalue correlations in unitary-invariant ensembles of large random matrices adhere to novel universal laws that only depend on a multicriticality of the bulk density of states near the soft edge of the spectrum. Our consideration is based on the previously unknown observation that genuine density of states and n−point correlation function are completely determined by the Dyson's density analytically continued onto the whole real axis.chao-dyn/9701006

Parametric spectral statistics in unitary random matrix ensembles: from distribution functions to intra-level correlations

“…The objective of this paper is to present a set of interrelated developments in the parametric RMT. In the next section we construct a formalism to determine the joint distribution functions for the eigenvalues of H and H ′ for perturbing matrices V of arbitrary rank r. In particular, we concentrate on the case when r is finite in the N → ∞ limit, although the results are equally applicable in the opposite limit where we reproduce the Gaussian transition kernel implicit in the earlier work on the subject [7,18].…”

“…Owing to the determinantal structure of R nm [17,7,18] (see also Section 3.1 below), the latter possess representations in terms of Fredholm determinants of certain parameterdependent integral kernels.…”

Multicritical microscopic spectral correlators of hermitian and complex matrices

“…Performing the usual large-N limit the recursion coefficients c n and the functions A n (λ) and B reg n (λ) are assumed to approach smooth functions. Under these conditions, with a generic potential V (λ), the function A n (λ) is directly related to the macroscopic spectral density ρ(λ) [13,17]:…”

“…From this we can derive the limiting forms of the orthogonal polynomials in the appropriate large-N scaling regions. A recent paper by Kanzieper and Freilikher [13] shows very elegantly how the derivation of this differential equation can be simplified. Their approach is in fact more powerful, and leads more easily to generalizations.…”

“…As derived in ref. [13], the crucial property of the differential equation is that it does not explicitly depend on the chosen matrix model potential V (M ) once the large-N limit is taken, but only on the associated macroscopic spectral density ρ(λ). As we shall see in section 2, this simplifying feature unfortunately does not survive in the particular microscopic large-N limit that is required to probe the multicritical domain near λ = 0.…”