Index distribution of random matrices with an application to disordered systems

Abstract

We compute the distribution of the number of negative eigenvalues (the index) for an ensemble of Gaussian random matrices, by means of the replica method. This calculation has important applications in the context of statistical mechanics of disordered systems, where the second derivative of the potential energy (the Hessian) is a random matrix whose negative eigenvalues measure the degree of instability of the energy surface. An analysis of the probability distribution of the Hessian index is therefore relevant for a geometric characterization of the energy landscape in disordered systems. The approach we use here is particularly suitable for this purpose, since it addresses the problem without any a priori assumption on the random matrix ensemble and can be naturally extended to more realistic, non-Gaussian distributions.

Index statistical properties of sparse random graphs

“…This result reveals that, in order to access the index fluctuations in this case, one needs to compute the next-order contribution to G N (μ,0) for large N . The same situation arises in the replica approach for the GOE ensemble [7]. We present in the next section explicit results for the mean and the variance of the index for specific random graph models in the regime |λ| > 0.…”

“…(1) is recovered in the regime of small fluctuations, with a variance that grows as σ 2 ln N for large N . The prefactor σ 2 is given by σ 2 = 1/π 2 for both Gaussian [7,[10][11][12] and Wishart [13] random matrices, independently of λ, while σ 2 = 2/π 2 for Cauchy random matrices [14]. This logarithmic behavior of the variance apparently reflects the repulsion between neighboring levels [16], which imposes a constraint on the total number of eigenvalues that fit in a finite region of the spectrum.…”

“…In this case, the Hessian belongs to the GOE ensemble of random matrices [2] and the index statistics has been studied originally in Ref. [7], using a fermionic version of the replica method.…”

“…In particular, the minima (maxima) of the potential energy are stationary points in which all Hessian eigenvalues are positive (negative). The index is a valuable tool to probe the energy landscape of systems as diverse as liquids [4,5], spin-glasses [6][7][8], synchronization models [9], and biomolecules [3].…”

Real space origin of temperature crossovers in supercooled liquids

“…[30], in analogy with what happens in mean field models [31,32]: the vanishing as T approaches T c of the mean intensive number of negative directions (intensive index) of stationary points of the potential energy, n s (T ). Numerical simulations [33,34,35,36,37,38] found that n s (T ) decreases with decreasing T , and fits were performed to show that n s (T c ) = 0 [33,34,35].…”