Large deviations of spectral determinants of matrix-valued random Schr\"odinger operators and Dyson Brownian motion in cubic potentials

TL;DR

This paper investigates the large deviations of spectral determinants of matrix-valued random Schrödinger operators, connecting their spectral properties to Dyson Brownian motion in cubic potentials, and provides insights into the density of states and complexity of stationary points.

Contribution

It introduces a novel mapping to a stochastic Ricatti equation linking spectral determinants to Dyson Brownian motion, and computes the barrier-crossing probability for finite N, revealing new large deviation behaviors.

Findings

Barrier-crossing probability scales as N(-E)^{3/2} at large negative energies.

Exponential tail of the density of states is estimated for matrix Schrödinger operators.

Provides an independent derivation of the complexity of stationary points for elastic strings in disorder.

Abstract

We study the moments of $\overline{|\det(H-E)|^q}$ and the associated large deviations of $\log |\det(H-E)|$ where $H$ are random matrix operators involving Laplace operators and random potentials. This includes as a special case Hessians of random elastic manifolds at a generic energy configuration. In one dimension $d=1$ these are $N \times N$ matrix valued random Schr\"odinger operators and $\log | \det(H-E) | $ is the sum of the $N$ associated Lyapunov exponents. Using a mapping to a stochastic matrix Ricatti equation we make a connection between the spectral properties of these operators and the total $N$ particle current of a Dyson Brownian motion (DBM) in a cubic potential. The latter model was studied by Allez and Dumaz [1] who showed that for $N=+\infty$ it exhibits a sharp transition between a phase with non-zero current and a confined (zero current) phase. We compute the barrier-crossing probability of the DBM at large but finite $N$, which gives an estimate of the exponential tail of the average density of states of a matrix Schrodinger operator below the edge of its spectrum. The barrier behaves as $\sim N (-E)^{3/2}$ at large negative energy and vanishes as $\sim N(E^*-E)^{5/4}$ near the edge. For $q=1$ the present work provides an independent derivation of the total complexity of stationary points for an elastic string embedded in $N$ dimension in presence of disorder.