Large deviations of spectral determinants of matrix-valued random Schrödinger operators and Dyson Brownian motion in cubic potentials

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 Schrödinger 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 $\sim N(E^{\ast }-E{)}^{5/4}$ 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.

27 pages