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

TL;DR

The paper analyzes large deviations of spectral determinants for matrix-valued random Schrödinger operators by computing growth rates $\Sigma_q$ of the reduced moments $\tilde{Y}_q = \overline{|\det({\cal H}-E)|^q}$ and, via a Legendre transform, the rate function $\Phi({\sf e})$ for the intensive energy variable. It develops two complementary methods: a continuum matrix Riccati reduction that links the determinant to Dyson Brownian motion in a cubic potential and a saddle-point scheme valid in any dimension $d$, yielding explicit barrier tails for the density of states below the spectral edge and connecting to the $q=1$ complexity in disordered elastic manifolds. The results include a detailed analysis in 1D, with a phase transition between confined and flowing DBM, a barrier $U$ giving Lifshitz-type tails $\rho_K(\alpha) \sim \exp(-N U/\tilde{J}^2)$ and explicit formulas for $d=0$ and $d=1$; the formalism unifies spectral-det determinant fluctuations with large-deviation theory and DOS tails, and clarifies the role of a zero-mode $\bar{\xi}$ in shaping the tails. The work provides new insights into the spectral statistics of Hessians of disordered elastic manifolds and relates the full distribution of $\log|\det({\cal H}-E)|$ to barrier-crossing dynamics in a cubic potential.

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