Work

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

Year: 2026

Type: article

Abstract: We study the moments of |det(H−E)|q̄ and the large deviations of the corresponding reduced intensive log spectral determinant e for random matrix operators with Laplacian and random-potential terms. For large N and real q > −1, we obtain a closed saddle-point characterization of the exponential growth rates Σq in arbitrary dimension. This yields the large-deviation form P(e)∼exp{−NLdΦ(e)} and an explicit expression for the rate function Φ(e). The framework includes Hessians of random elastic manifolds at generic energy configurations. In one dimension, these become N × N matrix-valued random Schrödinger operators, and log| det(H − E)| is the sum of the N Lyapunov exponents. A stochastic matrix Riccati mapping relates their spectral properties to the total N-particle current of Dyson Brownian motion (DBM) in a cubic potential. Allez and Dumaz [J. Stat. Phys. 160(3), 681–714 (2015)] showed that at N = ∞ this model has a sharp transition between a current-carrying and a confined phase. We compute the DBM barrier-crossing probability at large finite N, yielding the exponential tail of the mean density of states below the spectral edge. The barrier scales as ∼N(−E)3/2 at large negative energy and vanishes as ∼N(E*−E)5/4 near the edge. For q = 1 we also obtain an independent derivation of the total complexity of stationary points for a disordered elastic string embedded in N dimensions.

Source: Journal of Mathematical Physics

DOI: 10.1063/5.0351290

Authors: Yan V. Fyodorov, Pierre Le Doussal, Alexander Ossipov