Large deviations of spectral determinants of matrix-valued random Schrödinger operators and Dyson Brownian motion in cubic potentials
Abstract
We study the moments of | det (H -E)| q 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 Σ q in arbitrary dimension. This yields the large-deviation form P (e) ∼ exp {-NL d Φ(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.
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BibTeX
@article{fyodorov2026large,
title = {{Large deviations of spectral determinants of matrix-valued random Schrödinger operators and Dyson Brownian motion in cubic potentials}},
author = {Yan V Fyodorov and Pierre Le Doussal and Alexander Ossipov},
journal = {Journal of Mathematical Physics},
year = {2026},
month = oct,
doi = {10.1063/5.0351290},
eprint = {2511.00954},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05825130v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Large deviations of spectral determinants of matrix-valued random Schrödinger operators and Dyson Brownian motion in cubic potentials
AU - Yan V Fyodorov
AU - Pierre Le Doussal
AU - Alexander Ossipov
PY - 2026
DA - 2026/10/05
JO - Journal of Mathematical Physics
DO - 10.1063/5.0351290
UR - https://hal.science/hal-05825130v1
AB - We study the moments of | det (H -E)| q 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 Σ q in arbitrary dimension. This yields the large-deviation form P (e) ∼ exp {-NL d Φ(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.
SN - 0022-2488
ER -