Spectral Determinants of Random Schrödinger Operators Link to Dyson Brownian Motion and Current Transitions at Energy Configuration

The behaviour of random quantum systems presents a long-standing challenge in theoretical physics, and recent work by Yan Fyodorov, Pierre Le Doussal, and Alexander Ossipov from King’s College London and the Laboratoire de Physique de l’École Normale Supérieure addresses this by investigating the extreme fluctuations of spectral determinants in complex random potentials. The team establishes a surprising connection between the spectral properties of these matrix-valued Schrödinger operators and the behaviour of a ‘Dyson Brownian motion’, a model system describing particle diffusion in a cubic potential. This link allows them to calculate the probability of rare events in the Brownian motion, revealing crucial information about the density of states in the quantum system and providing an independent confirmation of theoretical predictions concerning the complexity of disordered elastic strings. The research significantly advances our understanding of how randomness impacts quantum systems and offers new insights into the behaviour of disordered materials.

Random Matrices, Statistical Physics, and Eigenvalues

This extensive compilation of references focuses on random matrix theory, statistical physics, mathematical physics, and related fields, covering core topics such as random matrix theory itself, connections to spin glasses and disordered systems, and the analysis of eigenvalue distributions. Researchers also explore quantum chaos, large deviations, high-dimensional random fields, and non-Hermitian systems, demonstrating the breadth of this research area, and highlighting the importance of Toeplitz and Hankel determinants and their applications within these fields. The collection reveals a strong emphasis on foundational work alongside emerging areas of research, including contributions from Dyson and Pastur, alongside more recent studies on multiplicative chaos and high-dimensional random fields. References are categorized to facilitate exploration, encompassing foundational works, studies of spin glasses and disordered systems, investigations of localization phenomena, and analyses of large deviations and extremal statistics. The bibliography demonstrates a growing focus on non-Hermitian random matrix theory, reflecting its increasing importance in modern physics, and emphasizes connections to probability and mathematical physics, highlighting the mathematical foundations of the field. The emergence of multiplicative chaos as a research area is also notable, suggesting a growing need for computational and numerical methods to address increasingly complex problems, and highlighting the interdisciplinary nature of random matrix theory, drawing on ideas from physics, mathematics, probability, and computer science.

Matrix Operators and Dyson Brownian Motion Dynamics

Researchers investigate the spectral properties of random matrix operators by connecting them to the dynamics of a Dyson Brownian motion in a cubic potential, developing a method to map the problem onto a stochastic Ricatti equation. This allows them to relate the density of states of matrix Schrödinger operators to the current exhibited by the Dyson Brownian motion, building upon earlier work demonstrating a phase transition in the Dyson Brownian motion, a shift between flowing and confined phases, and extending these findings to matrix-valued operators. Scientists analyzed the system using a continuum model in one dimension, defining a matrix operator with specific boundary conditions and expressing the associated functional determinant using a Gelfand-Yaglom formula, leading to a matrix Ricatti equation that determines the system’s behaviour. This equation describes the evolution of a matrix and is central to understanding the spectral properties of the operator, with researchers introducing the eigenvalues of this matrix and, through perturbation theory, deriving a stochastic evolution equation governing their dynamics, incorporating both deterministic and stochastic terms. To further analyze the system, the team defined the trace of the resolvent of the matrix as a Stiljies transform of the empirical density of eigenvalues, allowing them to derive an exact stochastic evolution equation for this trace, incorporating noise terms that account for the randomness of the system. Setting the initial condition for this equation to zero, researchers, in the large N limit, developed a framework for understanding the system’s behaviour and its connection to previous work, enabling the study of activated barrier crossing events and their relation to the density of states.

Spectral Determinants Link to Brownian Motion Currents

Researchers investigate the moments of spectral determinants arising from random matrix operators, including Laplace operators coupled with random potentials, which naturally appear when studying the Hessians of random elastic manifolds and also describe matrix-valued Schrödinger operators in quasi-one dimensional systems. The research establishes a connection between the spectral properties of these operators and the total particle current of a Dyson Brownian motion in a cubic potential. Experiments reveal that the barrier-crossing probability of the Dyson Brownian motion at large but finite energies provides an estimate of the exponential tail of the average density of states for a matrix Schrödinger operator below its spectrum edge, with the barrier behaving proportionally to negative energy at large negative values and vanishing near the edge of the spectrum. This work independently derives the total complexity of stationary points for an elastic string embedded in a disordered environment.

The study considers both discrete and continuous matrix-valued Schrödinger operators, representing disordered elastic manifolds of arbitrary dimension, with the discrete model utilizing a matrix defined by a Laplacian and random disorder, with correlations determined by a parameter controlling disorder strength. The continuous model employs a Gaussian white noise random matrix process, equivalent to the discrete model in one dimension, with researchers measuring the growth rates of moments of the modulus of the spectral determinant, finding they scale exponentially with the system size. The research demonstrates that these growth rates, termed Σq, can be determined to leading order at large matrix dimensions, building upon decades of research into random matrix theory, with connections to the Riemann zeta-function, Toeplitz determinants, and Gaussian free fields, and providing insights into the behaviour of disordered elastic systems and their glassy properties.

Dyson Motion and Random Schrödinger Operator Complexity

This research establishes a connection between the spectral properties of random matrix operators and the behaviour of a Dyson Brownian motion, a model exhibiting a transition between phases of differing particle current, computing the barrier-crossing probability of this motion, yielding insight into the exponential tail of the average density of states for random Schrödinger operators. The analysis reveals how this barrier scales at both large negative energies and near the edge of the spectrum, providing a new derivation of the complexity of stationary points for disordered elastic strings. The team extended this work to compute more general quantities related to Lyapunov exponents, which describe the growth rate of solutions to the random Schrödinger equation, obtaining the large deviation tail of the distribution of intensive free-energy through a Legendre transform, characterizing fluctuations in the sum of Lyapunov exponents at large matrix sizes. In one dimension, this approach allows for the definition of a generalized Lyapunov exponent, providing a means to study fluctuations in the sum of these exponents within a large deviation regime. The authors acknowledge that their calculations rely on a saddle-point approximation in the large system size limit and suggest that future work could explore corrections to this approximation.