Efficient circular Dyson Brownian motion algorithm

TL;DR

This paper introduces a novel, efficient algorithm for simulating circular Dyson Brownian motion in the unitary class, enabling large time evolution studies without approximation or high computational costs.

Contribution

The work presents an improved algorithm that allows large interval simulations of circular Dyson Brownian motion with fixed computational cost, without relying on perturbative approximations.

Findings

Algorithm enables large time evolution simulations efficiently.

No approximations are needed in the new method.

Applicable specifically to the unitary class (β=2).

Abstract

Circular Dyson Brownian motion describes the Brownian dynamics of particles on a circle (periodic boundary conditions), interacting through a logarithmic, long-range two-body potential. Within the log-gas picture of random matrix theory, it describes the level dynamics of unitary ("circular") matrices. A common scenario is that one wants to know about an initial configuration evolved over a certain interval of time, without being interested in the intermediate dynamics. Numerical evaluation of this is computationally expensive as the time-evolution algorithm is accurate only on short time intervals because of an underlying perturbative approximation. This work proposes an efficient and easy-to-implement improved circular Dyson Brownian motion algorithm for the unitary class (Dyson index $\beta = 2$, physically corresponding to broken time-reversal symmetry). The algorithm allows one to study time evolution over arbitrarily large intervals of time at a fixed computational cost, with no approximations being involved.