From maximal entropy exclusion process to unitary Dyson Brownian motion and free unitary hydrodynamics

TL;DR

This paper connects a discrete symmetric exclusion process with complex hydrodynamic limits to continuous models like the Unitary Dyson Brownian Motion and Free Unitary Brownian Motion, revealing algebraic and entropic structures.

Contribution

It introduces a spectral analysis of MESSEP using Schur polynomials, establishing a rigorous link between discrete exclusion dynamics and continuous free unitary hydrodynamics.

Findings

Rescaled MESSEP converges to Unitary Dyson Brownian Motion in low-density regime.

Hydrodynamic limit yields a nonlinear nonlocal transport equation.

As density vanishes, the equation matches that of Free Unitary Brownian Motion.

Abstract

We investigate the Maximal Entropy Simple Symmetric Exclusion Process (MESSEP) on a discrete ring with L sites and N indistinguishable particles. Its eigenfunctions are Schur polynomials evaluated at the L-th roots of unity, yielding an explicit spectral decomposition. The analysis relies on this eigenstructure and on the link between Schur polynomials and irreducible characters of the symmetric group, which forms the core algebraic tool for the scaling limits. In the low-density regime, where N is fixed and L tends to infinity, the rescaled dynamics converge to the Unitary Dyson Brownian Motion (UDBM). The electrostatic repulsion then emerges as an entropic force, providing a canonical microscopic derivation of the UDBM. In the hydrodynamic regime, where N is equivalent to $\alpha$L with $\alpha$ P p0, 1q, the empirical measure converges to a density solving a nonlinear, nonlocal transport equation. Its moment generating function satisfies a complex Burgers-type equation. As $\alpha$ tends to 0, this equation coincides with that governing the spectral distribution of the Free Unitary Brownian Motion (FUBM), thereby bridging discrete entropic exclusion dynamics and free unitary hydrodynamics. Overall, the MESSEP provides a unified canonical discrete framework connecting unitary Dyson motion and free unitary Brownian motion through nonlinear hydrodynamic limits, with Schur and character theory as the central algebraic structure.