Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yoann Offret

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 $α$L with $α$ 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 $α$ 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.

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

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 L with 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 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.
Paper Structure (47 sections, 47 theorems, 310 equations, 13 figures)

This paper contains 47 sections, 47 theorems, 310 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Overview of the main results and ideas
  3. Related results and perspectives
  4. Model and preliminary results
  5. The MESSEP on the ring
  6. The graph structure
  7. The MERW
  8. Extended dynamics
  9. A non-colliding random walk
  10. Full spectral decompositions and spectral gap estimates
  11. The graph spectrum
  12. The MESSEP spectrum
  13. Schur polynomial eigenfunctions
  14. Low-density limit: main results
  15. The scaling limits
  16. ...and 32 more sections

Key Result

Theorem 1.1

Let $(\Xi(t))_{t\geq 0}$ be the continuous-time linear interpolation of the MESSEP. Under standard assumptions, the following functional convergence holds: where $(\boldsymbol{X}(t))_{t\geq 0}$ is the solution of the system of stochastic differential equations

Figures (13)

  • Figure 1: A configuration $\xi \in \mathcal{C}_{L,N}$ with $\xi_1 < \cdots < \xi_N$ on the ring $\mathbb{T}_L$, identified with $\mathbb U_L$. Particle positions are ordered counterclockwise. Arrows indicate possible particle transitions.
  • Figure 2: Correspondence between $V_t$ and $\mathbb D$ under the conformal maps $\Phi_t$ and $w(t,\cdot)$. Here, boundary points match bijectively since $\partial V_t$ is a Jordan curve. Consequently, $f(t,x)$ is continuous on the unit circle. The density has four extremal points, two singular and two smooth.
  • Figure 3: Evolution of $f(t,x)$ from the step initial profile \ref{['eq:packed']} with $\alpha<1/2$, at times $t_1<t_\ast<t_2<t^\ast<t_3$. The density regularizes, converges to the uniform equilibrium $1/(2\pi)$, whereas the saturated plateau and support evolve.
  • Figure 4: Connections between MESSEP, UDBM and FUBM
  • Figure 5: Interface between a saturated region and a macroscopic void.
  • ...and 8 more figures

Theorems & Definitions (109)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Remark 2.2
  • Proposition 2.1
  • Remark 2.3
  • Proposition 2.2
  • ...and 99 more