Hydrodynamic limit for the two-dimensional finite-range exclusion process

Prove the existence of the hydrodynamic limit for the two-dimensional simple symmetric exclusion process with excluded volume specified by a finite particle shape A, particularly in the subcritical density regime, and rigorously identify the limiting nonlinear diffusion equation and its diffusion matrix.

Background

The paper establishes the hydrodynamic limit rigorously in one dimension, where the limiting equation is a porous medium equation. In two dimensions, the process is non-gradient when the exclusion shape is nontrivial, and the presence of hard-core constraints prevents direct application of available non-gradient methods. The authors therefore restrict their analysis to the subcritical regime, where they assume rather than prove that a deterministic hydrodynamic limit exists and is governed by a nonlinear diffusion equation.

The paper proves conditional structural results: the diffusion matrix is diagonal for rectangular exclusion zones and is a scalar multiple of the identity for square exclusion zones. Numerical simulations provide estimates of its entries, but neither the hydrodynamic-limit theorem nor the governing diffusion matrix is derived rigorously.

References

Since we cannot prove the existence of the hydrodynamic limit, we have to assume it.

— Hydrodynamic limits of finite range exclusion processes  (2610.21787 - Berrens et al., 15 Oct 2026) in Section 2, immediately before Section 2-dimensional analysis; Introduction

In conclusion, deriving PDE as diffusive limits from particle systems is a delicate matter, and even reasonable heuristic arguments can easily lead to the wrong effective equation. In particular, beyond the numerics that we presented and that we believe is trustworthy, we have no suggestion of a formula for the diffusivity in the case $A = {0,1}2$ or similar ones.

— Hydrodynamic limits of finite range exclusion processes  (2610.21787 - Berrens et al., 15 Oct 2026) in Remark at the end of Section 2-dimensional analysis