Canonical Local Equilibrium and Cutoff Profiles for the Symmetric Exclusion Process on Discrete Tori
Abstract
We prove a canonical (or fixed-population) local equilibrium theorem for the symmetric simple exclusion process on the discrete torus 𝕋ND, D≥2, at particle densities bounded away from $0$ and $1$, uniformly over all deterministic initial configurations with the prescribed particle number. At times tN(s)=(log(ND)+s)/2γN, γN=2−2cos(2π/N), the Radon--Nikodym density of the process relative to equilibrium converges in L2 to a canonical exponential tilt generated by the unique small mean-zero calibration field whose one-site marginals match the evolving one-particle heat profile. Consequently, whenever the profile coordinate converges, the corresponding total variation profile is a Gaussian shift. More precisely, for a deterministic initial sequence (SN), if the covariance-normalized squared amplitude qNSN(s) converges to a scalar q at a fixed s, then the distance to stationarity converges to $ 2Φ\!\left({\sqrt{\mathfrak q}}/2\right)-1 $, where Φ is the standard normal distribution function. The profile coordinate is asymptotically determined by the one-particle eigenspace corresponding to the smallest nonzero eigenvalue. The proof combines a calibrated canonical comparison, a fixed-degree comparison between independent and exclusion dynamics, and all-degree control obtained from pair energy estimates and preservation of the Strong--Rayleigh property.
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BibTeX
@misc{chen2026canonical,
title = {{Canonical Local Equilibrium and Cutoff Profiles for the Symmetric Exclusion Process on Discrete Tori}},
author = {Joe P. Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10304},
eprint = {2609.10304},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10304},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Canonical Local Equilibrium and Cutoff Profiles for the Symmetric Exclusion Process on Discrete Tori
AU - Joe P. Chen
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10304
UR - https://arxiv.org/abs/2609.10304
AB - We prove a canonical (or fixed-population) local equilibrium theorem for the symmetric simple exclusion process on the discrete torus 𝕋_N^D, D≥2, at particle densities bounded away from $0$ and $1$, uniformly over all deterministic initial configurations with the prescribed particle number. At times t_N(s)=(log(N^D)+s)/2γ_N, γ_N=2−2cos(2π/N), the Radon--Nikodym density of the process relative to equilibrium converges in L^2 to a canonical exponential tilt generated by the unique small mean-zero calibration field whose one-site marginals match the evolving one-particle heat profile. Consequently, whenever the profile coordinate converges, the corresponding total variation profile is a Gaussian shift. More precisely, for a deterministic initial sequence (S_N), if the covariance-normalized squared amplitude q_N^S_N(s) converges to a scalar q at a fixed s, then the distance to stationarity converges to $ 2Φ\!\left({\sqrt{\mathfrak q}}/2\right)-1 $, where Φ is the standard normal distribution function. The profile coordinate is asymptotically determined by the one-particle eigenspace corresponding to the smallest nonzero eigenvalue. The proof combines a calibrated canonical comparison, a fixed-degree comparison between independent and exclusion dynamics, and all-degree control obtained from pair energy estimates and preservation of the Strong--Rayleigh property.
ER -