Canonical Local Equilibrium and Cutoff Profiles for the Symmetric Exclusion Process on Discrete Tori

We prove a canonical (or fixed-population) local equilibrium theorem for the symmetric simple exclusion process on the discrete torus ${\mathbb{T}}_N^D$, $D\ge 2$, at particle densities bounded away from $0$ and $1$, uniformly over all deterministic initial configurations with the prescribed particle number. At times $$t_N(s)=\frac{\log (N^D)+s}{2{\gamma }_N},{\gamma }_N=2-2\cos \left(\frac{2\pi }{N}\right),$$ 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 ${\mathfrak{q}}_N^{S_N}(s)$ converges to a scalar $\mathfrak{q}$ at a fixed $s$, then the distance to stationarity converges to $2\mathrm{\Phi }\text{\hspace{0.17em}⁣}\left(\sqrt{\mathfrak{q}}\mathrm{/}2\right)-1$, where $\mathrm{\Phi }$ 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.