We show a diffusive upper bound on the transition probability of a tagged particle in the symmetric simple exclusion process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne–Varopoulos type.
"Heat kernel upper bounds for interacting particle systems." Ann. Probab. 47 (2) 1056 - 1095, March 2019. https://doi.org/10.1214/18-AOP1279