Sharp Error Term in Local Limit Theorems and Mixing for Lorentz Gases with Infinite Horizon

We obtain sharp error rates in the local limit theorem for the Sinai billiard map (one and two dimensional) with infinite horizon. This result allows us to further obtain higher order terms and thus, sharp mixing rates in the speed of mixing of dynamically Holder observables for the planar and tubular infinite horizon Lorentz gases in the map (discrete time) case. We also obtain an asymptotic estimate for the tail probability of the first return time to the initial cell. In the process, we study families of transfer operators for infinite horizon Sinai billiards perturbed with the free flight function and obtain higher order expansions for the associated families of eigenvalues and eigenprojectors.

Automated summary

The article discusses the local limit theorem for the Sinai billiard map, obtaining sharp error rates and mixing rates in the process. It also provides an asymptotic estimate for the tail probability of the first return time to the initial cell, while studying transfer operators and higher order expansions for eigenvalues and eigenprojectors.