Strong mixing for the periodic Lorentz gas flow with infinite horizon

TL;DR

This paper proves strong mixing for the infinite horizon periodic Lorentz gas flow with continuous observables, including those with points of infinite free flights, using new local limit theorems and tightness results.

Contribution

It introduces the first strong mixing results for the Lorentz gas flow with infinite horizon for a broad class of observables, including those with infinite free flights.

Findings

Proved strong mixing for the Lorentz gas flow with infinite horizon.

Established a mixing local limit theorem for Sinai billiard flow with infinite horizon.

Developed a tightness-type result controlling configurations with long free flights.

Abstract

We establish strong mixing for the $\mathbb Z^d$-periodic, infinite horizon, Lorentz gas flow for continuous observables with compact support. The essential feature of this natural class of observables is that their support may contain points with infinite free flights. Dealing with such a class of functions is a serious challenge and there is no analogue of it in the finite horizon case. The mixing result for the aforementioned class of functions is obtained via new results: 1) mixing for continuous observables with compact support consisting of configurations at a bounded time from the closest collision; 2) a tightness-type result that allows us to control the configurations with long free flights. To prove 1), we establish a mixing local limit theorem for the Sinai billiard flow with infinite horizon, previously an open question.