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We obtain sharp error rates in the local limit theorem for the Sinai billiardmap (one and two dimensional) with infinite horizon. This result allows us tofurther obtain higher order terms and thus, sharp mixing rates in the speed ofmixing of dynamically H\"older observables for the planar and tubular infinitehorizon Lorentz gases in the map (discrete time) case. We also obtain anasymptotic estimate for the tail probability of the first return time to theinitial cell. In the process, we study families of transfer operators forinfinite horizon Sinai billiards perturbed with the free flight function andobtain higher order expansions for the associated families of eigenvalues andeigenprojectors.

Sharp error term in local limit theorems and mixing for Lorentz gases with infinite horizon