Rare Events Statistics for $$\mathbb {Z}^d$$ Map Lattices Coupled by Collision

Understanding the statistics of collisions among locally confined gas particles poses a major challenge. In this work we investigate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow> <mml:mi>d</mml:mi></mml:msup> </mml:math> -map lattices coupled by collision with simplified local dynamics that offer significant insights for the above challenging problem. We obtain a first order approximation for the first collision rate at a site <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow> <mml:mo>∗</mml:mo></mml:msup> <mml:mo>∈</mml:mo> <mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow> <mml:mi>d</mml:mi></mml:msup> </mml:mrow> </mml:math> and we prove a distributional convergence for the first collision time to an exponential, with sharp error term. Moreover, we prove that the number of collisions at site <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow> <mml:mo>∗</mml:mo></mml:msup> </mml:math> converge in distribution to a compound Poisson distributed random variable. Key to our analysis in this infinite dimensional setting is the use of transfer operators associated with the <i>decoupled</i> map lattice at site <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow> <mml:mo>∗</mml:mo></mml:msup> </mml:math> .