A law of thin processes with neighbour-count thinning

Source abstract

We consider the superposition $S_n$ of $n$ i.i.d. simple point processes on $\mathbb{R}^d$ and apply a dependent thinning $T_n$, where the retention probability of a point depends on its local neighbour count within a radius $r_n$. While superpositions under independent thinning converge to Poisson processes, we show that under a critical geometric scaling $n v_d r_n^d \to τ\in (0,\infty)$, the local interactions are transformed in the limit into an inhomogeneous Poisson point process with modified intensity. We prove that the thinned sequence $T_n(S_n)$ converges to a Poisson process with a non-linearly modified intensity $\tildeλ(x) = λ(x)α(τλ(x))$.