A law of thin processes with neighbour-count thinning

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 $n{v}_d{r}_n^d\to \tau \in (0,\mathrm{\infty })$, 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 $\tilde{\lambda }(x)=\lambda (x)\alpha (\tau \lambda (x))$.