Strong spatial mixing for repulsive point processes

TL;DR

This paper proves strong spatial mixing for certain repulsive Gibbs point processes under specific activity bounds, leading to new identities and efficient algorithms for sampling and approximating pressures.

Contribution

It establishes strong spatial mixing for repulsive Gibbs point processes with finite-range interactions and derives new analytic identities and efficient algorithms for sampling and pressure approximation.

Findings

Proves strong spatial mixing for activities below a specific threshold.

Derives identities for infinite volume and surface pressure.

Develops algorithms with O(N log N) mixing time for sampling.

Abstract

We prove that a Gibbs point process interacting via a finite-range, repulsive potential $\phi$ exhibits a strong spatial mixing property for activities $\lambda < e/\Delta_{\phi}$, where $\Delta_{\phi}$ is the potential-weighted connective constant of $\phi$, defined recently in [MP21]. Using this we derive several analytic and algorithmic consequences when $\lambda$ satisfies this bound: (1) We prove new identities for the infinite volume pressure and surface pressure of such a process (and in the case of the surface pressure establish its existence). (2) We prove that local block dynamics for sampling from the model on a box of volume $N$ in $\mathbb R^d$ mixes in time $O(N \log N)$, giving efficient randomized algorithms to approximate the partition function and approximately sample from these models. (3) We use the above identities and algorithms to give efficient approximation algorithms for the pressure and surface pressure.