Exponential small-count tail bounds for Gibbs point processes (k ≥ 2)

Establish exponentially fast-decreasing upper bounds for small-count probabilities of stationary Gibbs point processes with bounded, translation-covariant Papangelou intensities as in Proposition p:voidGibbs, namely prove the existence of a function δ1(t) with exponential decay such that P(Φ(B_t) ≤ k−1) ≤ δ1(t) for all t > 0 and all integers k ≥ 2, where B_t denotes the Euclidean ball of radius t centered at the origin. This would verify condition (e:k) for these Gibbs point processes and thereby enable the hyperuniformity conclusions for the k-th–nearest-neighbor Voronoi-weighted measures developed in Example e:weightedVoronoi.

Background

In Example e:weightedVoronoi, the authors construct a Voronoi-weighted random measure by assigning to each point of a stationary point process Γ the volume of its k-th–nearest-neighbor Voronoi cell. Hyperuniformity of this random measure follows from Theorem tgen_mix_pert_ppLeb provided that Γ satisfies a small-count tail bound of the form P(Γ(B_t) ≤ k−1) ≤ δ1(t) with δ1 exponentially fast decreasing.

They prove such bounds in several cases (e.g., for Poisson and certain determinantal/permanental processes, and derive related void probability bounds for specific Gibbs processes in Proposition p:voidGibbs). However, for Gibbs processes covered by Proposition p:voidGibbs, they only establish (exponential) void probabilities (k = 1) and state that extending these bounds to P(Φ(B_t) ≤ k−1) for k ≥ 2 remains unproved.

Verifying these k ≥ 2 bounds would close the gap needed to apply the general transport-mixing theorems and conclude hyperuniformity for Voronoi-weighted measures generated from a broad class of Gibbs point processes.