Using random graphs to sample repulsive Gibbs point processes with arbitrary-range potentials
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We study computational aspects of Gibbs point processes that are defined by a fugacity λ∈ℝ_≥ 0 and a repulsive symmetric pair potential ϕ on bounded regions 𝕍 of a Polish space, equipped with a volume measure ν. We introduce a new approximate sampler for such point processes and a new randomized approximation algorithm for their partition functions Ξ_𝕍(λ, ϕ). Our algorithms have running time polynomial in the volume ν(𝕍) for all fugacities λ < e/C_ϕ, where C_ϕ is the temperedness constant of ϕ. In contrast to previous results, our approach is not restricted to finite-range potentials. Our approach is based on mapping repulsive Gibbs point processes to hard-core models on a natural family of geometric random graphs. Previous discretizations based on hard-core models used deterministic graphs, which limited the results to hard-constraint potentials and box-shaped regions in Euclidean space. We overcome both limitations by randomization. Specifically, we define a distribution ζ^(n)_𝕍, ϕ on graphs of size n, such that the hard-core partition function of graphs from this distribution concentrates around Ξ_𝕍(λ, ϕ). We show this by deriving a corollary of the Efron-Stein inequality, which establishes concentration for a function f of independent random inputs, given the output of f only exhibits small relative changes when an input is altered. Our approximation algorithm follows from approximating the hard-core partition function of a random graph from ζ^(n)_𝕍, ϕ. Further, we derive a sampling algorithm using an approximate sampler for the hard-core model on a random graph from ζ^(n)_𝕍, ϕ and prove that its density is close to the desired point process via Rényi-Mönch theorem.
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