Perfect Simulation of Infinite Range Gibbs Measures and Coupling with Their Finite Range Approximations

Abstract

In this paper we address the questions of perfectly sampling a Gibbs measure with infinite range interactions and of perfectly sampling the measure together with its finite range approximations. We solve these questions by introducing a perfect simulation algorithm for the measure and for the coupled measures. The algorithm works for general Gibbsian interaction under requirements on the tails of the interaction. As a consequence we obtain an upper bound for the error we make when sampling from a finite range approximation instead of the true infinite range measure.

Developments in Perfect Simulation of Gibbs Measures Through a New Result for the Extinction of Galton-Watson-Like Processes

“…Note that in literature more general definitions of interactions are considered but in our paper we will only use this more restrictive definition, as done also in [3]. For brevity of notation set χ B (σ ) = v∈B σ (v) for any B Z d and σ ∈ S. A probability measure π on (S, S) is said to be a Gibbs measure relative to the interaction J ∈ J if for all v ∈ Z d and for any ζ ∈ S π σ (v) = ζ(v)|σ (u) = ζ(u) ∀u = v = 1 1 + exp(−2 B: v∈B (J B χ B (σ ))) a.s. (2) which are called local specifications.…”

“…Now in [3] there is a construction of an auxiliary process that links the Glauber dynamics with the perfect sampling algorithm through decomposition (7).…”

Perfect simulation for the infinite random cluster model, Ising and Potts models at low or high temperature

“…One area of research concerns the Markov fields (see [3,12]); a second one concerns the processes with infinite memory (see [1,4,5,7]). Recently, these two areas of research have been in some sense unified by studying Gibbs measures with infinite interaction range (see [2,8]). Our paper is included in the latter context.…”

Modelagem estocástica de uma população de neurônios

“…Baseado no procedimento rascunho para trás, Galves et al (2010) abordaram o problema da simulação perfeita para a medida de Gibbs com interações de alcance infinito. Em , eles consideram um sistema de partícula em Z d com espaço de estados em R e interações de alcance infinito.…”