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

Abstract

This paper deals with the problem of perfect sampling from a Gibbs measure with infinite range interactions. We present some sufficient conditions for the extinction of processes which are like supermartingales when large values are taken. This result has deep consequences on perfect simulation, showing that local modifications on the interactions of a model do not affect the simulability. We also pose the question to optimize over a class of sequences of sets that influence the sufficient condition for the perfect simulation of the Gibbs measure. We completely solve this question both for the long range Ising models and for the spin models with finite range interactions

Spiking neurons : interacting Hawkes processes, mean field limits and oscillations

“…Notice how in the above theorem, condition (10) depends on the specific choice of growing neighborhoods (V i (k)) k≥0 . We refer the reader to De Santis and Lissandrelli (2012) [15] for a study (in the framework of stochastic chains having infinite memory) how to optimize over the choice of the sequence of growing neighborhoods in order to obtain the weakest sufficient condition (10).…”

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

“…• P (2) is the tessellation obtained by adding a vertex v σ in the barycentre of all 2-dimensional faces σ ∈ P 2 which are not a simplex and adding the edges joining v σ with the vertices of σ . Finally we set A (2) = A ∪ {v σ : σ ∩ A = ∅}.…”

“…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.…”

Stochastic Ising model with flipping sets of spins and fast decreasing temperature

“…[6,7,9,20]) or high temperature (see e.g. [6,8,19,21]). In particular, in [6], in the framework of random interactions, it is shown a different approach to the equilibrium measure in relation to the temperature.…”