On Equilibrium Distribution of a Reversible Growth Model

Abstract

We study a probabilistic model of interacting spins indexed by elements of a finite subset of the d-dimensional integer lattice, d ≥ 1. Conditions of time reversibility are examined. It is shown that the model equilibrium distribution converges to a limit distribution as the indexing set expands to the whole lattice. The occupied site percolation problem is solved for the limit distribution. Two models with similar dynamics are also discussed.

Long Term Behaviour of Locally Interacting Birth-and-Death Processes

“…If we assume that spins are bounded and consider the same birth-and-death dynamics then we will get a finite ergodic Markov chain whose equilibrium distribution is a Gibbs measure (see Remark 1). A particular case of the model with bounded spins, where α = β, Λ ⊂ Z d , was studied in [14]. For instance, if a spin takes values 0 and 1 only, and, in addition, α = β > 0, then we obtain a finite Markov chain whose equilibrium distribution is a Gibbs measure on {0, 1} Λ which is equivalent to a particular case of the famous Ising model on {−1, 1} Λ .…”

“…For instance, if a spin takes values 0 and 1 only, and, in addition, α = β > 0, then we obtain a finite Markov chain whose equilibrium distribution is a Gibbs measure on {0, 1} Λ which is equivalent to a particular case of the famous Ising model on {−1, 1} Λ . The main goal in [14] was to study the asymptotic behaviour of the stationary distribution as Λ ↑ Z d . In general, the asymptotic behaviour of such equilibrium distributions in thermodynamic limit, i.e.…”

A note on scaling limits for truncated birth-and-death processes with interaction

“…Models of interacting birth-and-death processes on integers provide a flexible mathematical framework for modelling such systems (e.g. see [9], [10], [12], [13] and references therein) that appear in biology, physics, queueing and other applications. Frequently, there are natural limitations on the system size (e.g.…”

“…A variant of this Markov chain was considered in [9], where transition rates were specified by interaction matrices A b = A = (α xy ) x,y∈Λ , such that α xy ≡ const, and A d ≡ 0, and graph Λ was a d-dimensional lattice cube. It was shown that a stationary distribution of the Markov chain converges to a Gibbs measure, as Λ expands to the whole lattice, and an occupied site percolation problem was solved for the limit distribution.…”

“…The long term behaviour of the Markov chain with non-negative and unbounded components (formally obtained by setting l = 0 and r = ∞) was studied in [10]. The transition rates in [10] were specified by interaction matrices A d ≡ 0 (as in [9]) and A b = αE + βI Λ , where α, β ∈ R, I Λ is the incidence matrix of graph Λ and E is the unit matrix. The main goal in [10] was to determine how the long term behaviour of the Markov chain depends on both the transition parameters and the structure of the underlying graph.…”