Critical attractive spin systems
type: Item
instance of: scholarly article
title: Critical attractive spin systems
zbMATH Open document ID: 0819.60094
author: Carol Bezuidenhout
author: Lawrence F. Gray
published in: The Annals of Probability
publication date: 1995-08-21T00:00:00Z
review text: Markovian random processes \(\{\xi_ t\}\) called attractive spin systems are investigated. The state space of the processes is the collection of subsets \(A\) in \(\mathbb{Z}^ d\), \(d = 2,3, \dots\) is the dimensionality, i.e. the random trajectories \(\xi_ t\) receive the values in \(2^{ \mathbb{Z}^ d}\). The time \(t\) is considered both discrete and continuous \((t \in \mathbb{R}\) or \(\mathbb{Z})\). Let \(N_ r = \{y \in \mathbb{Z}^ d : \max_ i | y_ i | \leq r\}\), \(r = 1,2, \dots, N_ r' = N_ r \backslash \{0\}\) and \(\beta = (\beta (\eta) : \eta \subseteq N_ r')\), \(\delta = (\delta (\eta) : \eta \subseteq N_ r')\) where \(\beta (\eta) \geq 0\), \(\delta (\eta) \geq 0\). The probabilistic measure \(\mathbb{P}_{\beta, \delta}\) on \(\Omega\) for each process of the process class called transition invariant finite-range spin systems is defined on the basis of the fixed function pair \((\beta, \delta)\). The vector is called attractive if \(\beta (\eta) \leq \beta (\eta')\) and \(\delta (\eta) \geq \delta (\eta')\) whenever \(\eta \leq \eta'\). The measure \(\mathbb{P}_{\beta, \delta}\) (the dynamics of spin systems) is defined by means of the conditional probabilities which, for example in the discrete time case, have the form \[ \mathbb{P}_{\beta, \delta} \bigl( x \notin \xi_{t+1} \mid (\xi_ t - x) \cap N_ r = \eta \cup \{x\} \bigr) = \delta (\eta), \] \[ \mathbb{P}_{\beta, \delta} \bigl( x \in \xi_{t+1} \mid (\xi_ t - x) \cap N_ r = \eta \bigr) = \delta (\eta) \] under the condition \(\beta (\eta) + \delta (\eta) \leq 1\). Let \(\xi_ t^ A\) be the random trajectory with initial condition \(\xi_ 0^ A = A \subseteq \mathbb{Z}^ d\). The process \(\xi^ A\) survives for subset \(A\) and for given sample trajectory \(\omega \in \Omega\) if \(\xi_ t^ A (\omega) \neq \emptyset\) for all \(t \geq 0\). If the survival probability \(\mathbb{P}_{\beta, \delta} (\xi^ A\) survives) is positive, then the process \(\xi^ A\) is called the viable one. The main result of the …
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 731979
zbMATH Keywords: Markovian process
zbMATH Keywords: interacting particles
zbMATH Keywords: attractive spin systems
zbMATH Keywords: critical phenomena
Wikidata QID: Q105584350
reviewed by: Yu. P. Virchenko
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988599
OpenAlex ID: W2038680818
DOI: 10.1214/AOP/1176988599
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description: scientific article; zbMATH DE number 731979