Critical attractive spin systems
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 paper is the following statement: For every \(r < \infty\) in discrete and continuous time the set \(\{(\beta, \delta) : \delta (N_ r') > 0, \xi^{\{0\}}\) is viable\} is open in the space of parameters \((\beta, \delta)\) corresponding to \(r\)-range attractive systems.
scientific article; zbMATH DE number 731979