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Stability in lattice systems with local prohibitions. (English. Russian original) Zbl 0573.60062

Sel. Math. Sov. 4, 73-92 (1985); translation from Interacting Markov processes and their applications to biology, Pushchino 1979, 74-99 (1979).
Let d be a given dimension and \(X_ 0\) a finite set, and E be a countable collection of cylindrical subsets of \(X=X_ 0^{{\mathbb{Z}}^ d}\) satisfying three conditions:
(a) if a set C belongs to E, then all the translates of C also belong to E;
(b) if every set is defined to be equivalent to its translates, then E will break up into only a finite number of equivalence classes;
(c) all elements of E are nonempty and distinct from X.
Then (E,X) will be called a system (with local prohibitions), and a configuration \(x\in X\) is said to be consistent in (E,X) if x belongs to the intersection of all the elements of E. Let M denote the class of probability measures on X. The author defines a subclass \(M_{\epsilon}\subset M\) for any \(\epsilon\in [0,1]:\) \(\mu \in M_{\epsilon}\) if for any \(k\in {\mathbb{Z}}_+\) and for any k distinct elements \(C_ 1,...,C_ k\) of E \(\mu (C_ 1\cup C_ 2\cup...\cup C_ k)\geq 1-\epsilon^ k\). For any configuration x, let N(x) be the collection of configurations which differ from x only on a finite number of points and for a point \(\nu\), define \(N_{\nu}(x)=\{y\in N(x):y_{\nu}\neq x_{\nu}\}\). Then a consistent configuration is called stable in the system (E,X) if \[ \lim_{\epsilon \to 0}\sup_{\mu \in M_{\epsilon},\nu \in {\mathbb{Z}}^ d}\mu (N_{\nu}(x))=0. \] This paper provides sufficient conditions for the stability of configurations in the above sense. Unfortunately they cannot be applied to Gibbs distributions, because no interesting measures in \(M_{\epsilon}\) could be found if \(d>1\).
Reviewer: W.Wreszinski

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60B99 Probability theory on algebraic and topological structures
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