Uniqueness of Gibbs Measure for Models With Uncountable Set of Spin Values on a Cayley Tree

We consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. It is known that the "splitting Gibbs measures" of the model can be described by solutions of a nonlinear integral equation. For arbitrary $k\geq 2$ we find a sufficient condition under which the integral equation has unique solution, hence under the condition the corresponding model has unique splitting Gibbs measure.Comment: 13 page

Prefixes and the entropy rate for long-range sources

The asymptotic a.s.-relation H = limn→∞ n log n ÷ Σi=1n Lin (X) is derived for any finite-valued stationary ergodic process X = (Xn, n ∈ Z) that satisfies a Doeblin-type condition: there exists r ≥ 1 such that essxinf P(Xn+r | x-∞,n) ≥ α > 0. Here, H is the entropy rate of the process X, and Lin(X) is the length of a shortest prefix in X which is initiated at time i and is not repeated among the prefixes initiated at times j, 1 ≤ i ≠ j ≤ n. The validity of this limiting result was established by Shields in 1989 for i.i.d. processes and also for irreducible aperiodic Markov chains. Under our new condition, we prove that this holds for a wider class of processes, that may have infinite memory