Cylinder Distribution of Thermodynamic Limit on Bethe Lattices

Abstract

We study the thermodynamic limit τ on Be the lattices when L_n → L for any finite spin set S. We show the cylinder values $\tau \left\{ {\sigma \in \Omega :\,{\sigma_{{{L_k}}}} = r\left| k \right|} \right\}$ are determined by vectors $\overline \psi = T{*^2}\overline \psi \overline \psi L = \mathbb{Z}$ which verify a period 2-equation: $\overline \psi L = \mathbb{Z}$. In the case q = 2 i.e. $L = \mathbb{Z}$ we arrive to a Perron — Frobenius equation, then to a unique distribution τ.