Lyapunov operator $\mathcal L$ with degenerate kernel and Gibbs measures

TL;DR

This paper explores the relationship between multiple interactions in spin models on a Cayley tree and Lyapunov integral equations, analyzing fixed points that correspond to translation-invariant Gibbs measures.

Contribution

It establishes a connection between complex spin interactions and Lyapunov operators with degenerate kernels, providing insights into Gibbs measures on Cayley trees.

Findings

Fixed points of Lyapunov operator correspond to Gibbs measures.

Analysis of degenerate kernels in Lyapunov operators.

Connection between multiple interactions and integral equations.

Abstract

In this paper we'll give a connection between four competing interactions (external field, nearest neighbor, second neighbors and triples of neighbors) of models with uncountable (i.e. $[0,1]$) set of spin values on the Cayley tree of order two and Lyapunov integral equation. Also we'll study fixed points of Lyapunov operator with degenerate kernel which each fixed point of the operator is correspond to a {\it translation-invariant} Gibbs measure.