A combinatorial proof of tree decay of semi-invariants

TL;DR

This paper presents a purely combinatorial proof for the exponential decay of semi-invariants in finite range Gibbs fields, extending applicability to disordered systems where traditional analyticity methods fail.

Contribution

It introduces a combinatorial approach to prove tree decay of semi-invariants, applicable beyond analyticity assumptions, including disordered systems in Griffiths' phase.

Findings

Proves exponential tree decay of semi-invariants using combinatorial methods.

Applicable to disordered systems where analyticity-based proofs do not hold.

Provides a new proof technique extending the understanding of Gibbs fields.

Abstract

We consider finite range Gibbs fields and provide a purely combinatorial proof of the exponential tree decay of semi--invariants, supposing that the logarithm of the partition function can be expressed as a sum of suitable local functions of the boundary conditions. This hypothesis holds for completely analytical Gibbs fields; in this context the tree decay of semi--invariants has been proven via analyticity arguments. However the combinatorial proof given here can be applied also to the more complicated case of disordered systems in the so called Griffiths' phase when analyticity arguments fail.