# Correlation Inequalities and Monotonicity Properties of the Ruelle Operator

## TL;DR

This paper establishes correlation inequalities and monotonicity properties for the Ruelle operator within Thermodynamic Formalism, extending classical results to low-regularity potentials and providing insights into phase transitions in statistical mechanics models.

## Contribution

It proves the FKG inequality for eigenmeasures of the Ruelle operator, extends monotonicity results to subshifts, and solves the eigenvalue problem for low-regularity potentials using involution kernels.

## Key findings

- FKG inequality holds for a class of eigenmeasures in Thermodynamic Formalism.
- Monotonicity properties of the Ruelle operator relate to phase transitions.
- Explicit upper bounds for the eigenvalue and pressure in Ising models with long-range interactions.

## Abstract

Let $X = \{1,-1\}^\mathbb{N}$ be the symbolic space endowed with the product order. A Borel probability measure $\mu$ over $X$ is said to satisfy the FKG inequality if for any pair of continuous increasing functions $f$ and $g$ we have $\mu(fg)-\mu(f)\mu(g)\geq 0$. In the first part of the paper we prove the validity of the FKG inequality on Thermodynamic Formalism setting for a class of eigenmeasures of the dual of the Ruelle operator, including several examples of interest in Statistical Mechanics. In addition to deducing this inequality in cases not covered by classical results about attractive specifications our proof has advantage of to be easily adapted for suitable subshifts. We review (and provide proofs in our setting) some classical results about the long-range Ising model on the lattice $\mathbb{N}$ and use them to deduce some monotonicity properties of the associated Ruelle operator and their relations with phase transitions. As is widely known, for some continuous potentials does not exists a positive continuous eigenfunction associated to the spectral radius of the Ruelle operator acting on $C(X)$. Here we employed some ideas related to the involution kernel in order to solve the main eigenvalue problem in a suitable sense - for a class of potentials having low regularity. From this we obtain an explicit tight upper bound for the main eigenvalue (consequently for the pressure) of the Ruelle operator associated to Ising models with $1/r^{2+\varepsilon}$ interaction energy. Extensions of the Ruelle operator to suitable Hilbert Spaces are considered and a theorem solving to the main eigenvalue problem (in a weak sense) is obtained by using the Lions-Lax-Milgram theorem.