Markov Renewal Theory for Transfer Operators and Point Processes on the Line

TL;DR

This paper establishes exponential decay of correlations in 1D stationary point processes with Markovian spacings, using a Markov renewal theorem and applying it to models in statistical mechanics.

Contribution

It introduces a Markov renewal theorem with exponential convergence and applies it to Gibbs point processes and harmonic chains, advancing understanding of correlation decay.

Findings

Proves exponential decay of pair correlations under specified conditions.

Develops a Markov renewal theorem with exponential convergence rate.

Applies results to models in statistical mechanics, including Gibbs processes and harmonic chains.

Abstract

We prove exponential decay of pair correlations for 1D stationary point processes when spacings satisfy a Markov condition, geometric ergodicity, and a condition on exponential moments. The conditions are phrased for stationary sequences of spacings (intervals between consecutive points) whose law comes from the Palm distribution of the point process. The key technical ingredient is a Markov renewal theorem with exponential convergence rate. The proofs combine classical regeneration techniques with the notion of geometric ergodicity for Markov chains with general state space. We apply the result to two models from statistical mechanics: (1) Gibbs point processes with a hard-core, finite-range pair potentials and (2) a harmonic chain of atoms, related to an autoregressive Gaussian process.