Matching with shift for one-dimensional Gibbs measures

TL;DR

This paper analyzes the maximal overlap in matching problems for Gibbsian sequences, revealing it scales logarithmically with sequence length and linking it to thermodynamic properties of the system.

Contribution

It provides a rigorous characterization of the maximal overlap behavior in Gibbsian sequences with shifts, connecting it to pressure and thermodynamic quantities.

Findings

Maximal overlap scales as c log n, with c explicitly determined.

Analysis applies to both equal and independent sequences.

Method uses first and second moment calculations of overlaps.

Abstract

We consider matching with shifts for Gibbsian sequences. We prove that the maximal overlap behaves as $c\log n$, where $c$ is explicitly identified in terms of the thermodynamic quantities (pressure) of the underlying potential. Our approach is based on the analysis of the first and second moment of the number of overlaps of a given size. We treat both the case of equal sequences (and nonzero shifts) and independent sequences.