Transformations of one-dimensional Gibbs measures with infinite range interaction

TL;DR

This paper investigates how one-dimensional Gibbs measures with infinite-range interactions behave under transformations, showing they preserve their Gibbsian property and analyzing how their potentials decay after transformation.

Contribution

It proves the preservation of Gibbsianness under transformations and provides quantitative decay estimates for transformed potentials with various decay rates.

Findings

Gibbsianness is conserved under the studied transformations.

Quantitative decay estimates are provided for transformed potentials.

Results apply to both exponential and power-law decaying potentials.

Abstract

We study single-site stochastic and deterministic transforma- tions of one-dimensional Gibbs measures in the uniqueness regime with infinite-range interactions. We prove conservation of Gibbsianness and give quantitative estimates on the decay of the transformed potential. As examples, we consider exponentially decaying potentials, and potentials decaying as a power-law.