Renormalization-group at criticality and complete analyticity of constrained models: a numerical study

TL;DR

This study investigates the Gibbsian nature of the renormalized measure for the 2D Ising model at criticality using numerical methods, confirming the validity of the Dobrushin-Shlosman Uniqueness condition for constrained models.

Contribution

It provides numerical evidence that the Dobrushin-Shlosman Uniqueness condition holds for constrained models at criticality, supporting the Gibbsian property of the renormalized measure.

Findings

DSU condition verified for large volumes in constrained models

Numerical evidence supports Gibbsian nature of renormalized measure

Monte Carlo algorithm effectively estimates Vasserstein distance

Abstract

We study the majority rule transformation applied to the Gibbs measure for the 2--D Ising model at the critical point. The aim is to show that the renormalized hamiltonian is well defined in the sense that the renormalized measure is Gibbsian. We analyze the validity of Dobrushin-Shlosman Uniqueness (DSU) finite-size condition for the "constrained models" corresponding to different configurations of the "image" system. It is known that DSU implies, in our 2--D case, complete analyticity from which, as it has been recently shown by Haller and Kennedy, Gibbsianness follows. We introduce a Monte Carlo algorithm to compute an upper bound to Vasserstein distance (appearing in DSU) between finite volume Gibbs measures with different boundary conditions. We get strong numerical evidence that indeed DSU condition is verified for a large enough volume $V$ for all constrained models.