Bootstrap, Markov Chain Monte Carlo, and LP/SDP Hierarchy for the Lattice Ising Model

TL;DR

This paper develops a bootstrap approach for the invariant measure of the stochastic Ising model, using LP and SDP hierarchies to establish convergence and relate different mathematical formulations of the model.

Contribution

It introduces a linear programming hierarchy for the invariant measure and connects it with a semidefinite programming hierarchy for the Gibbs measure, demonstrating convergence and relationships.

Findings

LP hierarchy converges explicitly to the invariant measure

Relation established between LP hierarchy and SDP hierarchy for Gibbs measure

Explicit construction of invariant measure from moments

Abstract

Bootstrap is an idea that imposing consistency conditions on a physical system may lead to rigorous and nontrivial statements about its physical observables. In this work, we discuss the bootstrap problem for the invariant measure of the stochastic Ising model defined as a Markov chain where probability bounds and invariance equations are imposed. It is described by a linear programming (LP) hierarchy whose asymptotic convergence is shown by explicitly constructing the invariant measure from the convergent sequence of moments. We also discuss the relation between the LP hierarchy for the invariant measure and a recently introduced semidefinite programming (SDP) hierarchy for the Gibbs measure of the statistical Ising model based on reflection positivity and spin-flip equations.