Bootstrap, Markov Chain Monte Carlo, and LP/SDP hierarchy for the lattice Ising model

• Published in: The journal of high energy physics Vol. 2023; no. 11; pp. 47 - 26
• Main Authors: Cho, Minjae, Sun, Xin
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2023; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms and Theoretical Developments; Asymptotic methods; Classical and Quantum Gravitation; Convergence; Elementary Particles; High energy physics; Invariants; Ising model; Lattice Quantum Field Theory; Linear programming; Markov analysis; Markov chains; Mathematical models; Other Lattice Field Theories; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Semidefinite programming; Statistical analysis; Stochastic Processes; String Theory; Lattice Quantum Field Theory; Algorithms and Theoretical Developments; Stochastic Processes; Other Lattice Field Theories
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP11(2023)047

A bstract 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.