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

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