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Lattice spin models in statistical physics are used to understand magnetism.Their Hamiltonians are a discrete form of a version of a Dirichlet energy,signifying a relationship to the Harmonic map heat flow equation. The Gibbsdistribution, defined with this Hamiltonian, is used in the Metropolis-Hastings(M-H) algorithm to generate dynamics tending towards an equilibrium state. Inthe limiting situation when the inverse temperature is large, we establish therelationship between the discrete M-H dynamics and the continuous Harmonic mapheat flow associated with the Hamiltonian. We show the convergence of the M-Hdynamics to the Harmonic map heat flow equation in two steps: First, with fixedlattice size and proper choice of proposal size in one M-H step, the M-Hdynamics acts as gradient descent and will be shown to converge to a system ofLangevin stochastic differential equations (SDE). Second, with proper scalingof the inverse temperature in the Gibbs distribution and taking the latticesize to infinity, it will be shown that this SDE system converges to thedeterministic Harmonic map heat flow equation. Our results are not unexpected,but show remarkable connections between the M-H steps and the SDE Stratonovichformulation, as well as reveal trajectory-wise out of equilibrium dynamics tobe related to a canonical PDE system with geometric constraints.

Limiting Behaviors of High Dimensional Stochastic Spin Ensembles