Limiting Behaviors of High Dimensional Stochastic Spin Ensembles

TL;DR

This paper investigates the limiting behavior of high-dimensional stochastic spin ensembles, demonstrating how Metropolis-Hastings dynamics converge to harmonic map heat flow through a series of scaling limits and stochastic differential equations.

Contribution

It establishes the connection between M-H dynamics and harmonic map heat flow in high dimensions, including convergence to PDEs and SDEs under specific scalings.

Findings

M-H dynamics act as gradient descent with fixed lattice size

Convergence of M-H to Langevin SDEs as lattice size is fixed

SDE system converges to harmonic map heat flow as temperature scales and lattice size grow

Abstract

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 Gibbs distribution, defined with this Hamiltonian, is used in the Metropolis-Hastings (M-H) algorithm to generate dynamics tending towards an equilibrium state. In the limiting situation when the inverse temperature is large, we establish the relationship between the discrete M-H dynamics and the continuous Harmonic map heat flow associated with the Hamiltonian. We show the convergence of the M-H dynamics to the Harmonic map heat flow equation in two steps: First, with fixed lattice size and proper choice of proposal size in one M-H step, the M-H dynamics acts as gradient descent and will be shown to converge to a system of Langevin stochastic differential equations (SDE). Second, with proper scaling of the inverse temperature in the Gibbs distribution and taking the lattice size to infinity, it will be shown that this SDE system converges to the deterministic Harmonic map heat flow equation. Our results are not unexpected, but show remarkable connections between the M-H steps and the SDE Stratonovich formulation, as well as reveal trajectory-wise out of equilibrium dynamics to be related to a canonical PDE system with geometric constraints.