Communications in Mathematical Sciences

Volume 19 (2021)

Number 2

Limiting behaviors of high dimensional stochastic spin ensembles

Pages: 453 – 494

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n2.a7

Authors

Yuan Gao (Department of Mathematics, University of North Carolina at Chapel Hill, N.C., U.S.A.)

Kay Kirkpatrick (Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Il., U.S.A.)

Jeremy Marzuola (Department of Mathematics, University of North Carolina at Chapel Hill, N.C., U.S.A.)

Jonathan Mattingly (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Katherine A. Newhall (Department of Mathematics, University of North Carolina at Chapel Hill, N.C., U.S.A.)

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.

Keywords

Metropolis-Hastings algorithm, harmonic map heat flow, Langevin equation, Stratonovich SDE

2010 Mathematics Subject Classification

58J65, 60H10, 60J10, 60J60, 65C05, 82C05

Full Text (PDF format)

Received 8 April 2019

Accepted 14 September 2020

Published 12 April 2021