Limiting behaviors of high dimensional stochastic spin ensembles

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