Yuan Gao (UNC Chapel Hill), GMA & Visions Seminar

Title: Limiting dynamics of Metropolis-Hastings algorithm applied to n-vector spin model

Abstract: The n-vector spin model is a simple system of interacting spins on a crystalline lattice with a Hamiltonian given by the interaction of nearest neighbors. The corresponding Gibbs distribution could be sampled by a Metropolis-Hastings (M-H) algorithm. It is also natural to think one could model the spin dynamics by a system of over-damped Langevin-type equations. When the number of spins goes to infinity, the limiting Hamiltonian is in the form of the energy to give harmonic map heat flow. In the case n equals 3, the model is used to describe ferromagnetism. It is also shown that the harmonic map heat flow equation on the unit sphere has the same form as Landau-Lifshitz equation of the ferromagnetic spin chain.

We are interested in understanding the relationship between these models and will show the result for the n=3 case from the 1D torus to the unit sphere. In the limit of decreasing proposal size, we show that the M-H dynamics converge to a Langevin type stochastic differential equation. (Here, a Stratonovich interpretation is used due to the geometric constraint of the sphere for each spin.) Then we will show that as the lattice size gets larger and a suitable scaling is chosen for the temperature in the Gibbs distribution and M-H proposal size, the M-H dynamics converge to the deterministic Landau-Lifshitz equation, a PDE describing the time evolution of magnetism.