# Limiting behaviors of high dimensional stochastic spin ensembles

@article{Gao2018LimitingBO, title={Limiting behaviors of high dimensional stochastic spin ensembles}, author={Yuan Gao and Kay L Kirkpatrick and Jeremy Louis Marzuola and Jonathan C. Mattingly and Katherine A. Newhall}, journal={Communications in Mathematical Sciences}, year={2018} }

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…

## References

SHOWING 1-10 OF 51 REFERENCES

### Optimal scaling for the transient phase of Metropolis Hastings algorithms: The longtime behavior

- Mathematics
- 2014

We consider the Random Walk Metropolis algorithm on $\R^n$ with Gaussian proposals, and when the target probability measure is the $n$-fold product of a one dimensional law. It is well-known (see…

### Stochastic Ferromagnetism: Analysis and Numerics

- Mathematics
- 2013

This monograph examines magnetization dynamics at elevated temperatures which can be described by the stochastic Landau-Lifshitz-Gilbert equation (SLLG) and proposes an implementable space-time discretization using random walks to construct a weak martingale solution of the corresponding stochastically partial differential equation which describes the magnetization process of infinite spin ensembles.

### Optimal scaling for the transient phase of the random walk Metropolis algorithm: The mean-field limit

- Mathematics
- 2015

We consider the random walk Metropolis algorithm on $\R^n$ with Gaussian proposals, and when the target probability is the $n$-fold product of a one dimensional law. In the limit $n \to \infty$, it…

### Struwe-like solutions for the Stochastic Harmonic Map flow

- Mathematics
- 2016

We give new results on the well-posedness of the two-dimensional Stochastic Harmonic Map flow, whose study is motivated by the Landau–Lifshitz–Gilbert model for thermal fluctuations in…

### Magnetic Elements at Finite Temperature and Large Deviation Theory

- PhysicsJ. Nonlinear Sci.
- 2005

It is shown how large deviation theory can be used to explain some properties of the astroids, like their shrinking and sharpening as the number of applied pulses is increased, and some issues and open questions regarding spatially nonuniform magnetization are discussed.

### Diffusion limits of the random walk metropolis algorithm in high dimensions

- Mathematics
- 2012

Diffusion limits of MCMC methods in high dimensions provide a useful theoretical tool for studying computational complexity. In particular, they lead directly to precise estimates of the number of…

### Finite-time singularity of the stochastic harmonic map flow

- MathematicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2019

We investigate the influence of an infinite dimensional Gaussian noise on the bubbling phenomenon for the stochastic harmonic map flow $u(t,\cdot ):\mathbb{D}^2\to\mathbb{S}^2$, from the…

### Weak convergence and optimal scaling of random walk Metropolis algorithms

- Mathematics
- 1997

This paper considers the problem of scaling the proposal distribution of a multidimensional random walk Metropolis algorithm in order to maximize the efficiency of the algorithm. The main result is a…

### Asymptotics of the Mean-Field Heisenberg Model

- Mathematics
- 2013

We consider the mean-field classical Heisenberg model and obtain detailed information about the total spin of the system by studying the model on a complete graph and sending the number of vertices…