### Book chapter

## Спиновые модели с динамически изменяемой геометрией

### In book

In the first part of the paper we consider a "random flight" process in \(R^d\) and obtain the weak limits under different transformations of the Poissonian switching times. In the second part we construct diffusion approximations for this process and investigate their accuracy. To prove the weak convergence result we use the approach of Stroock and Varadhan (1979). We consider more general model which may be called "random walk over ellipsoids in \(R^d\)". For this model we establish the Edgeworth type expansion. The main tool in this part is the parametrix method (Konakov (2012), Konakov and Mammen (2009)).

We discuss conditions for unique ergodicity of a collective random walk on a continuous circle. Individual particles in this collective motion perform independent (and different in general) random walks conditioned by the assumption that the particles cannot overrun each other. Additionally to sufficient conditions for the unique ergodicity we discover a new and unexpected way for its violation due to excessively large local jumps. Necessary and sufficient conditions for the unique ergodicity of the deterministic version of this system are obtained as well. Technically our approach is based on the interlacing property of the spin function which describes states of pairs of particles in coupled processes under study.

We study properties of Markov chain Monte Carlo simulations of classical spin models with local updates. We derive analytic expressions for the mean value of the acceptance rate of single-spin-flip algorithms for the one-dimensional Ising model. We find that for the Metropolis algorithm the average acceptance rate is a linear function of energy. We further provide numerical results for the energy dependence of the average acceptance rate for the three- and four-state Potts model, and the XY model in one and two spatial dimensions. In all cases, the acceptance rate is an almost linear function of the energy in the critical region. The variance of the acceptance rate is studied as a function of the specific heat. While the specific heat develops a singularity in the vicinity of a phase transition, the variance of the acceptance rate stays finite.