Torpid Mixing of Swendsen - Wang Dynamics on Z^d for Low-temperature Non-ferromagnetic Disordered Systems,

#### E. De Santis

2003, v.9, №1, 35-57

ABSTRACT

We study the Swendsen - Wang dynamics for disordered non-ferromagnetic Ising models on cubic subsets of the hypercubic lattice Z^d and we show that for all small values of the temperature parameter $\beta^{-1}$ the dynamics has a slow relaxation to equilibrium (it is torpid). Looking into this dynamics from the point of view of the Markov chains theory we can prove that the spectral radius goes to one when the size of the system goes to infinity. This means that, if we want to use the Swendsen - Wang dynamics for a computer simulation, we have a slow convergence to the stationary measure at low temperature. Also it is a good example of a non-local dynamics that relaxes slowly to the equilibrium measure.

Keywords: relaxation speed,spin-glass,Swendsen - Wang algorithm,spectral gap