Torpid mixing of Swendsen-Wang dynamics on Z^d for low-temperature non-ferromagnetic disordered systems.

We study the Swendsen-Wang (SW) dynamics for disordered non-ferromagnetic Ising models on cubic subsets of the lattice Z d . The SW dynamics is a reversible Markov chain having the Gibbs Ising measure as its stationary one. It was introduced for the simulation on a computer of the Ising or Potts model to obtain a more rapid simulation than the standard Monte Carlo methods (see references in the paper). The main result of the paper is that SW dynamics for disordered non-ferromagnetic Ising systems, on Z d , such as spin glass systems, has a slow convergence (it is torpid) to the equilibrium measure for small values of the temperature. More precisely the author proves that the spectral gap of the SW dynamics decreases to zero when the side length of the box goes to infinity if the inverse of temperature is large enough. The results are given in any dimension for all the Ising models where the interactions are random variables such that 0 belongs to the interior of the support (examples are the uniform measure on an interval that includes the origin or the Gaussian measures). The results are proven directly using the structure of the SW dynamics; arguments on the equilibrium measure or the phase transition of the Ising models are not used.