The dynamic behavior of cluster algorithms is analyzed in the classical mean-field limit. Rigorous analytical results below T c establish that the dynamic exponent has the value z SW ϭ1 for the Swendsen-Wang algorithm and z W ϭ0 for the Wolff algorithm. An efficient Monte Carlo implementation is introduced, adapted for using these algorithms for fully connected graphs. Extensive simulations both above and below T c demonstrate scaling and evaluate the finite-size scaling function by means of a rather impressive collapse of the data.

We study the spacial and temporal multiscale properties of complex systems. We present accelerated algorithms for dilute spin glasses and display explicitly their relation to the e ective dynamics of speci c collective degrees of freedom (macros).We discuss the di culties in applying multiscale-cluster algorithms (MCA) to general frustrated systems: MCA does not succeed to break the system into clusters. We relate these di culties to rigorous negative results in systems with an ultrametric space of ground states: the tunneling between vacua cannot be expressed into an algorithm acting upon independent macros. II. MULTISCALE-CLUSTER ALGORITHMS (MCA)An example of multiscale e ective dynamics and its related multiscale slowing down is the critical slowing down (CSD) at second order phase transitions. There, the relaxation time diverges with the systems size L as: L z Email: nathanp@vms.huji.ac.il sorin@vms.huji.ac.il 1 see for example of the projection by PTMG of exact lattice Atyiah-Singer modes 8].

In simulations of some infinite range spin glass systems with finite connectivity, it is found that for any resonable computational time, the saturated energy per spin that is achieved by a cluster algorithm is lowered in comparison to that achieved by Metropolis dynamics. The gap between the average energies obtained from these two dynamics is robust with respect to variations of the annealing schedule. For some probability distribution of the interactions the ground state energy is calculated analytically within the replica symmetry assumption and is found to be saturated by a cluster algorithm.

We present a new measure of the Dynamical Critical behavior: the "Multi-scale Dynamical Exponent (MDE)", z md . Using Dynamical RG concepts we study the relaxation times, τ Λ , of a family of space scales defined by Λ = 1 k . Assuming dynamical universality we argue, and verify numerically, that z md has the same value as the usually defined z. We measure z md in the 2D Ising model with the Metropolis and cluster dynamics and find z md met = 2.1 ± 0.1 and z md wolff = 0 ± 0.15, respectively. We note that in our approach z md is measured using a single temperature and a single lattice size. In addition, in the Metropolis case we present a new method which helps to overcome critical slowing down in the dynamical measurements themselves. * Emails: nathanp@vms