We study the static as well as the glassy or dynamical transition in the meanfield p-state Potts glass. By numerical solution of the saddle point equations we investigate the static and the dynamical transition for all values of p in the nonperturbative regime p > 4. The static and dynamical Edwards-Anderson parameter increase with p logarithmically. This makes the glassy transition temperature lie very close to the static one. We compare the main predictions of the theory with the numerical simulations.cond-mat/9410093 ROM2F/94/45 1
We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain coarsening. The continuous time process, whose state space consists of assignments of +1 or −1 to each site in Z 2 , is the zero-temperature limit of the stochastic homogeneous Ising ferromagnet (with Glauber dynamics): the initial state is chosen uniformly at random and then each site, at rate one, polls its 4 neighbors and makes sure it agrees with the majority, or tosses a fair coin in case of a tie. Among the main results (almost sure, with respect to both the process and initial state) are: clusters (maximal domains of constant sign) are finite for times t < ∞, but the cluster of a fixed site diverges (in diameter) as t → ∞; each of the two constant states is (positive) recurrent. We also present other results and conjectures concerning positive and null recurrence and the role of absorbing states.Mathematics Subject Classification 2000: 60K35, 82C22, 82C20, 60J25.
This paper is devoted to the perfect simulation of a stationary process with an at most countable state space. The process is specified through a kernel, prescribing the probability of the next state conditional to the whole past history. We follow the seminal work of Comets, Fernández and Ferrari (2002), who gave sufficient conditions for the construction of a perfect simulation algorithm. We definebackward coalescence timesfor these kind of processes, which allow us to construct perfect simulation algorithms under weaker conditions than in Comets, Fernández and Ferrari (2002). We discuss how to construct backward coalescence times (i) by means ofinformation depths, taking into account some a priori knowledge about the histories that occur; and (ii) by identifying suitablecoalescing events.
This paper deals with the problem of perfect sampling from a Gibbs measure with infinite range interactions. We present some sufficient conditions for the extinction of processes which are like supermartingales when large values are taken. This result has deep consequences on perfect simulation, showing that local modifications on the interactions of a model do not affect the simulability. We also pose the question to optimize over a class of sequences of sets that influence the sufficient condition for the perfect simulation of the Gibbs measure. We completely solve this question both for the long range Ising models and for the spin models with finite range interactions
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