Convergence to equilibrium for finite Markov processes, with application to the Random Energy Model

We estimate the distance in total variation between the law of a finite state Markov process at time t, starting from a given initial measure, and its unique invariant measure. We derive upper bounds for the time to reach the equilibrium. As an example of application we consider a special case of finite state Markov process in random environment: the Metropolis dynamics of the Random Energy Model. We also study the process of the environment as seen from the process

One-dimensional random field Kac's model: weak large deviations principle

We prove a quenched weak large deviations principle for the Gibbs measures of a Random Field Kac Model (RFKM) in one dimension. The external random magnetic field is given by symmetrically distributed Bernoulli random variables. The results are valid for values of the temperature, $\beta^{-1}$, and magnitude, $\theta$, of the field in the region where the free energy of the corresponding random Curie Weiss model has only two absolute minima $m_\beta$ and $Tm_\beta$. We give an explicit representation of the rate functional which is a positive random functional determined by two distinct contributions. One is related to the free energy cost ${\cal F}^*$ to undergo a phase change (the surface tension). The ${\cal F}^*$ is the cost of one single phase change and depends on the temperature and magnitude of the field. The other is a bulk contribution due to the presence of the random magnetic field. We characterize the minimizers of this random functional. We show that they are step functions taking values $m_\beta$ and $Tm_\beta$. The points of discontinuity are described by a stationary renewal process related to the $h-$extrema for a bilateral Brownian motion studied by Neveu and Pitman, where $h$ in our context is a suitable constant depending on the temperature and on magnitude of the random field. As an outcome we have a complete characterization of the typical profiles of RFKM (the ground states) which was initiated in [14] and extended in [16]

Distribution of overlap profiles in the one-dimensional Kac-Hopfield model

We study a one-dimensional version of the Hopfield model with long, but finite range interactions below the critical temperature. In the thermodynamic limit we obtain large deviation estimates for the distribution of the ``local'' overlaps, the range of the interaction, $\gamma^{-1}$, being the large parameter. We show in particular that the local overlaps in a typical Gibbs configuration are constant and equal to one of the mean-field equilibrium values on a scale o(\g^{-2}). We also give estimates on the size of typical ``jumps''. i.e. the regions where transitions from one equilibrium value to another take place. Contrary to the situation in the ferromagnetic Kac-model, the structure of the profiles is found to be governed by the quenched disorder rather than by entropy.Comment: 64pp, Plain Te

Correlation functions for the 2D random bonds Potts Models

We study the spin-spin and energy-energy correlation functions for the 2D Ising and 3-states Potts model with random bonds at the critical point. The procedure employed is the renormalisation group approach of the perturbation series around the conformal field theories representing the pure models. For the Ising model, we obtain a crossover in the amplitude for the correlation functions which doesn't change the critical exponent. For the $3$-state Potts model, we found a shift in the critical exponent produced by randomness. A comparison with numerical data is discussed briefly.Comment: To appear in the Proccedings of the Trieste Conference on Recent Developments in Statistical Mechanics and Quantum Field Theory, April 1995, 9 pages, latex, no figures, espcrc2.st

One-dimensional random field Kac's model: localization of the phases

We study the typical profiles of a one dimensional random field Kac model, for values of the temperature and magnitude of the field in the region of the two absolute minima for the free energy of the corresponding random field Curie Weiss model. We show that, for a set of realizations of the random field of overwhelming probability, the localization of the two phases corresponding to the previous minima is completely determined. Namely, we are able to construct random intervals tagged with a sign, where typically, with respect to the infinite volume Gibbs measure, the profile is rigid and takes, according to the sign, one of the two values corresponding to the previous minima. Moreover, we characterize the transition from one phase to the other

Critical interfaces in the random-bond Potts model

We study geometrical properties of interfaces in the random-temperature q-states Potts model as an example of a conformal field theory weakly perturbed by quenched disorder. Using conformal perturbation theory in q-2 we compute the fractal dimension of Fortuin Kasteleyn domain walls. We also compute it numerically both via the Wolff cluster algorithm for q=3 and via transfer-matrix evaluations. We obtain numerical results for the fractal dimension of spin cluster interfaces for q=3. These are found numerically consistent with the duality kappa(spin) * kappa(FK)= 16 as expressed in putative SLE parameters.Comment: 4 page

Randomly coupled minimal models

Using 1-loop renormalisation group equations, we analyze the effect of randomness on multi-critical unitary minimal conformal models. We study the case of two randomly coupled $M_p$ models and found that they flow in two decoupled $M_{p-1}$ models, in the infra-red limit. This result is then extend to the case with $M$ randomly coupled $M_p$ models, which will flow toward $M$ decoupled $M_{p-1}$.Comment: 12 pages, latex, 1 eps figures; new results adde