Anomalous diffusion in random media of any dimensionality

We show, through physical arguments and a renormalization group analysis, that in the presence of long-range correlated random forces, diffusions is anomalous in any dimension. We obtain in general surdiffusive behaviours, except when the random force is the gradient of a potential. In this last situation, with either short or long-range correlations, a subdiffusive behaviour with a disorder dependent exponent is found in the upper critical case (D = 2 for short-range correlations). This is because the β-function vanishes, which is explicitly proven at all orders of the perturbation theory. Apart from this case, a potential force is expected to lead to logarithmic diffusion (1/f noise), as suggested by simple arguments

Non-trivial fixed point structure of the two-dimensional +-J 3-state Potts ferromagnet/spin glass

The fixed point structure of the 2D 3-state random-bond Potts model with a bimodal ($\pm$J) distribution of couplings is for the first time fully determined using numerical renormalization group techniques. Apart from the pure and T=0 critical fixed points, two other non-trivial fixed points are found. One is the critical fixed point for the random-bond, but unfrustrated, ferromagnet. The other is a bicritical fixed point analogous to the bicritical Nishimori fixed point found in the random-bond frustrated Ising model. Estimates of the associated critical exponents are given for the various fixed points of the random-bond Potts model.Comment: 4 pages, 2 eps figures, RevTex 3.0 format requires float and epsfig macro

Strong disorder fixed points in the two-dimensional random-bond Ising model

The random-bond Ising model on the square lattice has several disordered critical points, depending on the probability distribution of the bonds. There are a finite-temperature multicritical point, called Nishimori point, and a zero-temperature fixed point, for both a binary distribution where the coupling constants take the values +/- J and a Gaussian disorder distribution. Inclusion of dilution in the +/- J distribution (J=0 for some bonds) gives rise to another zero-temperature fixed point which can be identified with percolation in the non-frustrated case (J >= 0). We study these fixed points using numerical (transfer matrix) methods. We determine the location, critical exponents, and central charge of the different fixed points and study the spin-spin correlation functions. Our main findings are the following: (1) We confirm that the Nishimori point is universal with respect to the type of disorder, i.e. we obtain the same central charge and critical exponents for the +/- J and Gaussian distributions of disorder. (2) The Nishimori point, the zero-temperature fixed point for the +/- J and Gaussian distributions of disorder, and the percolation point in the diluted case all belong to mutually distinct universality classes. (3) The paramagnetic phase is re-entrant below the Nishimori point, i.e. the zero-temperature fixed points are not located exactly below the Nishimori point, neither for the +/- J distribution, nor for the Gaussian distribution.Comment: final version to appear in JSTAT; minor change

Strong-disorder paramagnetic-ferromagnetic fixed point in the square-lattice +- J Ising model

We consider the random-bond +- J Ising model on a square lattice as a function of the temperature T and of the disorder parameter p (p=1 corresponds to the pure Ising model). We investigate the critical behavior along the paramagnetic-ferromagnetic transition line at low temperatures, below the temperature of the multicritical Nishimori point at T*= 0.9527(1), p*=0.89083(3). We present finite-size scaling analyses of Monte Carlo results at two temperature values, T=0.645 and T=0.5. The results show that the paramagnetic-ferromagnetic transition line is reentrant for T<T*, that the transitions are continuous and controlled by a strong-disorder fixed point with critical exponents nu=1.50(4) and eta=0.128(8), and beta = 0.095(5). This fixed point is definitely different from the Ising fixed point controlling the paramagnetic-ferromagnetic transitions for T>T*. Our results for the critical exponents are consistent with the hyperscaling relation 2 beta/nu - eta = d - 2 = 0.Comment: 32 pages, added refs and a discussion on hyperscalin

Exact properties of spin glasses. - I. 2D supersymmetry and Nishimori's result

We introduce an effective theory of interacting fermions and bosons in order to express the quenched internal energy of the 2D Ising spin glass. We show that an exact result derived by Nishimori appears, in this formulation, as a dimensional reduction due to the apparition of a supersymmetry. For a general Ising spin glass, this suggests new insights into the physical meaning of this exact result.Nous proposons une théorie effective de fermions et bosons en interaction afin d'exprimer l'énergie interne du verre de spin d'Ising bidimensionnel. Nous montrons qu'un résultat exact obtenu par Nishimori s'interprète, dans cette formulation, comme une réduction dimensionnelle due à l'apparition d'une supersymétrie. Ceci conduit à une meilleure compréhension de ce résultat exact pour un verre de spin d'Ising général

Diffusion anormale dans les milieux désordonnés : piégeage, corrélations et théorèmes de la limite centrale

Simple physical arguments are developed, allowing to predict the asymptotic behaviour of random walks in random media. It is shown that anomalous diffusion originates from distributions (for example of effective trapping times) with long tails, or from long-range correlations. This leads either to subdiffusive or superdiffusive behaviours.Nous présentons des arguments physiques simples permettant d'obtenir les comportements asymptotiques de marches aléatoires dans des milieux désordonnés. Nous montrons en particulier qu'un comportement non-diffusif est lié à l'impossibilité d'appliquer le théorème de la limite centrale de façon directe, en raison de l'apparition de lois de distribution « larges » (par exemple pour les temps effectifs de piégeage) ou de corrélations à longue portée, conduisant soit à une hypodiffusion, soit à une hyperdiffusion