Interface mapping in two-dimensional random lattice models

We consider two disordered lattice models on the square lattice: on the medial lattice the random field Ising model at T=0 and on the direct lattice the random bond Potts model in the large-q limit at its transition point. The interface properties of the two models are known to be related by a mapping which is valid in the continuum approximation. Here we consider finite random samples with the same form of disorder for both models and calculate the respective equilibrium states exactly by combinatorial optimization algorithms. We study the evolution of the interfaces with the strength of disorder and analyse and compare the interfaces of the two models in finite lattices.Comment: 7 pages, 6 figure

Rounding of first-order phase transitions and optimal cooperation in scale-free networks

We consider the ferromagnetic large-$q$ state Potts model in complex evolving networks, which is equivalent to an optimal cooperation problem, in which the agents try to optimize the total sum of pair cooperation benefits and the supports of independent projects. The agents are found to be typically of two kinds: a fraction of $m$ (being the magnetization of the Potts model) belongs to a large cooperating cluster, whereas the others are isolated one man's projects. It is shown rigorously that the homogeneous model has a strongly first-order phase transition, which turns to second-order for random interactions (benefits), the properties of which are studied numerically on the Barab\'asi-Albert network. The distribution of finite-size transition points is characterized by a shift exponent, $1/\tilde{\nu}'=.26(1)$, and by a different width exponent, $1/\nu'=.18(1)$, whereas the magnetization at the transition point scales with the size of the network, $N$, as: $m\sim N^{-x}$, with $x=.66(1)$.Comment: 8 pages, 6 figure

Density of critical clusters in strips of strongly disordered systems

We consider two models with disorder dominated critical points and study the distribution of clusters which are confined in strips and touch one or both boundaries. For the classical random bond Potts model in the large-q limit we study optimal Fortuin-Kasteleyn clusters by combinatorial optimization algorithm. For the random transverse-field Ising chain clusters are defined and calculated through the strong disorder renormalization group method. The numerically calculated density profiles close to the boundaries are shown to follow scaling predictions. For the random bond Potts model we have obtained accurate numerical estimates for the critical exponents and demonstrated that the density profiles are well described by conformal formulae.Comment: 9 pages, 9 figure

Birational Mappings and Matrix Sub-algebra from the Chiral Potts Model

We study birational transformations of the projective space originating from lattice statistical mechanics, specifically from various chiral Potts models. Associating these models to \emph{stable patterns} and \emph{signed-patterns}, we give general results which allow us to find \emph{all} chiral $q$-state spin-edge Potts models when the number of states $q$ is a prime or the square of a prime, as well as several $q$-dependent family of models. We also prove the absence of monocolor stable signed-pattern with more than four states. This demonstrates a conjecture about cyclic Hadamard matrices in a particular case. The birational transformations associated to these lattice spin-edge models show complexity reduction. In particular we recover a one-parameter family of integrable transformations, for which we give a matrix representationComment: 22 pages 0 figure The paper has been reorganized, splitting the results into two sections : results pertaining to Physics and results pertaining to Mathematic

Disorder induced rounding of the phase transition in the large q-state Potts model

The phase transition in the q-state Potts model with homogeneous ferromagnetic couplings is strongly first order for large q, while is rounded in the presence of quenched disorder. Here we study this phenomenon on different two-dimensional lattices by using the fact that the partition function of the model is dominated by a single diagram of the high-temperature expansion, which is calculated by an efficient combinatorial optimization algorithm. For a given finite sample with discrete randomness the free energy is a pice-wise linear function of the temperature, which is rounded after averaging, however the discontinuity of the internal energy at the transition point (i.e. the latent heat) stays finite even in the thermodynamic limit. For a continuous disorder, instead, the latent heat vanishes. At the phase transition point the dominant diagram percolates and the total magnetic moment is related to the size of the percolating cluster. Its fractal dimension is found d_f=(5+\sqrt{5})/4 and it is independent of the type of the lattice and the form of disorder. We argue that the critical behavior is exclusively determined by disorder and the corresponding fixed point is the isotropic version of the so called infinite randomness fixed point, which is realized in random quantum spin chains. From this mapping we conjecture the values of the critical exponents as \beta=2-d_f, \beta_s=1/2 and \nu=1.Comment: 12 pages, 12 figures, version as publishe

Nonequilibrium dynamics of fully frustrated Ising models at T=0

We consider two fully frustrated Ising models: the antiferromagnetic triangular model in a field of strength, $h=H T k_B$, as well as the Villain model on the square lattice. After a quench from a disordered initial state to T=0 we study the nonequilibrium dynamics of both models by Monte Carlo simulations. In a finite system of linear size, $L$, we define and measure sample dependent "first passage time", $t_r$, which is the number of Monte Carlo steps until the energy is relaxed to the ground-state value. The distribution of $t_r$, in particular its mean value, $$, is shown to obey the scaling relation, $\sim L^2 \ln(L/L_0)$, for both models. Scaling of the autocorrelation function of the antiferromagnetic triangular model is shown to involve logarithmic corrections, both at H=0 and at the field-induced Kosterlitz-Thouless transition, however the autocorrelation exponent is found to be $H$ dependent.Comment: 7 pages, 8 figure

Random walks and polymers in the presence of quenched disorder

After a general introduction to the field, we describe some recent results concerning disorder effects on both `random walk models', where the random walk is a dynamical process generated by local transition rules, and on `polymer models', where each random walk trajectory representing the configuration of a polymer chain is associated to a global Boltzmann weight. For random walk models, we explain, on the specific examples of the Sinai model and of the trap model, how disorder induces anomalous diffusion, aging behaviours and Golosov localization, and how these properties can be understood via a strong disorder renormalization approach. For polymer models, we discuss the critical properties of various delocalization transitions involving random polymers. We first summarize some recent progresses in the general theory of random critical points : thermodynamic observables are not self-averaging at criticality whenever disorder is relevant, and this lack of self-averaging is directly related to the probability distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples $(i)$ of size $L$. We describe the results of this analysis for the bidimensional wetting and for the Poland-Scheraga model of DNA denaturation.Comment: 17 pages, Conference Proceedings "Mathematics and Physics", I.H.E.S., France, November 200

Real topological entropy versus metric entropy for birational measure-preserving transformations

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