Many modern nanostructured materials and doped polymers are morphologically too complex to be interpreted by classical percolation theory. Here, we develop the concept of a hierarchical percolating (percolation-within-percolation) system to describe such complex materials and illustrate how to generalize the conventional percolation to double-level percolation. Based on Monte Carlo simulations,...

This paper addresses the problem of dielectric breakdown in composite materials. The dielectric breakdown model was generalized to describe dielectric breakdown patterns in conductor-loaded composites. Conducting particles are distributed at random in the insulating matrix, and the dielectric breakdown propagates according to new rules to take into account electrical properties and particle siz...

We show that the transience or recurrence of a random walk in certain random environments on an arbitrary infinite locally finite tree is determined by the branching number of the tree, which is a measure of the average number of branches per vertex. This generalizes and unifies previous work of the authors. It also shows that the point of phase transition for edge-reinforced random walk is lik...

We present study of finite-size scaling and universality of crossing probabilities for the q-state Potts model. Crossing probabilities of the Potts model are similar ones in percolation problem. We numerically investigated scaling of πs the probability of a system to percolate only in one direction for twodimensional site percolation, the Ising model, and the q-state Potts model for q = 3, 4, 5...

In this paper I present new data from possessive phrases in Chol (Mayan) and discuss the problem they present for standard analyses of pied-piping (see Heck 2004 and references cited therein). I argue that a theory of pied-piping in which features of a wh-word “percolate” up to a higher maximal projection is unable to straightforwardly account for the ordering facts found in interrogative posse...

We present preliminary results of our numerical study of the critical dynamics of percolation observables for the two-dimensional Ising model. We consider the (Monte-Carlo) short-time evolution of the system obtained with a local heat-bath method and with the global Swendsen-Wang algorithm. In both cases, we find qualitatively different dynamic behaviors for the magnetization and Ω, the order p...

We study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bondpercolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number o...

These lectures discuss the theory of phase transitions and critical phenomena in random systems. First a discussion is given of the basic principles, including averaging, the concept of frustration, and the Harris criterion for deciding whether weak randomness is relevant. This is followed by the study of several examples: weakly random ferromagnets, the percolation problem, the random eld prob...

When a quantum many-particle system exists on a randomly diluted lattice, its intrinsic thermal and quantum fluctuations coexist with geometric fluctuations due to percolation. In this paper, we explore how the interplay of these fluctuations influences the phase transition at the percolation threshold. While it is well known that thermal fluctuations generically destroy long-range order on the...