相关论文: Continuum Nonsimple Loops and 2D Critical Percolat…
Percolation, a paradigmatic geometric system in various branches of physical sciences, is known to possess logarithmic factors in its correlators. Starting from its definition, as the $Q\rightarrow1$ limit of the $Q$-state Potts model with…
The rigidity transition occurs when, as the density of microscopic components is increased, a disordered medium becomes able to transmit and ensure macroscopic mechanical stability, owing to the appearance of a space-spanning rigid…
The dynamical properties of the invasion percolation on the square lattice are investigated with emphasis on the geometrical properties on the growing cluster of infected sites. The exterior frontier of this cluster forms a critical loop…
We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on $Z^d \times Z_+$. In dimensions $d>6$, we obtain bounds on exit times, transition probabilities, and the range of the…
Scaling limits of critical percolation models show major differences between low and high dimensional models. The article discusses the formulation of the continuum limit for the former case. A mathematical framework is proposed for the…
We describe the critical window for percolation in the universality class of sparse growing random graphs. In our models, vertices arrive sequentially and connect independently to each earlier vertex $v$ with probability proportional to a…
Recently, the effective medium approach using 2x2 basic cluster of model lattice sites to predict the conductivity of interacting droplets has been presented by Hattori et al. To make a step aside from pure applications, we have studied…
This work extends the universal finite-size scaling framework for continuum percolation from two-dimensional (2D) to quasi-three-dimensional (Q3D) stick systems, in which sequentially deposited wires of finite diameter stack vertically on a…
We define a family of stochastic Loewner evolution-type processes in finitely connected domains, which are called continuous LERW (loop-erased random walk). A continuous LERW describes a random curve in a finitely connected domain that…
For independent nearest-neighbour bond percolation on Z^d with d >> 6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling…
We study the behavior of the random walk on the infinite cluster of independent long range percolation in dimensions $d=1,2$, where $x$ and $y$ a re connected with probability $\sim\beta/\|x-y\|^{-s}$. We show that when $d<s<2d$ the walk is…
Based on the field theoretic formulation of the general epidemic process we study logarithmic corrections to scaling in dynamic isotropic percolation at the upper critical dimension d=6. Employing renormalization group methods we determine…
We consider the dimer model on the square and hexagonal lattices with doubly periodic weights. The purpose of this paper is threefold: (a) we establish a rigourous connection with the massive SLE$_2$ constructed by Makarov and Smirnov (and…
We consider critical curves -- conformally invariant curves that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field…
This paper introduces the annulus SLE$_\kappa$ processes in doubly connected domains. Annulus SLE$_6$ has the same law as stopped radial SLE$_6$, up to a time-change. For $\kappa\not=6$, some weak equivalence relation exists between annulus…
Global physical properties of random media change qualitatively at a percolation threshold, where isolated clusters merge to form one infinite connected component. The precise knowledge of percolation thresholds is thus of paramount…
This is the first of two papers devoted to the proof of conformal invariance of the critical double random current model on the square lattice. More precisely, we show the convergence of loop ensembles obtained by taking the cluster…
Techniques of `dynamic renormalization', developed earlier for undirected percolation and the contact model, are adapted to the setting of directed percolation, thereby obtaining solutions of several problems for directed percolation on…
Stochastic Loewner evolution also called Schramm Loewner evolution (abbreviated, SLE) is a rigorous tool in mathematics and statistical physics for generating and studying scale invariant or fractal random curves in two dimensions. The…
We describe infinite clusters which arise in nearest-neighbour percolation for so-called cocycle measures on the square lattice. These measures arise naturally in the study of random transformations. We show that infinite clusters have a…