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We review some of the recent progress on the scaling limit of two-dimensional critical percolation; in particular, the convergence of the exploration path to chordal SLE(6) and the "full" scaling limit of cluster interface loops. The…
The BTW sandpile model is considered on three dimensional percolation lattice which is tunned with the occupation parameter $p$. Along with the three-dimensional avalanches, we study the energy propagation in two-dimensional cross-sections.…
Let $G=(V,E)$ be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on $G$. In recent work, we conjectured that if $G$ is nonamenable then the matrix of critical connection probabilities…
In this work we use the technique of the partial differential approximants to determine, from a pertubative supercritical series expansion for the ulimate survival probability, the critical line of the contact process model in one dimension…
In this paper, we consider inverse limits of [0,1] using upper semicontinuous set-valued bonding functions with the intermediate value property. Expanding on classical results by Barge and Martin, we explore the relationship between…
We study the conductivity of a class of disordered continuum systems represented by the Swiss-cheese model, where the conducting medium is the space between randomly placed spherical holes, near the percolation threshold. This model can be…
It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of…
We present an analytically solvable theory of BCS-type superconductivity in good metals which are confined along one of the three spatial directions, such as thin films. Closed-form expressions for the dependence of the superconducting…
We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasi-transitive subgroup. We prove that percolation on any such graph has a non-empty…
We study a contact process on a two-dimensional square lattice which is diluted by randomly removing bonds with probability p. For p<1/2 and varying birth rate $\lambda$ the model was shown to exhibit a continuous phase transition which…
We consider supercritical long-range percolation on transitive graphs of polynomial growth. In this model, any two vertices $x$ and $y$ of the underlying graph $G$ connect by a direct edge with probability $1-\exp(-\beta J(x,y))$, where…
We obtain the exact solution of the bond-percolation thresholds with inhomogenous probabilities on the square lattice. Our method is based on the duality analysis with real-space renormalization, which is a profound technique invented in…
In the Constrained-degree percolation model on a graph $(\mathbb{V},\mathbb{E})$ there are a sequence, $(U_e)_{e\in\mathbb{E}}$, of i.i.d. random variables with distribution $U[0,1]$ and a positive integer $k$. Each bond $e$ tries to open…
Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. We prove a law of large numbers for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. The value of the…
We study the confinement of vorticity for two-dimensional incompressible flows in an infinite cylinder. For Navier-Stokes solutions with non-negative and compactly supported initial vorticity, we derive quantitative decay estimates showing…
We prove that the heavy clusters are indistinguishable for Bernoulli percolation on quasi-transitive nonunimodular graphs. As an application, we show that the uniqueness threshold of any quasi-transitive graph is also the threshold for…
Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed…
The Euclidean first-passage percolation model of Howard and Newman is a rotationally invariant percolation model built on a Poisson point process. It is known that the passage time between 0 and $ne_1$ obeys a diffusive upper bound:…
We present an analytical study of one-dimensional semiconductor superlattices in external electric fields, which may be time-dependent. A number of general results for the (quasi)energies and eigenstates are derived. An equation of motion…
The values obtained experimentally for the conductivity critical exponent in numerous percolation systems, in which the interparticle conduction is by tunnelling, were found to be in the range of $t_0$ and about $t_0+10$, where $t_0$ is the…