Related papers: Backbone exponent for two-dimensional percolation
In two dimensions, the average electrical conductance from a point in a percolating network to the network boundary should be related by a conformal transformation to the conductance from one point to another in an unbounded network. We…
Based on our generalization of the Goulian-Li continuation in the power of the 2D cosmological term we construct the two and three-point correlation functions for Liouville exponentials with generic real coefficients. As a strong argument…
We prove that the Fourier transform of the properly-scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index \alpha>0 converges to e^{-C|k|^{\alpha\wedge2}} for some C\in(0,\infty) above…
We provide a new proof of the near-critical scaling relation $\beta=\xi_1\nu$ for Bernoulli percolation on the square lattice already proved by Kesten in 1987. We rely on a novel approach that does not invoke Russo's formula, but rather…
The Bernoulli convolution $\nu_\lambda$ with parameter $\lambda\in(0,1)$ is the probability measure supported on $\mathbf{R}$ that is the law of the random variable $\sum\pm\lambda^n$, where the $\pm$ are independent fair coin-tosses. We…
We consider oriented percolation on Z^d times Z_+ whose bond-occupation probability is pD(...), where p is the percolation parameter and D is a probability distribution on Z^d. Suppose that D(x) decays as |x|^{-d-\alpha} for some \alpha>0.…
The amplitude ratio of the susceptibility (or second size-moment) for two-dimensional percolation is calculated by two series methods and also by Monte-Carlo simulation. The first series method is a new approach based upon integrating…
We introduce a formula for translating any upper bound on the percolation threshold of a lattice \g into a lower bound on the exponential growth rate of lattice animals $a(G)$ and vice-versa. We exploit this to improve on the best known…
In this paper we present the proof of the convergence of the critical bond percolation exploration process on the square lattice to the trace of SLE$_{6}$. This is an important conjecture in mathematical physics and probability. The case of…
We discuss duality properties of critical Boltzmann planar maps such that the degree of a typical face is in the domain of attraction of a stable distribution with parameter $\alpha\in(1,2]$. We consider the critical Bernoulli bond…
The number of two-dimensional percolation clusters whose external hulls enclose an area greater than A, in a system of area Omega, behaves at the critical point as C \Omega /A for large A, where C = 1/(8 pi sqrt(3)). Here we show that away…
We study a percolation model on $\mathbb R^d$ called the random connection model. For $d$ large, we use the lace expansion to prove that the critical two-point connection probability decays like $|x|^{-(d-2)}$ as $|x| \to \infty$, with…
We present a review of the recent progress on percolation scaling limits in two dimensions. In particular, we will consider the convergence of critical crossing probabilities to Cardy's formula and of the critical exploration path to…
Building on the identification of the scaling limit of the critical percolation exploration process as a Schramm-Loewner Evolution, we derive a PDE characterization for the crossing probability of an annulus.
Consider Bernoulli bond percolation on a locally finite, connected graph $G$ and let $p_{\mathrm{cut}}$ be the threshold corresponding to a "first-moment method" lower bound. Kahn (\textit{Electron.\ Comm.\ Probab.\ Volume 8, 184-187.}…
We study the Diophantine properties of a new class of transcendental real numbers which contains, among others, Roy's extremal numbers, Bugeaud-Laurent Sturmian continued fractions, and more generally the class of Sturmian type numbers. We…
We study Bernoulli percolation on $\mathbb Z^d$ in dimensions ${d>6}$. We prove that a classical consequence of the van den Berg-Kesten inequality, often referred to as the Simon-Lieb inequality in the context of the Ising model, admits a…
We consider boundary roughness for the ``droplet'' created when supercritical two-dimensional Bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least $l^2$, for large $l$. The…
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…
We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex…