Related papers: Maximum Cluster Diameter in Non-Critical Bond Perc…
We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to L=512, and estimate the percolation thresholds to be $p_c ({\rm bond})=0.248\,811\,82(10)$ and $p_c ({\rm site})=0.311\,607\,7(2)$. By…
We consider high-dimensional percolation at the critical threshold. We condition the origin to be disjointly connected to two points, $x$ and $x'$, and subsequently take the limit as $|x|$, $|x'|$ as well as $|x-x'|$ diverge to infinity.…
We consider supercritical Bernoulli bond percolation on a large $b$-ary tree, in the sense that with high probability, there exists a giant cluster. We show that the size of the giant cluster has non-gaussian fluctuations, which extends a…
Under some general assumptions, we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site…
We study random subgraphs of the $n$-cube $\{0,1\}^n$, where nearest-neighbor edges are occupied with probability $p$. Let $p_c(n)$ be the value of $p$ for which the expected cluster size of a fixed vertex attains the value $\lambda…
We consider the Gaussian free field $\varphi$ on $\mathbb{Z}^d$, for $d\geq3$, and give sharp bounds on the probability that the radius of a finite cluster in the excursion set $\{\varphi \geq h\}$ exceeds a large value $N$, for any height…
We consider the cardinality of supercritical oriented bond percolation in two dimensions. We show that, whenever the origin is conditioned to percolate, the process appropriately normalized converges asymptotically in distribution to the…
We present the explicit forms of the maximum eigenvalue and the corresponding eigenfunction for the second-order reduced density matrix (RDM2) of the BCS superconducting state (SS). Using these quantities, we deal with two topics in the…
We prove that for Bernoulli bond percolation on $\mathbb{Z}^d$, $d\geq 2$ the percolation density is an analytic function of the parameter in the supercritical interval $(p_c,1]$. This answers a question of Kesten from 1981.
We consider invasion percolation on the square lattice. It has been proved by van den Berg, Peres, Sidoravicius and Vares, that the probability that the radius of a so-called pond is larger than n, differs at most a factor of order log n…
We give a new construction of the incipient infinite cluster (IIC) associated with high-dimensional percolation in a broad setting and under minimal assumptions. Our arguments differ substantially from earlier constructions of the IIC; we…
We study site percolation on Angel & Schramm's uniform infinite planar triangulation. We compute several critical and near-critical exponents, and describe the scaling limit of the boundary of large percolation clusters in all regimes…
We consider Bernoulli first-passage percolation on the triangular lattice in which sites have 0 and 1 passage times with probability $p$ and $1-p$, respectively. For each $p\in(0,p_c)$, let $\mathcal {B}(p)$ be the limit shape in the…
We study bond percolation on the Hamming hypercube {0,1}^m around the critical probability p_c. It is known that if p=p_c(1+O(2^{-m/3})), then with high probability the largest connected component C_1 is of size Theta(2^{2m/3}) and that…
The probability that the cluster of the origin in critical site percolation on the triangular grid has diameter larger than $R$ is proved to decay like $R^{-5/48}$ as $R\to\infty$.
We establish a general formula for the maximum size of finite length block codes with minimum pairwise distance no less than $d$. The achievability argument involves an iterative construction of a set of radius-$d$ balls, each centered at a…
Central limit theorems (CLTs) for high-dimensional random vectors with dimension possibly growing with the sample size have received a lot of attention in the recent times. Chernozhukov et al. (2017) proved a Berry--Esseen type result for…
In this note we study some properties of infinite percolation clusters on non-amenable graphs. In particular, we study the percolative properties of the complement of infinite percolation clusters. An approach based on mass-transport is…
We prove upper bounds on the average kissing number $k(\mathcal{P})$ and contact number $C(\mathcal{P})$ of an arbitrary finite non-congruent sphere packing $\mathcal{P}$, and prove an upper bound on the packing density…
In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A \subset V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected…