Related papers: Short-range correlations in percolation at critica…
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…
In this paper we investigate the critical probability $p_c(Q_n,r)$ for bootstrap percolation with the infection threshold $r$ on the $n$-dimensional hypercube $Q_n$ with vertex set $V(Q_n)=\{0,1\}^n$ and edges connecting the pairs at…
We study a generalization of two-dimensional site percolation by assigning an energy cost $\varepsilon$ to bonds between nearest-neighbor occupied sites. This leads to a competition between entropy-driven cluster growth and energetic…
We study critical percolation on a regular planar lattice. Let $E_G(n)$ be the expected number of open clusters intersecting or hitting the line segment $[0,n]$. (For the subscript $G$ we either take $\mathbb{H}$, when we restrict to the…
In previous work with Scullard, we defined a graph polynomial P_B(q,T) that gives access to the critical temperature T_c of the q-state Potts model on a general two-dimensional lattice L. It depends on a basis B, containing n x m unit cells…
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…
We present the results of a study of the three-dimensional $XY$-model on a simple cubic lattice using the single cluster updating algorithm combined with improved estimators. We have measured the susceptibility and the correlation length…
Models of percolation processes on networks currently assume locally tree-like structures at low densities, and are derived exactly only in the thermodynamic limit. Finite size effects and the presence of short loops in real systems however…
We define the correlation of holes on the triangular lattice under periodic boundary conditions and study its asymptotics as the distances between the holes grow to infinity. We prove that the joint correlation of an arbitrary collection of…
In a recent article, we described how the microscopic structure of density-density correlations in the fluid interfacial region, for systems with short-ranged forces, can be understood by considering the resonances of the local structure…
We consider the bond percolation problem on a transient weighted graph induced by the excursion sets of the Gaussian free field on the corresponding cable system. Owing to the continuity of this setup and the strong Markov property of the…
We consider spread-out models of lattice trees and lattice animals on $\mathbb Z^d$, for $d$ above the upper critical dimension $d_{\mathrm c}=8$. We define a correlation length and prove that it diverges as $(p_c-p)^{-1/4}$ at the critical…
The site and bond percolation problems are conventionally studied on (hyper)cubic lattices, which afford straightforward numerical treatments. The recent implementation of efficient simulation algorithms for high-dimensional systems now…
We study critical bond percolation on periodic four-dimensional (4D) and five-dimensional (5D) hypercubes by Monte Carlo simulations. By classifying the occupied bonds into branches, junctions and non-bridges, we construct the whole, the…
Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0,1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized…
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…
In the random $r$-neighbour bootstrap percolation process on a graph $G$, a set of initially infected vertices is chosen at random by retaining each vertex of $G$ independently with probability $p\in (0,1)$, and "healthy" vertices get…
Let $G_1,\dots, G_m$ be independent Bernoulli random subgraphs of the complete graph ${\cal K}_n$ having variable sizes $X_1,\dots, X_m\in \{0,1,2,\dots\}$ and densities $Q_1,\dots, Q_m\in [0,1]$. Letting $n,m\to+\infty$ we establish the…
We consider critical percolation on the triangular lattice in a bounded simply connected domain with boundary conditions that force an interface between two prescribed boundary points. We say the interface forms a "near-loop" when it comes…
In 1994, Kranakis et al. published a conjecture about the minimum link-length of every rectilinear covering path for the $k$-dimensional grid $P(n,k) := \{0,1, \dots, n-1\} \times \{0,1, \dots, n-1\} \times \cdots \times \{0,1, \dots,…