相关论文: Asymptotic expansion of the critical point for ori…

We use the lace expansion to prove that the critical values for nearest-neighbour bond percolation on the $n$-cube $\{0,1\}^n$ and on $\mathbb{Z}^n$ have asymptotic expansions, with rational coefficients, to all orders in powers of…

The Hamming graph $H(d,n)$ is the Cartesian product of $d$ complete graphs on $n$ vertices. Let $m=d(n-1)$ be the degree and $V = n^d$ be the number of vertices of $H(d,n)$. Let $p_c^{(d)}$ be the critical point for bond percolation on…

We expand the critical point for site percolation on the $d$-dimensional hypercubic lattice in terms of inverse powers of $2d$, and we obtain the first three terms rigorously. This is achieved using the lace expansion.

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 provide a complete proof of the diagrammatic bounds on the lace-expansion coefficients for oriented percolation, which are used in [arXiv:math/0703455] to investigate critical behavior for long-range oriented percolation above…

We consider self-avoiding walk and percolation in $\Zd$, oriented percolation in $\Zd\times\Zp$, and the contact process in $\Zd$, with $p D(\cdot)$ being the coupling function whose range is denoted by $L<\infty$. For percolation, for…

We consider nearest-neighbor self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on ${\mathbb{Z}}^d$. The two-point functions of these models are respectively the generating function for self-avoiding walks from…

Let $p_c(\mathbb{Q}_n)$ and $p_c(\mathbb{Z}^n)$ denote the critical values for nearest-neighbour bond percolation on the $n$-cube $\mathbb{Q}_n = \{0,1\}^n$ and on $\Z^n$, respectively. Let $\Omega = n$ for $\mathbb{G} = \mathbb{Q}_n$ and…

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.…

We consider Lipschitz percolation in $d+1$ dimensions above planes tilted by an angle $\gamma$ along one or several coordinate axes. In particular, we are interested in the asymptotics of the critical probability as $d \to \infty$ as well…

We consider percolation on the discrete torus $\mathbb{Z}_n^d$ at $p_c(\mathbb{Z}^d)$, the critical value for percolation on the corresponding infinite lattice $\mathbb{Z}^d$, and within the scaling window around it. We assume that $d$ is a…

We consider supercritical bond percolation in $\mathbb{Z}^d$ for $d \geq 3$. The origin lies in a finite open cluster with positive probability, and, when it does, the diameter of this cluster has an exponentially decaying tail. For each…

We prove $|x|^{-2}$ decay of the critical two-point function for the continuous-time weakly self-avoiding walk on $\mathbb{Z}^d$, in the upper critical dimension $d=4$. This is a statement that the critical exponent $\eta$ exists and is…

We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in high dimensions, or when d>6 for sufficient spread-out percolation. We use a relatively simple…

We study biased random walks on dynamical percolation in $\mathbb{Z}^d$, which were recently introduced by Andres et al. We provide a second order expansion for the asymptotic speed and show for $d \ge 2$ that the speed of the biased random…

We show that the critical probability for percolation on a d-regular non-amenable graph of large girth is close to the critical probability for percolation on an infinite d-regular tree. This is a special case of a conjecture due to O.…

In this paper we are concerned with contact processes on open clusters of oriented percolation in $Z^d$, where the disease spreads along the direction of open edges. We show that the two critical infection rates in the quenched and annealed…

We introduce a method for translating any upper bound on the percolation threshold of a lattice $G$ into a lower bound on the exponential growth rate $a(G)$ of lattice animals and vice-versa. We exploit this in both directions. We improve…

We consider the Gaussian free field on $\mathbb{Z}^d$, $d$ greater or equal to $3$, and prove that the critical density for percolation of its level sets behaves like $1/d^{1 + o(1)}$ as $d$ tends to infinity. Our proof gives the principal…

Many real-world networks exhibit the so-called small-world phenomenon: their typical distances are much smaller than their sizes. One mathematical model for this phenomenon is a long-range percolation graph on a $d$-dimensional box $\{0, 1,…