Related papers: The one-arm exponent for mean-field long-range per…

Consider critical site percolation on $\mathbb{Z}^d$ with $d \geq 2$. Cerf (2015) pointed out that from classical work by Aizenman, Kesten and Newman (1987) and Gandolfi, Grimmett and Russo (1988) one can obtain that the two-arms exponent…

We consider long-range percolation, Ising model, and self-avoiding walk on $\mathbb{Z}^d$, with couplings decaying like $|x|^{-(d+\alpha)}$ where $0 < \alpha \le 2$, above the upper critical dimensions. In the spread-out setting where the…

For two-dimensional percolation at criticality, we discuss the inequality $\alpha_4 > 1$ for the polychromatic four-arm exponent (and stronger versions, the strongest so far being $\alpha_4 \geq 1 + \frac{\alpha_2}{2}$, where $\alpha_2$…

We prove upper bounds on the one-arm exponent $\eta_1$ for a class of dependent percolation models which generalise Bernoulli percolation; while our main interest is level set percolation of Gaussian fields, the arguments apply to other…

In long-range percolation on $\mathbb{Z}^d$, points $x$ and $y$ are connected by an edge with probability $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $\alpha>0$ is fixed and $\beta \geq 0$ is a parameter. As $d$ and $\alpha$ vary, the model…

We consider the following oriented percolation model of $\mathbb {N} \times \mathbb{Z}^d$: we equip $\mathbb {N}\times \mathbb{Z}^d$ with the edge set $\{[(n,x),(n+1,y)] | n\in \mathbb {N}, x,y\in \mathbb{Z}^d\}$, and we say that each edge…

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…

The 1-arm exponent $\rho$ for the ferromagnetic Ising model on $\mathbb{Z}^d$ is the critical exponent that describes how fast the critical 1-spin expectation at the center of the ball of radius $r$ surrounded by plus spins decays in powers…

We investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in ${G}$ have polynomial volume growth with growth…

We study the random walk on dynamical percolation of $\mathbb{Z}^d$ (resp., the two-dimensional triangular lattice $\mathcal{T}$), where each edge (resp., each site) can be either open or closed, refreshing its status at rate $\mu\in…

We consider the standard site percolation model on the $d$-dimensional lattice. A direct consequence of the proof of the uniqueness of the infinite cluster of Aizenman, Kesten and Newman [Comm. Math. Phys. 111 (1987) 505-531] is that the…

For the critical level-set of the Gaussian free field on the metric graph of $\mathbb Z^d$, we consider the one-arm probability $\theta_d(N)$, i.e., the probability that the boundary of a box of side length $2N$ is connected to the center.…

We study the probability that the origin is connected to the sphere of radius r (an arm event) in critical percolation in high dimensions, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out.…

In long-range percolation on $\mathbb{Z}^d$, we connect each pair of distinct points $x$ and $y$ by an edge independently at random with probability $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $\alpha>0$ is fixed and $\beta\geq 0$ is a…

In long-range percolation on $\mathbb{Z}^d$, points $x$ and $y$ are connected by an edge with probability $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $\alpha>0$ is fixed and $\beta \geq 0$ is a parameter. As $d$ and $\alpha$ vary, the model…

This article proposes a new way of deriving mean-field exponents for sufficiently spread-out Bernoulli percolation in dimensions $d>6$. We obtain an upper bound for the full-space and half-space two-point functions in the critical and…

We consider a model of long-range first-passage percolation on the $d$ dimensional square lattice $Z^d$ in which any two distinct vertices $x, y \in Z^d$ are connected by an edge having exponentially distributed passage time with mean…

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 study limit laws for simple random walks on supercritical long range percolation clusters on $\Z^d, d \geq 1$. For the long range percolation model, the probability that two vertices $x, y$ are connected behaves asymptotically as…

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…