We develop a new approach to study Bernoulli percolation in dimensions $d>6$. The key idea is to approximate open paths at probability $p'>p$ by a Markov chain of pointed $p$-open clusters. This allows us to transfer sharp information on the two point function from $p$ to $p'$. This inductively gives good asymptotic estimates on the two point function as $p\uparrow p_c$. As a main application, we show that having critical meanfield behavior is a semi-decidable problem. Along the way, we also prove sharp asymptotics for the susceptibility and the average radius of gyration, and prove a local central limit theorem for the slightly subcritical two-point function.
We show that the percolation model induced by the double random current representation of the Ising model on $\mathbb Z^d$, with $d\geq 3$, satisfies local uniqueness of the macroscopic cluster with high probability for every $\beta>\beta_c$, uniformly in boundary conditions (i.e. sources). Such a result is sometimes c...
Trishen S. Gunaratnam, C. Panagiotis, Romain Panis et al.· 1 citation
We give an estimate for the distribution tail of the passage time between $0$ and $\infty$ in first-passage percolation on $\mathbb Z^d$, when the weight distribution is bounded and has a supercritical atom at $0$. A special case involves the number of disjoint cutsets between $0$ and $\infty$ in supercritical bond per...
We consider the level sets of the Gaussian free field and the vacant set of random interlacements, both defined on a typical realization of the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z}^d$, $d \geq 3$. We prove that in the entire supercritical regime of Bernoulli bond percolation, both...
Alberto Chiarini, Zhi-Zhou Liu, Maximilian Nitzschner· 0 citations
We solve a longstanding problem in Bernoulli percolation: we derive sharp Ornstein--Zernike asymptotics for the probability that two distant points are connected by a finite cluster in the whole supercritical regime. Our proof has two main ingredients. First, the renewal structure from Fridbergh and Hammond (2024) yiel...
We consider long-range percolation (LRP) on $\mathbb{Z}^d, d \ge 1$ with connection probabilities $\mathbf{p}(x, y) \approx \frac{\beta}{\|x-y\|^s}$ for $\beta>0$ and $s>\min(2d, d+2)$. Our main result is that the mixing time of the random walk on the giant component of supercritical LRP inside a box of side length $n$...
We prove that the correlation length of FK-percolation and Potts models on $\mathbb{Z}^d$ is analytic in the whole subcritical regime of parameters. Our proof makes use of the identification of the polar dual of the mass with the solution of an equation involving the leading eigenvalue of the FK--Ruelle operator, intro...
Lucas D'Alimonte, Loïc Gassmann· 0 citations
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