Developments in High Dimensional Percolation and Branching Random Walks

Large-scale connectivity in high-dimensional percolation

We discuss a few recent results precisely quantifying large-scale behavior in high-dimensional critical percolation. One of these provides an exact limiting law for intrinsic distances in open clusters, and another gives a sharp asymptotic for the probability that k distant points lie in the same cluster. We will also prove a mixing result on which our other arguments depend: a convergence result for the law of large open clusters which is robust to different definitions of "large".

Supercritical sharpness of percolation I

Given an infinite transitive graph (such as the lattice Z^d), consider the random subgraph obtained by independently retaining each edge with probability p. This model undergoes a phase transition as p varies across a critical value p_c that marks the emergence of an infinite component. A classical result states that for each p < p_c, the probability that a given vertex belongs to a component of size at least n decays exponentially in n. Sahar Diskin, Ritvik Ramanan Radhakrishnan, Benny Sudakov, Vincent Tassion, and I have recently proved the analogous result for p > p_c.

Supercritical Sharpness of Percolation II

Dimension dependence of critical phenomena in percolation

It is conjectured that many models of statistical mechanics have a rich, fractal-like behaviour at and near their points of phase transition, with power-law scaling governed by critical exponents that are expected to depend on the dimension but not on the small-scale details of the model such as the choice of lattice. This is now reasonably well understood in two dimensions and in high dimensions, but remains poorly understood in intermediate dimensions (e.g. d=3). I will overview the conjectures around this area and describe recent progress on related problems for models with long-range interactions.

Large-scale connectivity in high-dimensional percolation

We discuss a few recent results precisely quantifying large-scale behavior in high-dimensional critical percolation. One of these provides an exact limiting law for intrinsic distances in open clusters, and another gives a sharp asymptotic for the probability that k distant points lie in the same cluster. We will also prove a mixing result on which our other arguments depend: a convergence result for the law of large open clusters which is robust to different definitions of "large".

Infinite and bi-infinite incipient clusters in high dimension

The incipient infinite cluster is a critical percolation cluster that is conditioned on being infinitely large. Even though the condition has probability zero, Kesten demonstrated the first such construction for critical two-dimensional percolation in 1986 through a suitable limiting scheme. Similar constructions for critical high-dimensional percolation have been carried out by van der Hofstad and Jaraí in 2004. In this talk, I will present a construction of an incipient cluster with *two* disjoint connections to infinity, which we call bi-infinite incipient cluster. Based on joint work with Manuel Cabezas, Alexander Fribergh, and Antal Jaraí.

(Near-)critical behavior of a strongly correlated percolation model and Hausdorff dimension of the critical clusters

For (near-)critical independent Bernoulli percolation, particularly profound results have been obtained in the high-dimensional setting as well as on planar lattices. We consider a strongly correlated percolation model — the level sets of the metric graph Gaussian free field — where significant understanding has also been developed regarding its (near-)critical behavior in intermediate dimensions. We will explain the origin of the model's integrability, and discuss its implications for the associated universality class. A particular focus will be on recent results for the Hausdorff dimension of the critical connected components.

Percolation of loop-soups, loop-soups in high-dimensional critical percolation

Abstract: I will discuss the following three types of results: (a) a switching result for Brownian loop-soups on any cable graph that roughly says that when a cluster contains a cycle around a “hole” then the total index of the Brownian loops around this hole is odd with probability 1/2. (b) In d dimensions when d>6, half of the long cycles in loop soup clusters are made of actual Brownian loops while the other half are created by chains of much smaller loops (j.w. with Titus Lupu), (c) For critical Bernoulli percolation in high dimensions, the collection of large cycles in clusters tends to a Brownian loop-soup in the scaling limit (joint and ongoing work with Amelia Carpenter).

Large-scale connectivity in high-dimensional percolation

We discuss a few recent results precisely quantifying large-scale behavior in high-dimensional critical percolation. One of these provides an exact limiting law for intrinsic distances in open clusters, and another gives a sharp asymptotic for the probability that k distant points lie in the same cluster. We will also prove a mixing result on which our other arguments depend: a convergence result for the law of large open clusters which is robust to different definitions of "large".

Supercritical sharpness for Ising double random currents

We show that the double random current representation of the Ising model satisfies local uniqueness of macroscopic cluster with high probability in the supercritical regime, uniformly in boundary conditions (i.e. sources). Such a statement is the strongest version of what is sometimes called supercritical sharpness in percolation theory, and yields a detailed description of the model via standard renormalization techniques. In particular, this easily implies the exponential decay or truncated correlations for Ising and exponential mixing for the FK-Ising measure, results that had already been obtained by Duminil-Copin, Goswami and Raoufi. However, our proof is shorter and simpler, as we avoid the use of a highly technical multi-valued map principle by using a sprinkling argument instead. Based on joint work with Trishen Gunaratnam, Romain Panis and Franco Severo.