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

TBA

Abstract text goes here...

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".

TBA

Abstract text goes here...

Height correlations in Abelian sandpiles in dimensions d > 4

Motivated by questions about the structure of large avalanches in the driven-dissipative Abelian sandpile model above its critical dimension, we study the height correlations in dimensions d > 4. We show that for any two height values at sites o and x, these decay as |x|^{-2d} as |x| goes to infinity, thereby extending an old result of Dhar and Majumdar concerning the correlations between vacant sites. We also study correlations between nearby sites.

TBA

Abstract text goes here...

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.