Federico Camia

We relate various concepts of fractal dimension of the limiting set C in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in C (the "dust"). In two dimensions, we also show that the set consisting of connected components larger than one point is a.s. the union of non-trivial H\"older continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1.

It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of the plane are surrounded by arbitrarily large loops and every deterministic point is almost surely surrounded by a countably infinite family of nested loops with radii going to zero, and (3) an intermediate one, in which every deterministic point of the plane is almost surely surrounded by a largest loop and by a countably infinite family of nested loops with radii going to zero. We show how one can prove this using elementary arguments, with the help of known scaling relations for percolation. The trivial limit corresponds to subcritical and supercritical percolation, as well as to the case when the density p approaches the critical probability, p_c, sufficiently slowly as the lattice spacing is sent to zero. The second type corresponds to critical percolation and to a faster approach of p to p_c. The third, or near-critical, type of limit corresponds to an intermediate speed of approach of p to p_c. The fact that in the near-critical case a deterministic point is a.s. surrounded by a largest loop demonstrates the persistence of a macroscopic correlation length in the scaling limit and the absence of scale invariance.

We provide a representation for the scaling limit of the d = 2 critical Ising magnetization field as a (conformal) random field by using Schramm–Loewner Evolution clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical Fortuin–Kasteleyn clusters and the random field is a convergent sum of the area measures with random signs. Extensions to off-critical scaling limits, to d = 3, and to Potts models are also considered.

We provide a representation for the scaling limit of the d=2 critical Ising magnetization field as a (conformal) random field using SLE (Schramm-Loewner Evolution) clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. Extensions to off-critical scaling limits, to d=3 and to Potts models are also considered.

We present a review of the recent progress on percolation scaling limits in two dimensions. In particular, we will consider the convergence of critical crossing probabilities to Cardy's formula and of the critical exploration path to chordal SLE(6), the full scaling limit of critical cluster boundaries, and near-critical scaling limits.

We consider a type of dependent percolation introduced by Aizenman and Grimmett, who showed that certain "enhancements" of independent (Bernoulli) percolation, called essential, make the percolation critical probability strictly smaller. In this paper we first prove that, for two-dimensional enhancements with a natural monotonicity property, being essential is also a necessary condition to shift the critical point. We then show that (some) critical exponents and the scaling limit of crossing probabilities of a two-dimensional percolation process are unchanged if the process is subjected to a monotonic enhancement that is not essential. This proves a form of universality for all dependent percolation models obtained via a monotonic enhancement (of Bernoulli percolation) that does not shift the critical point. For the case of site percolation on the triangular lattice, we also prove a stronger form of universality by showing that the full scaling limit is not affected by any monotonic enhancement that does not shift the critical point.