Federico Camia

Two-Dimensional Critical Percolation: The Full Scaling Limit

We use SLE6 paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice – that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.

The Scaling Limit Geometry of Near-Critical 2D Percolation

We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for p = pc+λδ1/ν, with ν = 4/3, as the lattice spacing δ → 0. Our proposed framework extends previous analyses for p = pc, based on SLE6. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (critical) loop process. The double points are exactly the continuum limits of “macroscopically pivotal” lattice sites and the marked ones are those that actually change state as λ varies. This structure is rich enough to yield a one-parameter family of near-critical loop processes and their associated connectivity probabilities as well as related processes describing, e.g., the scaling limit of 2D minimal spanning trees.

Continuum nonsimple loops and 2D critical percolation

Substantial progress has been made in recent years on the 2D critical percolation scaling limit and its conformal invariance properties. In particular, chordal SLE6(the Stochastic Loewner Evolution with parameter κ=6) was, in the work of Schramm and of Smirnov, identified as the scaling limit of the critical percolation “exploration process.” In this paper we use that and other results to construct what we argue is the fullscaling limit of the collection of allclosed contours surrounding the critical percolation clusters on the 2D triangular lattice. This random process or gas of continuum nonsimple loops in Bbb R2is constructed inductively by repeated use of chordal SLE6. These loops do not cross but do touch each other—indeed, any two loops are connected by a finite “path” of touching loops.

Planar Ising magnetization field I. Uniqueness of the critical scaling limit

The aim of this paper is to prove the following result. Consider the critical Ising model on the rescaled grid

Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits

a\mathbb{Z}^2

Ising (conformal) fields and cluster area measures

, then the renormalized magnetization field \[\Phi^a:=a^{15/8}\sum_{x\in a\mathbb{Z}^2}\sigma_x\delta_x,\] seen as a random distribution (i.e., generalized function) on the plane, has a unique scaling limit as the mesh size

A particular bit of universality : scaling limits of some dependent percolation models

a\searrow0

Conformal correlation functions in the Brownian loop soup

. The limiting field is conformally covariant.

Trivial, Critical and Near-critical Scaling Limits of Two-dimensional Percolation

In [CGN12], we proved that the renormalized critical Ising magnetization fields

Non-Backtracking Loop Soups and Statistical Mechanics on Spin Networks

\Phi^a:= a^{15/8} \sum_{x\in a\, \Z^2} \sigma_x \, \delta_x