Scaling Limits of Two-Dimensional Percolation: an Overview

Communications in Mathematical Physics, 2006

We use SLE 6 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.

Arxiv preprint math/0504036, 2005

Abstract: We use SLE (6) 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 ...

Journal of statistical physics, 2006

We analyze the geometry of scaling limits of near-critical 2D percolation, ie, for p= p c+ λδ 1/ν, with ν= 4/3, as the lattice spacing δ→ 0. Our proposed framework extends previous analyses for p= pc, based on SLE 6. It combines the continuum nonsimple loop ...

Journal of Statistical Physics, 2009

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

2015

This chapter is based on [10] with Rob van den Berg. We consider (near-)critical percolation on the square lattice. Let M n be the size of the largest open cluster contained in the box [−n, n] 2 , and let π(n) be the probability that there is an open path from O to the boundary of the box. It is well-known (see [17]) that for all 0 < a < b the probability that M n is smaller than an 2 π(n) and the probability that M n is larger than bn 2 π(n) are bounded away from 0 as n → ∞. It is a natural question, which arises for instance in the study of so-called frozenpercolation processes, if a similar result holds for the probability that M n is between an 2 π(n) and bn 2 π(n). By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is armative. The`sublinearity' of 1/π(n) appears to be essential for the argument. percolation and FK-Ising This chapter is based on [20] with Federico Camia and Demeter Kiss. Under some general assumptions we construct the scaling limit of open clusters and their associated counting measures in a class of two-dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice. We also provide conditional results for the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. Applications such as the scaling limit of the largest cluster in a bounded domain and a geometric representation of the magnetization eld for the critical Ising model are presented.

Physics Procedia

We summarize several decades of work in finding values for the percolation threshold p c for site percolation on the square lattice, the universal correction-to-scaling exponent Ω, and the susceptibility amplitude ratio C + /C − , in two dimensions. Recent studies have yielded the precise values p c = 0.59274602(4), Ω = 72/91 ≈ 0.791, and C + /C − = 161.5(2.0), resolving long-standing controversies about the last two quantities and verifying the widely used value p c = 0.592746 for the first.

Electronic Communications in Probability, 2012

We consider (near-)critical percolation on the square lattice. Let Mn be the size of the largest open cluster contained in the box [−n, n] 2 , and let π(n) be the probability that there is an open path from O to the boundary of the box. It is well-known (see [BCKS01]) that for all 0 < a < b the probability that Mn is smaller than an 2 π(n) and the probability that Mn is larger than bn 2 π(n) are bounded away from 0 as n → ∞. It is a natural question, which arises for instance in the study of so-called frozenpercolation processes, if a similar result holds for the probability that Mn is between an 2 π(n) and bn 2 π(n). By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is affirmative. The 'sublinearity' of 1/π(n) appears to be essential for the argument.

2015

We consider critical site percolation on the triangular lattice in the upper half-plane. Let $u_1, u_2$ be two sites on the boundary and $w$ a site in the interior of the half-plane. It was predicted by Simmons, Kleban and Ziff in a paper from 2007 that the ratio $\mathbb{P}(nu_1 \leftrightarrow nu_2 \leftrightarrow nw)^{2}\,/\,\mathbb{P}(nu_1 \leftrightarrow nu_2)\cdot\mathbb{P}(nu_1 \leftrightarrow nw)\cdot\mathbb{P}(nu_2 \leftrightarrow nw)$ converges to $K_F$ as $n \to \infty$, where $x\leftrightarrow y$ denotes the event that $x$ and $y$ are in the same open cluster, and $K_F$ is an explicitly known constant. Beliaev and Izyurov proved in a paper in 2012 an analog of this factorization in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for the probability $\mathbb{P}(nu_2 \leftrightarrow [nu_1,nu_1+s];\, nw \leftrightarrow [nu_1,nu_1+s])$, where $s>0$.

Physical Review E, 2003

We present Monte Carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 4 to 13 dimensions. For d < 6 they are preliminary, for d ≥ 6 they are between 20 to 10 4 times more precise than the best previous estimates. This was achieved by three ingredients: (i) simple and fast hashing which allowed us to simulate clusters of millions of sites on computers with less than 500 MB memory; (ii) a histogram method which allowed us to obtain information for several p values from a single simulation; and (iii) a variance reduction technique which is especially efficient at high dimensions where it reduces error bars by a factor up to ≈ 30 and more. Based on these data we propose a new scaling law for finite cluster size corrections.

Journal of Statistical Physics, 2003

Several formulas for crossing functions arising in the continuum limit of critical two-dimensional percolation models are studied. These include Watts&#39;s formula for the horizontal-vertical crossing probability and Cardy&#39;s new formula for the expected number of crossing clusters. It is shown that for lattices where conformal invariance holds, they simplify when the spatial domain is taken to be the interior of an equilateral triangle. The two crossing functions can be expressed in terms of an equianharmonic elliptic function with a triangular rotational symmetry. This suggests that rigorous proofs of Watts&#39;s formula and Cardy&#39;s new formula will be easiest to construct if the underlying lattice is triangular. The simplification in a triangular domain of Schramm&#39;s “bulk Cardy&#39;s formula” is also studied.