The scaling limit geometry of near-critical 2D percolation

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 ...

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.

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

Statistica Neerlandica, 2008

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.

Journal of Statistical Physics, 2004

Substantial progress has been made in recent years on the 2D critical percolation scaling limit and its conformal invariance properties. In particular, chordal SLE 6 (the Stochastic Löwner 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 full scaling limit of the collection of all closed contours surrounding the critical percolation clusters on the 2D triangular lattice. This random process or gas of continuum nonsimple loops in R 2 is constructed inductively by repeated use of chordal SLE 6. These loops do not cross but do touch each other-indeed, any two loops are connected by a finite "path" of touching loops.

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.

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.

Physical Review E, 2014

We derive the critical nearest-neighbor connectivity g n as 3/4, 3(7 − 9p tri c)/4(5 − 4p tri c), and 3(2 + 7p tri c)/ 4(5 − p tri c) for bond percolation on the square, honeycomb, and triangular lattice, respectively, where p tri c = 2 sin(π/18) is the percolation threshold for the triangular lattice, and confirm these values via Monte Carlo simulations. On the square lattice, we also numerically determine the critical next-nearest-neighbor connectivity as g nn = 0.687 500 0(2), which confirms a conjecture by Mitra and Nienhuis [J. Stat. Mech. (2004) P10006], implying the exact value g nn = 11/16. We also determine the connectivity on a free surface as g surf n = 0.625 000 1(13) and conjecture that this value is exactly equal to 5/8. In addition, we find that at criticality, the connectivities depend on the linear finite size L as ∼ L yt −d , and the associated specific-heat-like quantities C n and C nn scale as ∼ L 2yt −d ln(L/L 0), where d is the lattice dimensionality, y t = 1/ν the thermal renormalization exponent, and L 0 a nonuniversal constant. We provide an explanation of this logarithmic factor within the theoretical framework reported recently by Vasseur et al.

Physical Review C, 1997

We examine the average cluster distribution as a function of lattice probability for a very small (Lϭ6) lattice and determine the scaling function of three-dimensional percolation. The behavior of the second moment, calculated from the average cluster distribution of Lϭ6 and Lϭ63 lattices, is compared to power-law behavior predicted by the scaling function. We also examine the finite-size scaling of the critical point and the size of the largest cluster at the critical point. This analysis leads to estimates of the critical exponent and the ratio of critical exponents ␤/. ͓S0556-2813͑97͒02703-9͔