It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE(6). We provide here a detailed proof, which relies on Smirnov's theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy's formula). The version of convergence to SLE(6) that we prove suffices for the Smirnov-Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops.
Building on the identification of the scaling limit of the critical percolation exploration process as a Schramm-Loewner Evolution, we derive a PDE characterization for the crossing probability of an annulus.
We review some of the recent progress on the scaling limit of two-dimensional critical percolation; in particular, the convergence of the exploration path to chordal SLE(6) and the "full" scaling limit of cluster interface loops. The results given here on the full scaling limit and its conformal invariance extend those presented previously. For site percolation on the triangular lattice, the results are fully rigorous. We explain some of the main ideas, skipping most technical details.
Let $Q$ be a free Boltzmann quadrangulation with simple boundary decorated by a critical ($p=3/4$) face percolation configuration. We prove that the chordal percolation exploration path on $Q$ between two marked boundary edges converges in the scaling limit to chordal SLE$_6$ on an independent $\sqrt{8/3}$-Liouville quantum gravity disk (equivalently, a Brownian disk). The topology of convergence is the Gromov-Hausdorff-Prokhorov-uniform topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces. We also obtain analogous scaling limit results for fa…
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. In the main body of the paper these results are proved while assuming, as argued by Schramm and Smirnov, that the percolation exploration path converges in distribution to the trace of chordal SLE(6). Then, i…
We set the foundation for a series of works aimed at proving strong relations between uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a bijective encoding of site-percolated planar triangulations by certain 2D lattice paths. Our bijection parallels in the discrete setting the \emph{mating-of-trees} framework of LQG and Schramm-Loewner evolutions (SLE) introduced by Duplantier, Miller, and Sheffield. Combining these two correspondences allows us to relate uniform site-percolated triangulations to $\sqrt{8/3}$-LQG and SLE$_6$. In particular, we establish the …
This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLE(k)), a one-parameter family of conformally invariant random curves discovered by Schramm (2000). The article is organized around the aim of proving the result, due to Smirnov (2001) and to Camia and Newman (2007), that the percolation exploration path…
Following the approach outlined in [26], convergence to SLE$_6$ of the Exploration Processes for the correlated bond-triangular type models studied in [11] is established. This puts the said models in the same universality class as the standard site percolation model on the triangular lattice [27]. In the context of these models, the result is proven for all domains with boundary Minkowski dimension less than two. Moreover, the proof of convergence applies in the context of general critical 2D percolation models and for general domains, under the stipulation that Cardy's Formula can be establi…
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 k=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 continuu…
We consider a particular type of $\sqrt{8/3}$-Liouville quantum gravity surface called a doubly marked quantum disk (equivalently, a Brownian disk) decorated by an independent chordal SLE$_6$ curve $η$ between its marked boundary points. We obtain descriptions of the law of the quantum surfaces parameterized by the complementary connected components of $η([0,t])$ for each time $t \geq 0$ as well as the law of the left/right $\sqrt{8/3}$-quantum boundary length process for $η$.
We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results and $SLE$ techniques, and in principle should provide a new approach to establishing conformal invariance of percolation.
We study the possible scaling limits of percolation interfaces in two dimensions on the triangular lattice. When one lets the percolation parameter p(N) vary with the size N of the box that one is considering, three possibilities arise in the large-scale limit. It is known that when p(N) does not converge to 1/2 fast enough, then the scaling limits are degenerate, whereas if p(N) - 1 / 2 goes to zero quickly, the scaling limits are SLE(6) as when p=1/2. We study some properties of the (non-void) intermediate regime where the large scale behavior is neither SLE(6) nor degenerate. We prove that …
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 prove that the interface of critical site percolation on the triangular lattice converges to SLE$_6$ in its natural parametrization, where the discrete interface is parametrized such that each edge is crossed in one unit of time, while the limiting curve is parametrized by the $7/4$-dimensional Minkowski content. We also prove that the scaling limit of counting measure on the pivotal points, which was proved to exist by Garban, Pete, and Schramm (2013), is the $3/4$-dimensional Minkowski content up to a deterministic multiplicative constant.
We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK-Ising percolation to chordal SLE$_κ( κ-6)$ with $κ=16/3$. Our proof follows the classical excursion-construction of SLE$_κ(κ-6)$ processes in the continuum and we are thus led to introduce suitable cut-off stopping times in order to analyse the behaviour of the driving function of the discrete system when Dobrushin boundary condition collapses to a single point. Our proof is very different from [KS15, KS16] as it only relies on the convergence to the chordal SLE$_κ$ process in Dobrushin…
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.
We define radial exploration processes from $a$ to $b$ and from $b$ to $a$ in a domain $D$ of hexagons where $a$ is a boundary point and $b$ is an interior point. We prove the reversibility: the time-reversal of the process from $b$ to $a$ has the same distribution as the process from $a$ to $b$. We show the scaling limit of such an exploration process is a radial SLE$_6$ in $D$. As a consequence, the distribution of the last hitting point with the boundary of any radial SLE$_6$ is harmonic measure. We also prove the scaling limit of a similar exploration process defined in the full complex pl…
Recent works have shown that random triangulations decorated by critical ($p=1/2$) Bernoulli site percolation converge in the scaling limit to a $\sqrt{8/3}$-Liouville quantum gravity (LQG) surface (equivalently, a Brownian surface) decorated by SLE$_6$ in two different ways: 1. The triangulation, viewed as a curve-decorated metric measure space equipped with its graph distance, the counting measure on vertices, and a single percolation interface converges with respect to a version of the Gromov-Hausdorff topology. 2. There is a bijective encoding of the site-percolated triangulation by means …
We study the diffusion front for a natural two-dimensional model where many particles starting at the origin diffuse independently. It turns out that this model can be described using properties of near-critical percolation, and provides a natural example where critical fractal geometries spontaneously arise.
We show that the laws of scaling limits of nearcritical percolation exploration paths with different parameters are singular with respect to each other. This generalises a result of Nolin and Werner, using a similar technique. As a corollary, the singularity can even be detected from an infinitesimal initial segment. Moreover, nearcritical scaling limits of exploration paths are mutually singular under scaling maps.
Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on $\mathbb{Z}^2$ is $\mathrm{SLE}_2$. We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into $\mathbb{C}$ so that edges do not cross one another). We show that if the scaling limit of the random walk is planar Brownian motion, then the scaling limit of its loop-erasure is $\mathrm{SLE}_2$. Our main contribution is showing that for such graphs, the discrete Poisson kernel can be approximated by the continuous one. One example is the infinite component of …
By analogy with Carleson's observation on Cardy's formula describing crossing probabilities for the scaling limit of critical percolation, we exhibit ``privileged geometries'' for Stochastic Loewner Evolutions with various parameters, for which certain hitting distributions are uniformly distributed. Consequences for limiting probabilities of events concerning various critical plane discrete models are then examined.
In this article we show the convergence of a loop ensemble of interfaces in the FK Ising model at criticality, as the lattice mesh tends to zero, to a unique conformally invariant scaling limit. The discrete loop ensemble is described by a canonical tree glued from the interfaces, which then is shown to converge to a tree of branching SLEs. The loop ensemble contains unboundedly many loops and hence our result describes the joint law of infinitely many loops in terms of SLE type processes, and the result gives the full scaling limit of the FK Ising model in the sense of random geometry of the …
In the second article of this series, we establish the convergence of the loop ensemble of interfaces in the random cluster Ising model to a conformal loop ensemble (CLE) --- thus completely describing the scaling limit of the model in terms of the random geometry of interfaces. The central tool of the present article is the convergence of an exploration tree of the discrete loop ensemble to a branching SLE$(16/3,-2/3)$. Such branching version of the Schramm's SLE not only enjoys the locality property, but also arises logically from the Ising model observables.
Following the approach outlined in [17], convergence to SLE$_6$ of the Exploration Processes for the correlated bond-triangular type models studied in [6] is established in [3] and the present work. In this second part, we focus on establishing Cardy's Formula for general domains.
We review some of the results that have been derived in the last years on conformal invariance, scaling limits and properties of some two-dimensional random curves. In particular, we describe the intuitive ideas that lead to the definition of the Schramm-Loewner evolutions SLE, we define these objects, study its various properties, show how to compute (probabilities, critical exponents) using SLE, relate SLE to planar Brownian motions (i.e. the determination of the critical exponents), planar self-avoiding walks, critical percolation, loop-erased random walks and uniform spanning trees.
The peeling process, which describes a step-by-step exploration of a planar map, has been instrumental in addressing percolation problems on random infinite planar maps. Bond and face percolation on maps with faces of arbitrary degree are conveniently studied via so-called lazy-peeling explorations. During such explorations distinct vertices on the exploration contour may at later stage be identified, making the process less suited to the study of site percolation. To tackle this situation and to explicitly identify site-percolation thresholds, we come back to the alternative "simple" peeling …
The Ising model at inverse temperature $β$ and zero external field can be obtained via the Fortuin-Kasteleyn (FK) random-cluster model with $q=2$ and density of open edges $p=1-e^{-β}$ by assigning spin +1 or -1 to each vertex in such a way that (1) all the vertices in the same FK cluster get the same spin and (2) +1 and -1 have equal probability. We generalize the above procedure by assigning spin +1 with probability $r$ and -1 with probability $1-r$, with $r \in [0,1]$, while keeping condition (1). For fixed $β$, this generates a dependent (spin) percolation model with parameter $r$. We show…
In this paper we present the proof of the convergence of the critical bond percolation exploration process on the square lattice to the trace of SLE$_{6}$. This is an important conjecture in mathematical physics and probability. The case of critical site percolation on the hexagonal lattice was established in the seminal work of Smirnov via proving Cardy's formula. Our proof uses a series of transformations and conditioning to construct a pair of paths: the $+\partial$CBP and the $-\partial$CBP. The convergence in the site percolation case on the hexagonal lattice allows us to obtain certain e…
This work is the first in a series of papers devoted to the construction and study of scaling limits of dynamical and near-critical planar percolation and related objects like invasion percolation and the Minimal Spanning Tree. We show here that the counting measure on the set of pivotal points of critical site percolation on the triangular grid, normalized appropriately, has a scaling limit, which is a function of the scaling limit of the percolation configuration. We also show that this limit measure is conformally covariant, with exponent 3/4. Similar results hold for the counting measure o…
It is known that Schramm-Loewner Evolutions (SLEs) have a.s. frontier points if $κ>4$ and a.s. cutpoints if $4<κ<8$. If $κ>4$, an appropriate version of $\SLE(κ)$ has a renewal property: it starts afresh after visiting its frontier. Thus one can give an excursion decomposition for this particular $\SLE(κ)$ ``away from its frontier''. For $4<κ<8$, there is a two-sided analogue of this situation: a particular version of $\SLE(κ)$ has a renewal property w.r.t its cutpoints; one studies excursion decompositions of this $\SLE$ ``away from its cutpoints''. For $κ=6$, this overlaps Virág's results on…
SLE is a random growth process based on Loewner's equation with driving parameter a one-dimensional Brownian motion running with speed $κ$. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions. The present paper attempts a first systematic study of SLE. It is proved that for all $κ\ne 8$ the SLE trace is a path; for $κ\in[0,4]$ it is a simple path; for $κ\in(4,8)$ it is a self-intersecting path; and for $κ>8$ it is …
We prove that, after centering and diffusively rescaling space and time, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on the space-time lattice Z^2_{even}:={(x,i) in Z^2: x+i is even} converges in distribution to the Brownian web. This proves a conjecture of Wu and Zhang. Our key observation is that each rightmost infinite open path can be approximated by a percolation exploration cluster, and different exploration clusters evolve independently before they intersect.
We study limit laws for simple random walks on supercritical long range percolation clusters on $\Z^d, d \geq 1$. For the long range percolation model, the probability that two vertices $x, y$ are connected behaves asymptotically as $\|x-y\|_2^{-s}$. When $s\in(d, d+1)$, we prove that the scaling limit of simple random walk on the infinite component converges to an $α$-stable Lévy process with $α= s-d$ establishing a conjecture of Berger and Biskup. The convergence holds in both the quenched and annealed senses. In the case where $d=1$ and $s>2$ we show that the simple random walk converges to…
We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE(6) and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We explain how these marked loops should yield continuum versions of near-critical percolation, dynamical percolation, minimal spanning trees and related plane filling curves, and invasion percolation. We show that this yields for some of the continuum objects a conformal covariance property that generalizes the conformal invariance of critical systems. It is an op…
We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for $p=p_c+λδ^{1/ν}$, with $ν=4/3$, as the lattice spacing $δ\to 0$. Our proposed framework extends previous analyses for $p=p_c$, based on $SLE_6$. 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 struc…
We prove that crossing probabilities for the critical planar Ising model with free boundary conditions are conformally invariant in the scaling limit, a phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin. We do so by establishing the convergence of certain exploration processes towards SLE$(3,\frac{-3}2,\frac{-3}2)$. We also construct an exploration tree for free boundary conditions, analogous to the one introduced by Sheffield.
Let L_n denote the lowest crossing of the 2n \times 2n square box B(n) centered at the origin for critical site percolation on Z^2 or critical site percolation on the triangular lattice imbedded in Z^2, and denote by Q_n the set of pivotal sites along this crossing. On the event that a pivotal site exists, denote the minimum height that a pivotal site attains above the bottom of B(n) by M_n:= min{m:(x,-n+m)\in Q_n for some -n\le x\le n}. Else, define M_n = 2n. We prove that P(M_n < m) \asymp m/n, uniformly for 1\le m\le n. This relation extends Theorem 1 of van den Berg and Jarai (2003) who ha…
We study "tricolor percolation" on the regular tessellation of R^3 by truncated octahedra, which is the three-dimensional analog of the hexagonal tiling of the plane. We independently assign one of three colors to each cell according to a probability vector p = (p_1, p_2, p_3) and define a "tricolor edge" to be an edge incident to one cell of each color. The tricolor edges form disjoint loops and/or infinite paths. These loops and paths have been studied in the physics literature, but little has been proved mathematically. We show that each p belongs to either the compact phase (in which the l…
Following the approach outlined in [18], convergence to SLE6 of the Exploration Processes for the correlated bond-triangular type models studied in [7] is established. This puts the said models in the same universality class as the standard site percolation model on the triangular lattice [19]. The result is proven for all domains with boundary (upper) Minkowski dimension less than two. Moreover, the proof of convergence applies in the general context of critical 2D percolation models, under the stipulation that Cardy's Formula can be established.
We consider metric graph Gaussian free field (GFF) defined on polygons of $δ\mathbb{Z}^2$ with alternating boundary data. The crossing probabilities for level-set percolation of metric graph GFF have scaling limits. When the boundary data is well-chosen, the scaling limits of crossing probabilities can be explicitly constructed as "fusion" of multiple SLE$_4$ pure partition functions.
We study the giant component problem slightly above the critical regime for percolation on Poissonian random graphs in the scale-free regime, where the vertex weights and degrees have a diverging second moment. Critical percolation on scale-free random graphs have been observed to have incredibly subtle features that are markedly different compared to those in random graphs with converging second moment. In particular, the critical window for percolation depends sensitively on whether we consider single- or multi-edge versions of the Poissonian random graph. In this paper, and together with ou…
Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has its roots in string theory and conformal field theory from the 1980s and 1990s. The second is the Brownian map, which has its roots in planar map combinatorics from the 1960s together with recent scaling limit results. This article surveys a series of works with Sheffield in which it is shown that Liouville quantum gravity (LQG) with parameter $γ=\sqrt{8/3}$ is equivalent to the Brownian map. We also briefly describe a series of works with …
In a groundbreaking work, Duplantier, Miller and Sheffield showed that subcritical Liouville quantum gravity (LQG) coupled with Schramm-Loewner evolutions (SLE) can be described by the mating of two continuum random trees. In this paper, we consider the counterpart of their result for critical LQG and SLE, i.e., for the case when $γ^2=κ=16/κ=4$. We prove that as one sends $κ\downarrow 4$ in the subcritical setting, the space-filling SLE$_κ$ in a disk degenerates to the CLE$_4$ exploration introduced by Werner and Wu, along with a collection of i.i.d.\ coin tosses indexed by the branch points o…
Making use of a recent complete calculation of a chiral six-point correlation function C(z) in a rectangle we calculate various quantities of interest for percolation (SLE parameter κ= 6) and many other two-dimensional critical points. In particular, we specify the density at z of critical clusters conditioned to touch either or both vertical sides of the rectangle, with these sides 'wired,' i.e. constrained to be in a single cluster, and the horizontal sides free. These quantities probe the structure of various cluster configurations, including those that contribute to the crossing probabilit…
We study the alternating $k$-arm incipient infinite cluster (IIC) of site percolation on the triangular lattice $\mathbb{T}$. Using Camia and Newman's result that the scaling limit of critical site percolation on $\mathbb{T}$ is CLE$_6$, we prove the existence of the scaling limit of the $k$-arm IIC for $k=1,2,4$. Conditioned on the event that there are open and closed arms connecting the origin to $\partial \mathbb{D}_R$, we show that the winding number variance of the arms is $(3/2+o(1))\log R$ as $R\rightarrow \infty$, which confirms a prediction of Wieland and Wilson (2003). Our proof uses…
We study two closely related processes on the triangular lattice: frozen percolation, where connected components of occupied vertices freeze (they stop growing) as soon as they contain at least $N$ vertices, and forest fire processes, where connected components burn (they become entirely vacant) at rate $ζ> 0$. In this paper, we prove that when the density of occupied sites approaches the critical threshold for Bernoulli percolation, both processes display a striking phenomenon: the appearance of near-critical "avalanches". More specifically, we analyze the avalanches, all the way up to the na…
