Related papers: Pivotal, cluster and interface measures for critic…
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…
We study site percolation on Angel & Schramm's uniform infinite planar triangulation. We compute several critical and near-critical exponents, and describe the scaling limit of the boundary of large percolation clusters in all regimes…
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…
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…
We study gradient percolation for site percolation on the triangular lattice. This is a percolation model where the percolation probability depends linearly on the location of the site. We prove the results predicted by physicists for this…
We consider critical percolation on the triangular lattice in a bounded simply connected domain with boundary conditions that force an interface between two prescribed boundary points. We say the interface forms a "near-loop" when it comes…
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…
We prove existence of the scaling limit of the invasion percolation cluster (IPC) on a regular tree. The limit is a random real tree with a single end. The contour and height functions of the limit are described as certain diffusive…
The talk presented at ICMP 97 focused on the scaling limits of critical percolation models, and some other systems whose salient features can be described by collections of random lines. In the scaling limit we keep track of features seen…
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…
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…
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…
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…
We prove Tsirelson's conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential…
This is a short (and somewhat informal) contribution to the proceedings of the XVIth International Congress on Mathematical Physics, Prague, 2009, written up by the second author. We describe how the recent proof of the existence and…
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…
We extend Smirnov's proof of the existence and conformal invariance of the scaling limit of critical site-percolation on the triangular lattice to particular sequences of periodic graphs with more arbitrary large-scale structure, obtained…
We study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either…
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…
We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for $p=p_c+\lambda\delta^{1/\nu}$, with $\nu=4/3$, as the lattice spacing $\delta \to 0$. Our proposed framework extends previous analyses for $p=p_c$, based…