Related papers: Exploration processes and SLE$_6$
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 aim at finding the reversal of radial SLE and proving the reversibility of whole-plane SLE. For this purpose, we define annulus SLE$(\kappa,\Lambda)$ processes in doubly connected domains with one marked boundary point. We derive some…
We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK-Ising percolation to chordal SLE$_\kappa( \kappa-6)$ with $\kappa=16/3$. Our proof follows the classical excursion-construction of…
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 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 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…
Suppose that D is a planar Jordan domain and x and y are distinct boundary points of D. Fix \kappa \in (4,8) and let \eta\ be an SLE_\kappa process from x to y in D. We prove that the law of the time-reversal of \eta is, up to…
We consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its…
A Bessel excursion is a Bessel process that begins at the origin and first returns there at some given time $T$. We study the distribution of the area under such an excursion, which recently found application in the context of laser…
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 extend the peeling exploration introduced in arxiv:1506.01590 to the setting of Boltzmann planar maps coupled to a rigid $O(n)$ loop model. Its law is related to a class of discrete Markov processes obtained by confining random walks to…
We consider multiple radial SLE curves with various time parameterizations and possible spiraling behavior. We construct them by tilting independent radial SLEs with a suitable local martingale, generalizing the earlier construction by…
We consider an infinite-dimensional stochastic clustering model on $\mathbb{R}$. In discrete time, each point of a unit-intensity simple point process moves halfway toward either of its left or right neighbors, chosen uniformly at random.…
We derive some geometric properties of chordal SLE$(\kappa;\vec{\rho})$ processes. Using these results and the method of coupling two SLE processes, we prove that the outer boundary of the final hull of a chordal SLE$(\kappa;\vec{\rho})$…
The harmonic explorer is a random grid path. Very roughly, at each step the harmonic explorer takes a turn to the right with probability equal to the discrete harmonic measure of the left-hand side of the path from a point near the end of…
This paper introduces the annulus SLE$_\kappa$ processes in doubly connected domains. Annulus SLE$_6$ has the same law as stopped radial SLE$_6$, up to a time-change. For $\kappa\not=6$, some weak equivalence relation exists between annulus…
Given a simply connected planar domain D, distinct points x,y \in \partial D, and \kappa >0, the Schramm-Loewner evolution SLE_\kappa is a random continuous non-self-crossing path in the closure of D from x to y. The…
We consider the behavior of spatial point processes when subjected to a class of linear transformations indexed by a variable T. It was shown in Ellis [Adv. in Appl. Probab. 18 (1986) 646-659] that, under mild assumptions, the transformed…
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…
Starting with a transient irreducible diffusion process $X^0$ on a locally compact separable metric space $(D, d)$, one can construct a canonical symmetric reflected diffusion process $\bar X$ on a completion $D^*$ of $(D, d)$ through the…