Related papers: N-sided Radial Schramm-Loewner Evolution
We find optimal (up to constant) bounds for the following measures for the regularity of the Schramm-Loewner evolution (SLE): variation regularity, modulus of continuity, and law of the iterated logarithm. For the latter two we consider the…
We disclose the origin of anisotropic percolation perimeters in terms of the Stochastic Loewner Evolution (SLE) process. Precisely, our results from extensive numerical simulations indicate that the perimeters of multi-layered and directed…
We study a generalization of the Schramm-Loewner evolution loop measure to pairs of non-intersecting Jordan curves on the Riemann sphere. We also introduce four equivalent definitions for a two-loop Loewner potential: respectively…
In recent work we have shown that loop-erased random walk (LERW) connecting two boundary points of a domain converges to the chordal Schramm-Loewner evolution (SLE(2)) in the sense of curves parametrized by Minkowski content. In this note…
This work introduces a construction of conformal processes that combines the theory of branching processes with chordal Loewner evolution. The main novelty lies in the choice of driving measure for the Loewner evolution: given a finite…
We numerically show that the statistical properties of the shortest path on critical percolation clusters are consistent with the ones predicted for Schramm-Loewner evolution (SLE) curves for $\kappa=1.04\pm0.02$. The shortest path results…
The backward chordal Schramm-Loewner Evolution naturally defines a conformal welding homeomorphism of the real line. We show that this homeomorphism is invariant under the automorphism $x\mapsto -1/x$, and conclude that the associated…
We study the Loewner evolution whose driving function is $W_t = B_t^1 + i B_t^2$, where $(B^1,B^2)$ is a pair of Brownian motions with a given covariance matrix. This model can be thought of as a generalization of Schramm-Loewner evolution…
The Rohde--Schramm theorem states that Schramm--Loewner Evolution with parameter $\kappa$ (or SLE$_\kappa$ for short) exists as a random curve, almost surely, if $\kappa \neq 8$. Here we give a new and concise proof of the result, based on…
Loop-erased random walk and it's scaling limit, Schramm--Loewner evolution, have found numerous applications in mathematics and physics. We present a 2 dimensional analogue of LERW, the loop erased random surface. We do this by defining a 2…
Schramm-Loewner Evolutions (SLEs) describe a one-parameter family of growth processes in the plane that have particular conformal invariance properties. For instance, SLE can define simple random curves in a simply connected domain. In this…
In previous work [AHP24], we proved a finite-time large deviation principle in the Hausdorff metric for multiradial Schramm-Loewner evolution, SLE$(\kappa)$, as $\kappa \to 0$, with good rate function being the multiradial Loewner energy.…
The Schramm-Loewner evolution (SLE) describes the continuum limit of domain walls at phase transitions in two dimensional statistical systems. We consider here the SLEs in the self-dual Z(N) spin models at the critical point. For N=2 and…
SLE is a random growth process based on Loewner's equation with driving parameter a one-dimensional Brownian motion running with speed $\kappa$. This process is intimately connected with scaling limits of percolation clusters and with the…
Stochastic Loewner evolution (SLE) is a differential equation driven by a one-dimensional Brownian motion (BM), whose solution gives a stochastic process of conformal transformation on the upper half complex-plane $\H$. As an evolutionary…
We consider multiple radial SLE as the number of curves tends to infinity. We give conditions that imply the tightness of the associated processes given by the Loewner equation. In the case of equal weights, the infinite-slit limit is…
Existence of Loewner trace is revisited. We identify finite energy paths (the "skeleton of Wiener measure") as natural class of regular drivers for which we find simple and natural estimates in terms of their (Cameron--Martin) norm.…
Schramm--Loewner evolution (SLE) has been one of the central topics in the probabilistic study of two-dimensional critical systems. It is a random curve in two dimensions to which a cluster interface in a critical lattice system is…
This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions…
Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models.…