Related papers: Intersection probabilities for a chordal SLE path …

We show that, for $\kappa\le 4$, the integral of the two-sided radial SLE$_\kappa$ measures over all interior points is chordal SLE$_\kappa$ biased by the path's natural length, which is its $(1+\kappa/8)$-dimensional Minkowski content.

Linear structural equation models postulate noisy linear relationships between variables of interest. Each model corresponds to a path diagram, which is a mixed graph with directed edges that encode the domains of the linear functions and…

The focus of this paper is on a thin obstacle problem where the obstacle is defined on the intersection between a hyper-plane $\Gamma$ in $\mathbb{R}^n$ and a periodic perforation $\mathcal{T}_\varepsilon$ of $\mathbb{R}^n$, depending on a…

A covering path for a finite set $P$ of points in the plane is a polygonal path such that every point of $P$ lies on a segment of the path. The vertices of the path need not be at points of $P$. A covering path is plane if its segments do…

We give sharp, uniform estimates for the probability that a random walk of n steps on the reals avoids a half-line [y,infinity) given that it ends at the point x. The estimates hold for general continuous or lattice distributions provided…

Schramm-Loewner evolution (SLE$_\kappa$) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by $\sqrt{\kappa}$ times Brownian motion. This yields a (half-plane) valued random field $\gamma = \gamma…

We give a new, short computation of pairing probabilities for multiple chordal interfaces in the critical Ising model, the harmonic explorer, and for multiple level lines of the Gaussian free field. The core of the argument are the known…

We study the topology of $SLE$ curves for $\kappa > 4$. More precisely, we show that, a.s., there is no homeomorphism $\Phi: \overline{\mathbb{H}} \rightarrow \overline{\mathbb{H}}$, taking the range of one independent $SLE$ curve to…

We show the intersection of a compact almost complex subvariety of dimension $4$ and a compact almost complex submanifold of codimension $2$ is a $J$-holomorphic curve. This is a generalization of positivity of intersections for…

This paper is concerned with the asymptotics of resonances in the semiclassical limit $h\to 0^+$ for $2\times 2$ matrix Schr\"odinger operators in one dimension. We study the case where the two underlying classical Hamiltonian trajectories…

We prove that for every $\varepsilon > 0$ there is $c_0$ such that if $G\sim G(n,c/n)$, $c\ge c_0$, then with high probability $G$ can be covered by at most $(1+\varepsilon)\cdot \frac{1}{2}ce^{-c} \cdot n$ vertex disjoint paths, which is…

For a compact surface $S$ with constant negative curvature $-\kappa$ (for some $\kappa>0$) and genus $g\geq2$, we show that the tails of the distribution of $i(\alpha,\beta)/l(\alpha)l(\beta)$ (where $i(\alpha,\beta)$ is the intersection…

We consider global 2-SLE$_{\kappa}$ $(\eta_1, \eta_2)$ in a topological rectangle with $\kappa\in (4,8)$. We derive the law of a random hitting point of the curves and show that, conditional on this random hitting point, the pair of two…

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…

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…

We consider an ensemble of $N$ discrete nonintersecting paths starting from equidistant points and ending at consecutive integers. Our first result is an explicit formula for the correlation kernel that allows us to analyze the process as…

Given $\omega\ge 1$, let $Z^2_{(\omega)}$ be the graph with vertex set $Z^2$ in which two vertices are joined if they agree in one coordinate and differ by at most $\omega$ in the other. (Thus $Z^2_{(1)}$ is precisely $Z^2$.) Let…

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for…

Whole-plane SLE$_\kappa$ is a random fractal curve between two points on the Riemann sphere. Zhan established for $\kappa \leq 4$ that whole-plane SLE$_\kappa$ is reversible, meaning invariant in law under conformal automorphisms swapping…

We prove that, for $\kappa\in(0,4)$ and $\rho\ge (\kappa-4)/2$, the chordal SLE$(\kappa;\rho)$ trace started from $(0;0^+)$ or $(0;0^-)$ satisfies the reversibility property. And we obtain the equation for the reversal of the chordal…