Related papers: Intersection probabilities for a chordal SLE path …
A path system $\mathscr{P}$ in a graph $G=(V,E)$ is a collection of paths, with exactly one path between any two vertices in $V$. A path system is said to be consistent if it is closed under subpaths. We say that a path system $\mathscr{P}$…
We prove that the chordal SLE$_{\kappa}$ trace is reversible for $\kappa\in(0,4]$.
An E_0-semigroup acting on B(H) is called pure if the intersection of the ranges $\alpha_t(B(H))$, $t>0$, is the algebra of scalars. We determine all pure E_0-semigroups which have a weakly continuous invariant state $\omega$ and which are…
We show that, for $\kappa\in(0,8)$, the integral of the laws of two-sided radial SLE$_\kappa$ curves through different interior points against a measure with SLE$_\kappa$ Green function density is the law of a chordal SLE$_\kappa$ curve,…
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…
The numerical condition of the problem of intersecting a fixed $m$-dimensional irreducible complex projective variety $Z\subseteq\mathbb{P}^n$ with a varying linear subspace $L\subseteq\mathbb{P}^n$ of complementary dimension $s=n-m$ is…
In this paper we consider the problem of embedding almost-spanning, bounded degree graphs in a random graph. In particular, let $\Delta\geq 5$, $\varepsilon > 0$ and let $H$ be a graph on $(1-\varepsilon)n$ vertices and with maximum degree…
Off-shell supersymmetry, which restricts sparticles to appear only off-shell, solves the gauge hierarchy problem and unifies the gauge couplings in the usual way. Without introducing any new interactions or exacerbating the naturalness,…
Assume two finite families $\mathcal A$ and $\mathcal B$ of convex sets in $\mathbb{R}^3$ have the property that $A\cap B\ne \emptyset$ for every $A \in \mathcal A$ and $B\in \mathcal B$. Is there a constant $\gamma >0$ (independent of…
This work derives an upper bound on the maximum cardinality of a family of graphs on a fixed number of vertices, in which the intersection of every two graphs in that family contains a subgraph that is isomorphic to a specified graph H.…
The hole probability, i.e., the probability that a region is void of particles, is a benchmark of correlations in many body systems. We compute analytically this probability $P(R)$ for a spherical region of radius $R$ in the case of $N$…
Suppose $S$ is a closed, oriented surface of genus at least two. This paper investigates the geometry of the homology multicurve complex, $\mathcal{HC}(S,\alpha)$, of $S$; a complex closely related to complexes studied by…
We study graphs that are formed by independently-positioned needles (i.e., line segments) in the unit square. To mathematically characterize the graph structure, we derive the probability that two line segments intersect and determine…
In this paper, we construct a compact subset of the real plane whose path-connectedness equivalence relation is Borel bireducible to a nonsmooth hyperfinite Borel equivalence relation. This answers a question of \cite{bec98}.
The scaling limit of planar loop-erased random walks is described by a stochastic Loewner evolution with parameter kappa=2. In this note SLE(2) in the upper half-plane H minus a simply-connected compact subset K of H is studied. As a main…
In the previous article (S. Matsutani and Y. Shimosako and Y. Wang, Physica A \bf{391} (2012) 5802-5809) we numerically investigated an electric potential problem with high contrast local conductivities ($\gamma_0$ and $\gamma_1$,…
For an even set of points in the plane, choose a max-sum matching, that is, a perfect matching maximizing the sum of Euclidean distances of its edges. For each edge of the max-sum matching, consider the ellipse with foci at the edge's…
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…
We consider a model of two (fully) compact polymer chains, coupled through an attractive interaction. These compact chains are represented by Hamiltonian paths (HP), and the coupling favors the existence of common bonds between the chains.…
Let $P : \Sigma \rightarrow S$ be a finite degree covering map between surfaces. Rafi and Schleimer show that there is an induced quasi-isometric embedding $\Pi : \mathcal{C}(S) \rightarrow \mathcal{C}(\Sigma)$ between the associated curve…