Searcharxiv⌕ Search

arXiv subjects

Oded Schramm

Publications and source records attributed to Oded Schramm.

69 records · Page 4Linked to original sources
arXiv2000

Values of Brownian intersection exponents II: Plane exponents

We derive the exact value of intersection exponents between planar Brownian motions or random walks, confirming predictions from theoretical physics by Duplantier and Kwon. Let B and B' be independent Brownian motions (or simple random walks) in the plane, started from distinct points. We prove that the probability that the paths B[0,t] and B'[0,t] do not intersect decays like t^{-5/8}. More precisely, there is a constant c>0 such that if |B(0) - B'(0)| =1, for all t \ge 1, c^{-1} t^{-5/8} \le ¶[ B[0,t] \cap B'[0,t] = \emptyset ] \le c t^{-5/8}. One consequence is that the set of cut-points of B[0,1] has Hausdorff dimension 3/4 almost surely. The values of other exponents are also derived. Using an analyticity result, which is to be established in a forthcoming paper, it follows that the Hausdorff dimension of the outer boundary of B[0,1] is 4/3, as conjectured by Mandelbrot. The proofs are based on a study of SLE_6 (stochastic Loewner evolution with parameter 6), a recently discovered process which conjecturally is the scaling limit of critical percolation cluster boundaries. The exponents of SLE_6 are calculated, and they agree with the physicists' predictions for the exponents for critical percolation and self-avoiding walks. From the SLE_6 exponents the Brownian intersection exponents are then derived.

math.PR↗
arXiv2000

Percolation in the Hyperbolic Plane

This is a study of percolation in the hyperbolic plane and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and Poisson-Voronoi-Bernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first phase, $p\in(0, p_c]$, there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, $p\in(p_c,p_u)$, there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, $p\in [p_u,1)$, there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of $p_c$ in the Poisson-Voronoi-Bernoulli percolation process on the intensity of the underlying Poisson process.

math.PR↗
arXiv2000

Values of Brownian intersection exponents I: Half-plane exponents

This paper proves conjectures originating in the physics literature regarding the intersection exponents of Brownian motion in a half-plane. For instance, suppose that B and B' are two independent planar Brownian motions started from distinct points in a half-plane H. Then, as $t \to \infty$, $$ P [ B[0,t] \cap B'[0,t] = \emptyset, B[0,t] \cup B'[0,t] \subset H ] = t^{-5/3 + o(1)}.$$ The proofs use ideas and tools developped by the authors in previous papers. We prove that one of the stochastic Loewner evolution processes (with parameter 6, that we will call SLE_6 and which has been conjectured to correspond to the scaling limit of critical percolation cluster boundaries) satisfies the ``conformal restriction property''. We establish a generalization of Cardy's formula (for crossings of a rectangle by a percolation cluster) for SLE_6, from which the exact values of intersection exponents for SLE_6 followd. Since this process satisfies the conformal restriction property, the Brownian intersection exponents can be determined from the SLE_6 intersection exponents. Results about intersection exponents in the whole plane appear in subsequent papers.

math.PR↗
arXiv2000

The Dimension of the Planar Brownian Frontier is 4/3

In a series of recent preprints, we have proven that with probability one the Hausdorff dimension on the outer boundary of planar Brownian motion is 4/3, confirming a conjecture by Mandelbrot. It is also shown that the Hausdorff dimension of the set of cut points is a.s. 3/4. The present paper is an expository outline of the arguments involved in the proof of these and related results.

math.PR↗
arXiv2000

Analyticity of intersection exponents for planar Brownian motion

We show that the intersection exponents for planar Brownian motions are analytic. More precisely, let $B$ and $B'$ be independent planar Brownian motions started from distinct points, and define the exponent $ξ(1, λ)$ by $$ E[P[B[0,t] \cap B'[0,t] = \emptyset | B[0,t]]^λ] \approx t^{-ξ(1, λ)/2}, t \to \infty. $$ Then the mapping $λ\mapsto ξ(1, λ)$ is real analytic in $(0,\infty)$. The same result is proved for the exponents $ξ(k, λ)$ where $k$ is a positive integer. In combination with the determination of $ξ(k, λ)$ for integer $k \ge 1$ and real $λ\ge 1$ in our previous papers, this gives the value of $ξ(k, λ)$ also for $λ\in (0,1)$ and the disconnection exponents $\lim_{λ\searrow 0} ξ(k, λ)$. In particular, it shows that $\lim_{λ\searrow 0} ξ(2, λ) = 2/3$ and concludes the proof of the following result that had been conjectured by Mandelbrot: the Hausdorff dimension of the outer boundary of $B[0,1]$ is 4/3 almost surely.

math.PR↗
arXiv2000

Values of Brownian intersection exponents III: Two-sided exponents

This paper determines values of intersection exponents between packs of planar Brownian motions in the half-plane and in the plane that were not derived in our first two papers. For instance, it is proven that the exponent $ξ(3,3)$ describing the asymptotic decay of the probability of non-intersection between two packs of three independent planar Brownian motions each is $(73-2 \sqrt {73}) / 12$. More generally, the values of $ξ(w_1, >..., w_k)$ and $\tx (w_1', ..., w_k')$ are determined for all $ k \ge 2$, $w_1, w_2\ge 1$, $w_3, ...,w_k\in[0,\infty)$ and all $w_1',...,w_k'\in[0,\infty)$. The proof relies on the results derived in our first two papers and applies the same general methods. We first find the two-sided exponents for the stochastic Loewner evolution processes in a half-plane, from which the Brownian intersection exponents are determined via a universality argument.

math.PR↗
arXiv2000

On the cover time of planar graphs

The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all the vertices. It is known that the cover time on any n-vertex, connected graph is at least (1+o(1)) n log(n) and at most (1+o(1))(4/27)n^3. This paper proves that for bounded-degree planar graphs the cover time is at least c n(log n)^2, and at most 6n^2, where c is a positive constant depending only on the maximal degree of the graph. The lower bound is established via use of circle packings.

math.PR↗
arXiv2000

Noise Sensitivity of Boolean Functions and Applications to Percolation

It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority functions are shown to be noise-stable. Several necessary and sufficient conditions for noise sensitivity and stability are given. Consider, for example, bond percolation on an $n+1$ by $n$ grid. A configuration is a function that assigns to every edge the value 0 or 1. Let $ω$ be a random configuration, selected according to the uniform measure. A crossing is a path that joins the left and right sides of the rectangle, and consists entirely of edges $e$ with $ω(e)=1$. By duality, the probability for having a crossing is 1/2. Fix an $ε\in(0,1)$. For each edge $e$, let $ω'(e)=ω(e)$ with probability $1-ε$, and $ω'(e)=1-ω(e)$ with probability $ε$, independently of the other edges. Let $p(τ)$ be the probability for having a crossing in $ω$, conditioned on $ω'=τ$. Then for all $n$ sufficiently large, $P\{τ: |p(τ)-1/2|>ε\}<ε$.

math.PR↗
arXiv1999

Stationary Measures for Random Walks in a Random Environment with Random Scenery

Let $Γ$ act on a countable set V with only finitely many orbits. Given a $Γ$-invariant random environment for a Markov chain on V and a random scenery, we exhibit, under certain conditions, an equivalent stationary measure for the environment and scenery from the viewpoint of the random walker. Such theorems have been very useful in investigations of percolation on quasi-transitive graphs.

math.PR↗
arXiv1999

Indistinguishability of Percolation Clusters

We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to non-decay of connectivity (a.k.a. long-range order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products, and inequalities for $p_u$.

math.PR↗
arXiv1999

Scaling limits of loop-erased random walks and uniform spanning trees

The uniform spanning tree (UST) and the loop-erased random walk (LERW) are related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of the subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane. The scaling limits of these processes are conjectured to be conformally invariant in 2 dimensions. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the Loewner differential equation ${\partial f\over\partial t} = z {ζ(t)+z \over ζ(t)-z} {\partial f\over\partial z}$ with boundary values $f(z,0)=z$, in the range $z\in\U=\{w\in\C\st |w|<1\}$, $t\le 0$. We choose $ζ(t):= \B(-2t)$, where $\B(t)$ is Brownian motion on $\partial \U$ starting at a random-uniform point in $\partial \U$. Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to $\partial\U$ has the same law as that of the path $f(ζ(t),t)$. We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters.

math.PR↗
arXiv1998

Percolation Perturbations in Potential Theory and Random Walks

We show that on a Cayley graph of a nonamenable group, almost surely the infinite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has positive speed, and that these clusters admit bounded harmonic functions. A principal new finding on which these results are based is that such clusters admit invariant random subgraphs with positive isoperimetric constant. We also show that percolation clusters in any amenable Cayley graph almost surely admit no nonconstant harmonic Dirichlet functions. Conversely, on a Cayley graph admitting nonconstant harmonic Dirichlet functions, almost surely the infinite clusters of $p$-Bernoulli percolation also have nonconstant harmonic Dirichlet functions when $p$ is sufficiently close to 1. Many conjectures and questions are presented.

math.PR↗
arXiv1994

Average kissing numbers for non-congruent sphere packings

The Koebe circle packing theorem states that every finite planar graph can be realized as the nerve of a packing of (non-congruent) circles in R^3. We investigate the average kissing number of finite packings of non-congruent spheres in R^3 as a first restriction on the possible nerves of such packings. We show that the supremum k of the average kissing number for all packings satisfies 12.566 ~ 666/53 <= k < 8 + 4*sqrt(3) ~ 14.928 We obtain the upper bound by a resource exhaustion argument and the upper bound by a construction involving packings of spherical caps in S^3. Our result contradicts two naive conjectures about the average kissing number: That it is unbounded, or that it is supremized by an infinite packing of congruent spheres.

math.MG↗