arXiv subjects

Oded Schramm

Publications and source records attributed to Oded Schramm.

At least 37 records · Page 2Linked to original sources

arXiv2008

Visibility to infinity in the hyperbolic plane, despite obstacles

Suppose that $Z$ is a random closed subset of the hyperbolic plane $\H^2$, whose law is invariant under isometries of $\H^2$. We prove that if the probability that $Z$ contains a fixed ball of radius 1 is larger than some universal constant $p<1$, then there is positive probability that $Z$ contains (bi-infinite) lines. We then consider a family of random sets in $\H^2$ that satisfy some additional natural assumptions. An example of such a set is the covered region in the Poisson Boolean model. Let $f(r)$ be the probability that a line segment of length $r$ is contained in such a set $Z$. We show that if $f(r)$ decays fast enough, then there are almost surely no lines in $Z$. We also show that if the decay of $f(r)$ is not too fast, then there are almost surely lines in $Z$. In the case of the Poisson Boolean model with balls of fixed radius $R$ we characterize the critical intensity for the almost sure existence of lines in the covered region by an integral equation. We also determine when there are lines in the complement of a Poisson process on the Grassmannian of lines in $\H^2$.

math.PR↗

arXiv2008

KPZ in one dimensional random geometry of multiplicative cascades

We prove a formula relating the Hausdorff dimension of a subset of the unit interval and the Hausdorff dimension of the same set with respect to a random path matric on the interval, which is generated using a multiplicative cascade. When the random variables generating the cascade are exponentials of Gaussians, the well known KPZ formula of Knizhnik, Polyakov and Zamolodchikov from quantum gravity appears. This note was inspired by the recent work of Duplantier and Sheffield proving a somewhat different version of the KPZ formula for Liouville gravity. In contrast with the Liouville gravity setting, the one dimensional multiplicative cascade framework facilitates the determination of the Hausdorff dimension, rather than some expected box count dimension.

math.PR↗

arXiv2008

Ends in Uniform Spanning Forests

It has hitherto been known that in a transitive unimodular graph, each tree in the wired spanning forest has only one end a.s. We dispense with the assumptions of transitivity and unimodularity, replacing them with a much broader condition on the isoperimetric profile that requires just slightly more than uniform transience.

math.PR↗

arXiv2008

Growth of the Number of Spanning Trees of the Erdös-Rényi Giant Component

The number of spanning trees in the giant component of the random graph $\G(n, c/n)$ ($c>1$) grows like $\exp\big\{m\big(f(c)+o(1)\big)\big\}$ as $n\to\infty$, where $m$ is the number of vertices in the giant component. The function $f$ is not known explicitly, but we show that it is strictly increasing and infinitely differentiable. Moreover, we give an explicit lower bound on $f'(c)$. A key lemma is the following. Let $\PGW(λ)$ denote a Galton-Watson tree having Poisson offspring distribution with parameter $λ$. Suppose that $λ^*>λ>1$. We show that $\PGW(λ^*)$ conditioned to survive forever stochastically dominates $\PGW(λ)$ conditioned to survive forever.

math.PR↗

arXiv2008

Contour lines of the two-dimensional discrete Gaussian free field

We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant lambda > 0 such that when h is an interpolation of the discrete Gaussian free field on a Jordan domain -- with boundary values -lambda on one boundary arc and lambda on the complementary arc -- the zero level line of h joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are -a < 0 on the first arc and b > 0 on the complementary arc, then the convergence is to SLE(4;a/lambda-1,b/lambda-1), a variant of SLE(4).

math.PR↗

arXiv2008

Tug-of-war and the infinity Laplacian

We prove that every bounded Lipschitz function F on a subset Y of a length space X admits a tautest extension to X, i.e., a unique Lipschitz extension u for which Lip_U u = Lip_{boundary of U} u for all open subsets U of X that do not intersect Y. This was previously known only for bounded domains R^n, in which case u is infinity harmonic, that is, a viscosity solution to Delta_infty u = 0. We also prove the first general uniqueness results for Delta_infty u = g on bounded subsets of R^n (when g is uniformly continuous and bounded away from zero), and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of u. Let u^epsilon(x) be the value of the following two-player zero-sum game, called tug-of-war: fix x_0=x \in X minus Y. At the kth turn, the players toss a coin and the winner chooses an x_k with d(x_k, x_{k-1})< epsilon. The game ends when x_k is in Y, and player one's payoff is F(x_k) - (epsilon^2/2) sum_{i=0}^{k-1} g(x_i) We show that the u^εconverge uniformly to u as epsilon tends to zero. Even for bounded domains in R^n, the game theoretic description of infinity-harmonic functions yields new intuition and estimates; for instance, we prove power law bounds for infinity-harmonic functions in the unit disk with boundary values supported in a delta-neighborhood of a Cantor set on the unit circle.

math.AP↗

arXiv2008

Dynamical sensitivity of the infinite cluster in critical percolation

In dynamical percolation, the status of every bond is refreshed according to an independent Poisson clock. For graphs which do not percolate at criticality, the dynamical sensitivity of this property was analyzed extensively in the last decade. Here we focus on graphs which percolate at criticality, and investigate the dynamical sensitivity of the infinite cluster. We first give two examples of bounded degree graphs, one which percolates for all times at criticality and one which has exceptional times of nonpercolation. We then make a nearly complete analysis of this question for spherically symmetric trees with spherically symmetric edge probabilities bounded away from 0 and 1. One interesting regime occurs when the expected number of vertices at the nth level that connect to the root at a fixed time is of order n(\log n)^α. R. Lyons (1990) showed that at a fixed time, there is an infinite cluster a.s. if and only if α>1. We prove that the probability that there is an infinite cluster at all times is 1 if α> 2, while this probability is 0 if 1<α\le 2. Within the regime where a.s. there is an infinite cluster at all times, there is yet another type of ``phase transition'' in the behavior of the process: if the expected number of vertices at the nth level connecting to the root at a fixed time is of order n^θwith θ> 2, then the number of connected components of the set of times in [0,1] at which the root does not percolate is finite a.s., while if 1<θ< 2, then the number of such components is infinite with positive probability.

math.PR↗

arXiv2008

Poisson Matching

Suppose that red and blue points occur as independent homogeneous Poisson processes in R^d. We investigate translation-invariant schemes for perfectly matching the red points to the blue points. For any such scheme in dimensions d=1,2, the matching distance X from a typical point to its partner must have infinite d/2-th moment, while in dimensions d>=3 there exist schemes where X has finite exponential moments. The Gale-Shapley stable marriage is one natural matching scheme, obtained by iteratively matching mutually closest pairs. A principal result of this paper is a power law upper bound on the matching distance X for this scheme. A power law lower bound holds also. In particular, stable marriage is close to optimal (in tail behavior) in d=1, but far from optimal in d>=3. For the problem of matching Poisson points of a single color to each other, in d=1 there exist schemes where X has finite exponential moments, but if we insist that the matching is a deterministic factor of the point process then X must have infinite mean.

math.PR↗

arXiv2008

Every Minor-Closed Property of Sparse Graphs is Testable

Suppose $G$ is a graph with degrees bounded by $d$, and one needs to remove more than $εn$ of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of $G$ is far from the statistics of local neighborhoods around vertices of any planar graph $G'$ with the same degree bound. In fact, a similar result is proved for any minor-closed property of bounded degree graphs. As an immediate corollary of the above result we infer that many well studied graph properties, like being planar, outer-planar, series-parallel, bounded genus, bounded tree-width and several others, are testable with a constant number of queries, where the constant may depend on $ε$ and $d$, but not on the graph size. None of these properties was previously known to be testable even with $o(n)$ queries.

math.CO↗

arXiv2007

Boundary proximity of SLE

This paper examines how close the chordal $\SLE_κ$ curve gets to the real line asymptotically far away from its starting point. In particular, when $κ\in(0,4)$, it is shown that if $β>β_κ:=1/(8/κ-2)$, then the intersection of the $\SLE_κ$ curve with the graph of the function $y=x/(\log x)^β$, $x>e$, is a.s. bounded, while it is a.s. unbounded if $β=β_κ$. The critical $\SLE_4$ curve a.s. intersects the graph of $y=x^{-(\log\log x)^α}$, $x>e^e$, in an unbounded set if $α\le 1$, but not if $α>1$. Under a very mild regularity assumption on the function $y(x)$, we give a necessary and sufficient integrability condition for the intersection of the $\SLE_κ$ path with the graph of $y$ to be unbounded. We also prove that the Hausdorff dimension of the intersection set of the $\SLE_κ$ curve and real axis is $2-8/κ$ when $4<κ<8$.

math.PR↗

arXiv2007

Hyperfinite graph limits

Gábor Elek introduced the notion of a hyperfinite graph family: a collection of graphs is hypefinite if for every $ε>0$ there is some finite $k$ such that each graph $G$ in the collection can be broken into connected components of size at most $k$ by removing a set of edges of size at most $ε|V(G)|$. We presently extend this notion to a certain compactification of finite bounded-degree graphs, and show that if a sequence of finite graphs converges to a hyperfinite limit, then the sequence itself is hyperfinite.

math.PR↗

arXiv2007

Combinatorically Prescribed Packings and Applications to Conformal and Quasiconformal Maps

The Andreev-Thurston Circle Packing Theorem is generalized to packings of convex bodies in planar simply connected domains. This turns out to be a useful tool for constructing conformal and quasiconformal mappings with interesting geometric properties. We attempt to illustrate this with a few results about uniformizations of finitely connected planar domains. For example, the following variation of a theorem by Courant, Manel and Shiffman is proved and generalized. If $G$ is an $n+1$-connected bounded planar domain, $H$ is a simply connected bounded planar domain, and $P_1,P_2,...,P_n$ are (compact) planar convex bodies, then sets $P_j'$ can be found so that $G$ is conformally equivalent to $H-\cup_{j=1}^n P_j'$, and each $P_j'$ is either a point, or is positively homothetic to $P_j$.

math.CV↗

arXiv2007

Compositions of random transpositions

Let $Y=(y_1,y_2,...)$, $y_1\ge y_2\ge...$, be the list of sizes of the cycles in the composition of $c n$ transpositions on the set $\{1,2,...,n\}$. We prove that if $c>1/2$ is constant and $n\to\infty$, the distribution of $f(c)Y/n$ converges to PD(1), the Poisson-Dirichlet distribution with paramenter 1, where the function $f$ is known explicitly. A new proof is presented of the theorem by Diaconis, Mayer-Wolf, Zeitouni and Zerner stating that the PD(1) measure is the unique invariant measure for the uniform coagulation-fragmentation process.

math.PR↗

arXiv2007

First Passage Percolation Has Sublinear Distance Variance

Let $0 1$ the distance $dist_ω(0,v)$ from the origin to a vertex $v$, $|v|>2$, has variance bounded by $C |v|/\log|v|$, where $C=C(a,b,d)$ is a constant which may only depend on $a$, $b$ and $d$. Some related variants are also discussed

math.PR↗

arXiv2007

Quantitative noise sensitivity and exceptional times for percolation

One goal of this paper is to prove that dynamical critical site percolation on the planar triangular lattice has exceptional times at which percolation occurs. In doing so, new quantitative noise sensitivity results for percolation are obtained. The latter is based on a novel method for controlling the "level k" Fourier coefficients via the construction of a randomized algorithm which looks at random bits, outputs the value of a particular function but looks at any fixed input bit with low probability. We also obtain upper and lower bounds on the Hausdorff dimension of the set of percolating times. We then study the problem of exceptional times for certain "k-arm" events on wedges and cones. As a corollary of this analysis, we prove, among other things, that there are no times at which there are two infinite "white" clusters, obtain an upper bound on the Hausdorff dimension of the set of times at which there are both an infinite white cluster and an infinite black cluster and prove that for dynamical critical bond percolation on the square grid there are no exceptional times at which three disjoint infinite clusters are present.

math.PR↗

arXiv2006

Minimal spanning forests

Minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs. These limits can be taken with free or wired boundary conditions and are denoted FMSF (free minimal spanning forest) and WMSF (wired minimal spanning forest), respectively. The WMSF is also the union of the trees that arise from invasion percolation started at all vertices. We show that on any Cayley graph where critical percolation has no infinite clusters, all the component trees in the WMSF have one end a.s. In $\mathbb{Z}^d$ this was proved by Alexander [Ann. Probab. 23 (1995) 87--104], but a different method is needed for the nonamenable case. We also prove that the WMSF components are ``thin'' in a different sense, namely, on any graph, each component tree in the WMSF has $p_{\mathrm{c}}=1$ a.s., where $p_{\mathrm{c}}$ denotes the critical probability for having an infinite cluster in Bernoulli percolation. On the other hand, the FMSF is shown to be ``thick'': on any connected graph, the union of the FMSF and independent Bernoulli percolation (with arbitrarily small parameter) is a.s. connected. In conjunction with a recent result of Gaboriau, this implies that in any Cayley graph, the expected degree of the FMSF is at least the expected degree of the FSF (the weak limit of uniform spanning trees). We also show that the number of infinite clusters for Bernoulli($p_{\mathrm{u}}$) percolation is at most the number of components of the FMSF, where $p_{\mathrm{u}}$ denotes the critical probability for having a unique infinite cluster. Finally, an example is given to show that the minimal spanning tree measure does not have negative associations.

math.PR↗

arXiv2006

Random-Turn Hex and other selection games

The game of Hex has two players who take turns placing stones of their respective colors on the hexagons of a rhombus-shaped hexagonal grid. Black wins by completing a crossing between two opposite edges, while White wins by completing a crossing between the other pair of opposite edges. Although ordinary Hex is famously difficult to analyze, Random-Turn Hex--in which players toss a coin before each turn to decide who gets to place the next stone--has a simple optimal strategy. It belongs to a general class of random-turn games--called selection games--in which the expected payoff when both players play the random-turn game optimally is the same as when both players play randomly. We also describe the optimal strategy and study the expected length of the game under optimal play for Random-Turn Hex and several other selection games.

math.PR↗

arXiv2006

SLE coordinate changes

The purpose of this note is to describe a framework which unifies radial, chordal and dipolar SLE. When the definition of SLE(kappa;rho) is extended to the setting where the force points can be in the interior of the domain, radial SLE(kappa) becomes chordal SLE(kappa;rho), with rho=kappa-6, and vice versa. We also write down the martingales describing the Radon-Nykodim derivative of SLE(kappa;rho_1,...,rho_n) with respect to SLE(kappa).

math.PR↗