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Itai Benjamini

Publications and source records attributed to Itai Benjamini.

153 records · Page 9Linked to original sources
arXiv2002

Determining the Genus of a Map by Local Observation of a Simple Random Process

Given a graph embedded in an orientable surface, a process consisting of random excitations and random node and face balancing is constructed and analyzed. It is shown that given a priori bounds g' on the genus and n' on the number of nodes, one can determine the genus of the surface from local observations of the process restricted to any connected subgraph which cannot be separated from the rest of the graph by fewer than 16g' nodes. The observation time and the computation time are polynomial in n'^g'. The process constructs slightly perturbed random ``discrete analytic functions'' on the surface, and the key fact in the analysis is that such a function cannot vanish on a large piece of the surface.

math.PR↗
arXiv2001

Recurrence of Distributional Limits of Finite Planar Graphs

Suppose that $G_j$ is a sequence of finite connected planar graphs, and in each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit $G$ of such graphs. Assume that the vertex degrees of the vertices in $G_j$ are bounded, and the bound does not depend on $j$. Then after passing to a subsequence, the limit exists, and is a random rooted graph $G$. We prove that with probability one $G$ is recurrent. The proof involves the Circle Packing Theorem. The motivation for this work comes from the theory of random spherical triangulations.

math.PR↗
arXiv2001

percolation on finite graphs

The asymptotic study of percolation on finite transitive graphs is considered. Several questions and very few answers regarding percolation on finite graphs are presented.

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

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↗
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↗
arXiv1997

Paths with exponential intersection tails and oriented percolation

We show that oriented percolation occurs whenever a condition is satisfied called "exponential intersection tails". This condition says that a measure on paths exists for which the probability of two independent paths intersecting in more than k sites is exponentially small in k.

math.PR↗