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## Recurrence of Distributional Limits of Finite Planar Graphs

- I. Benjamini, O. Schramm
- Mathematics
- 2 November 2000

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… Expand

## Noise sensitivity of Boolean functions and applications to percolation

- I. Benjamini, G. Kalai, O. Schramm
- Mathematics
- 26 November 1998

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… Expand

## Uniform spanning forests

- I. Benjamini, R. Lyons, Y. Peres, O. Schramm
- Mathematics
- 1 February 2001

We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF) or wired (WSF)… Expand

## Percolation Beyond $Z^d$, Many Questions And a Few Answers

- I. Benjamini, O. Schramm
- Mathematics
- 10 August 1996

A comprehensive study of percolation in a more general context than the usual $Z^d$ setting is proposed, with particular focus on Cayley graphs, almost transitive graphs, and planar graphs. Results… Expand

## Group-invariant Percolation on Graphs

- I. Benjamini, R. Lyons, Y. Peres, O. Schramm
- Mathematics
- 1 March 1999

Abstract. Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes… Expand

## First Passage Percolation Has Sublinear Distance Variance

- I. Benjamini, G. Kalai, O. Schramm
- Mathematics
- 25 March 2002

Let 0 1, the distance dist ω (0, ν) from the origin to a vertex ν, | ν | > 2, has variance bounded by C |ν|/ log |ν|, where C = C (a, b, d) is a constant which may only depend on a, b and d. Some… Expand

## Percolation in the hyperbolic plane

- I. Benjamini, O. Schramm
- Mathematics
- 30 December 1999

The purpose of this paper is to study percolation in the hyperbolic plane and in transitive planar graphs that are quasi-isometric to the hyperbolic plane.

## Non-backtracking random walks mix faster

- N. Alon, I. Benjamini, Eyal Lubetzky, S. Sodin
- Mathematics
- 18 October 2006

We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as… Expand

## Excited Random Walk

- I. Benjamini, D. Wilson
- Mathematics
- 22 February 2003

A random walk on $\mathbb{Z}^d$ is excited if the first time it visits a vertex there is a bias in one direction, but on subsequent visits to that vertex the walker picks a neighbor uniformly at… Expand

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