Itai Benjamini

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.

Percolation on finite graphs and isoperimetric inequalities

Consider a uniform expanders family Gn with a uniform bound on the degrees. It is shown that for any p and c>0, a random subgraph of Gn obtained by retaining each edge, randomly and independently, with probability p, will have at most one cluster of size at least c|Gn|, with probability going to one, uniformly in p. The method from Ajtai, Komlos and Szemeredi [Combinatorica 2 (1982) 1–7] is applied to obtain some new results about the critical probability for the emergence of a giant component in random subgraphs of finite regular expanding graphs of high girth, as well as a simple proof of a result of Kesten about the critical probability for bond percolation in high dimensions. Several problems and conjectures regarding percolation on finite transitive graphs are presented.

EXCITED RANDOM WALK

A random walk on

KPZ in One Dimensional Random Geometry of Multiplicative Cascades

\mathbb{Z}^d

NON-BACKTRACKING RANDOM WALKS MIX FASTER

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 random. We show that excited random walk on

Every minor-closed property of sparse graphs is testable

\mathbb{Z}^d

Critical Percolation on Any Nonamenable Group has no Infinite Clusters

is transient iff

Every Graph with a Positive Cheeger Constant Contains a Tree with a Positive Cheeger Constant

d > 1

Geometry of the uniform spanning forest: Transitions in dimensions 4, 8, 12, …

.

Distinguishing sceneries by observing the scenery along a random walk path

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 [KPZ88] appears. This note was inspired by the recent work of Duplantier and Sheffield [DS08] 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.