Jeff Kahn

The influence of variables on Boolean functions

Methods from harmonic analysis are used to prove some general theorems on Boolean functions. These connections with harmonic analysis viewed by the authors are very promising; besides the results on Boolean functions they enable them to prove theorems on the rapid mixing of the random walk on the cube and in the extremal theory of finite sets.<<ETX>>

On the second eigenvalue of random regular graphs

The following is an extended abstract for two papers, one written by Kahn and Szemeredi, the other written by Friedman, which have been combined at the request of the STOC committee. The introduction was written jointly, the second section by Kahn and Szemeredi, and the third by Friedman, Let G be a d-regular (i.e. each vertex has degree d) undirected graph on n nodes. It’s adjacency matrix is symmetric, and therefore has real eigenvalues Ar = d 2 x2 >_ *-. >_ X, with IX,] 5 d. Graphs for which X2 and

A counterexample to Borsuk's conjecture

Let f(d) be the smallest number so that every set in R d of diameter I can be partitioned into f(d) sets of diameter smaller than 1. Borsuks conjecture was that f(d) = d + 1. We prove that f(d) ≥ (1.2)√ d for large d

A topological approach to evasiveness

The complexity of a digraph property is the number of entries of the vertex adjacency matrix of a digraph which must be examined in worst case to determine whether the graph has the property. Rivest and Vuillemin proved the result (conjectured by Aanderaa and Rosenberg) that every graph property that is monotone (preserved by addition of edges) and nontrivial (holds for some but not all graphs) has complexity Ω(v2) wherev is the number of vertices. Karp conjectured that every such property is evasive, i.e., requires that every entry of the incidence matrix be examined. In this paper the truth of Karp’s conjecture is shown to follow from another conjecture concerning group actions on topological spaces. A special case of the conjecture is proved which is applied to prove Karp’s conjecture for the case of properties of graphs on a prime power number of vertices.

On the probability that a random ±1-matrix is singular

We report some progress on the old problem of estimating the probability, Pn, that a random n× n ± 1 matrix is singular: Theorem. There is a positive constant ε for which Pn < (1− ε)n. This is a considerable improvement on the best previous bound, Pn = O(1/ √ n), given by Komlós in 1977, but still falls short of the often-conjectured asymptotical formula Pn = (1 + o(1))n 221−n. The proof combines ideas from combinatorial number theory, Fourier analysis and combinatorics, and some probabilistic constructions. A key ingredient, based on a Fourier-analytic idea of Halász, is an inequality (Theorem 2) relating the probability that a ∈ Rn is orthogonal to a random ε ∈ {±1}n to the corresponding probability when ε is random from {−1, 0, 1}n with Pr(εi = −1) = Pr(εi = 1) = p and εi’s chosen independently.

An Entropy Approach to the Hard-Core Model on Bipartite Graphs

We use entropy ideas to study hard-core distributions on the independent sets of a finite, regular bipartite graph, specifically distributions according to which each independent set I is chosen with probability proportional to λ∣I∣ for some fixed λ > 0. Among the results obtained are rather precise bounds on occupation probabilities; a ‘phase transition’ statement for Hamming cubes; and an exact upper bound on the number of independent sets in an n-regular bipartite graph on a given number of vertices.

The influence of variables in product spaces

AbstractLetX be a probability space and letf: Xn → {0, 1} be a measurable map. Define the influence of thek-th variable onf, denoted byIf(k), as follows: Foru=(u1,u2,…,un−1) ∈Xn−1 consider the setlk(u)={(u1,u2,...,uk−1,t,uk,…,un−1):t ∈X}.

Entropy and sorting

I_f (k) = \Pr (u \in X^{n - 1} :f is not constant on l_k (u)).