János Komlós

The eigenvalues of random symmetric matrices

AbstractLetA=(aij) be ann ×n matrix whose entries fori≧j are independent random variables andaji=aij. Suppose that everyaij is bounded and for everyi>j we haveEaij=μ,D2aij=σ2 andEaii=v.E. P. Wigner determined the asymptotic behavior of the eigenvalues ofA (semi-circle law). In particular, for anyc>2σ with probability 1-o(1) all eigenvalues except for at mosto(n) lie in the intervalI=(−c√n,c√n).We show that with probability 1-o(1)all eigenvalues belong to the above intervalI if μ=0, while in case μ>0 only the largest eigenvalue λ1 is outsideI, andn

On optimal matchings

lambda _1 = frac{{Sigma _{i,j} a_{ij} }}{n} + frac{{sigma ^2 }}{mu } + Oleft( {frac{I}{{sqrt n }}} right)

A Dense Infinite Sidon Sequence

n i.e. λ1 asymptotically has a normal distribution with expectation (n−1)μ+v+(σ2/μ) and variance 2σ2 (bounded variance!).

Largest random component of a k-cube

We give a sorting network withcn logn comparisons. The algorithm can be performed inc logn parallel steps as well, where in a parallel step we comparen/2 disjoint pairs. In thei-th step of the algorithm we compare the contents of registersRj(i), andRk(i), wherej(i), k(i) are absolute constants then change their contents or not according to the result of the comparison.

The longest path in a random graph

Givenn random red points on the unit square, the transportation cost between them is tipically √n logn.

Extremal Uncrowded Hypergraphs

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.

Linear verification for spanning trees

But the task of constructing a denser sequence has so far resisted all efforts, both constructive and random methods. Here we use a random construction for giving a sequence that is slightly denser than the above trivial one. (However Erdos conjectures that even fs(n» n!-e is possible.) Lemma 2 is of independent interest for a graph-theorist. We also remark that using random construction Erdos and Renyi [4] proved the existence of an infinite sequence S with fs(n) > cn l-e (for all n) such that the number of solutions of the equation

ON TURAN'S THEOREM FOR SPARSE GRAPHS

AbstractLetCk denote the graph with vertices (ɛ1, ...,ɛk),ɛi=0,1 and vertices adjacent if they differ in exactly one coordinate. We callCk thek-cube.LetG=Gk, p denote the random subgraph ofCk defined by lettingn