We show that there exists a family of groups $G_n$ and nontrivial irreducible representations $rho_n$ such that, for any constant $t$, the average of $rho_n$ over $t$ uniformly random elements $g_1, ldots, g_t in G_n$ has operator norm $1$ with proba
bility approaching 1 as $n rightarrow infty$. More quantitatively, we show that there exist families of finite groups for which $Omega(log log |G|)$ random elements are required to bound the norm of a typical representation below $1$. This settles a conjecture of A. Wigderson.
We show the existence of regular combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise permutations.
In all cases, the sizes of the objects are optimal up to polynomial overhead. The proof of existence is probabilistic. We show that a randomly chosen structure has the required properties with positive yet tiny probability. Our method allows also to give rather precise estimates on the number of objects of a given size and this is applied to count the number of orthogonal arrays, t-designs and regular hypergraphs. The main technical ingredient is a special local central limit theorem for suitable lattice random walks with finitely many steps.
We prove a Chernoff-like large deviation bound on the sum of non-independent random variables that have the following dependence structure. The variables $Y_1,...,Y_r$ are arbitrary Boolean functions of independent random variables $X_1,...,X_m$, mod
ulo a restriction that every $X_i$ influences at most $k$ of the variables $Y_1,...,Y_r$.
Minimizing the discrepancy of a set system is a fundamental problem in combinatorics. One of the cornerstones in this area is the celebrated six standard deviations result of Spencer (AMS 1985): In any system of n sets in a universe of size n, there
always exists a coloring which achieves discrepancy 6sqrt{n}. The original proof of Spencer was existential in nature, and did not give an efficient algorithm to find such a coloring. Recently, a breakthrough work of Bansal (FOCS 2010) gave an efficient algorithm which finds such a coloring. His algorithm was based on an SDP relaxation of the discrepancy problem and a clever rounding procedure. In this work we give a new randomized algorithm to find a coloring as in Spencers result based on a restricted random walk we call Edge-Walk. Our algorithm and its analysis use only basic linear algebra and is truly constructive in that it does not appeal to the existential arguments, giving a new proof of Spencers theorem and the partial coloring lemma.
