Abstract. Let Qn denote a random symmetric n by n matrix, whose upper diagonal entries are i.i.d. Bernoulli random variables (which take values 0 and 1 with probability 1/2). We prove that Qn is non-singular with probability 1 − O(n −1/8+δ ) for any fixed δ > 0. The proof uses a quadratic version of Littlewood-Offord type results concerning the concentration functions of random variables and can be extended for more general models of random matrices.

ABSTRACT:We show that almost surely the rank of the adjacency matrix of the Erdős-Rényi random graph G(n, p) equals the number of nonisolated vertices for any c ln n/n < p < 1/2, where c is an arbitrary positive constant larger than 1/2. In particular, the adjacency matrix of the giant component (a.s.) has full rank in this range.

If f (x 1 , . . . , xn) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version of the classical Littlewood-Offord problem: Given nonzero constants a 1 , . . . an, what is the maximum number of sums of the form ±a 1 ± a 2 · · · ± an which take on any single value? Here we consider the case where f is either a bilinear form or a quadratic form. For the bilinear case, we show that the only forms having concentration significantly larger than n −1 are those which are in a certain sense very close to being degenerate. For the quadratic case, we show that no form having many nonzero coefficients has concentration significantly larger than n −1/2 . In both cases the results are nearly tight.

Abstract. We show that the absolute value of the determinant of a matrix with random independent (but not necessarily iid) entries is strongly concentrated around its mean.As an application, we show that Godsil-Gutman and Barvinok estimators for the permanent of a strictly positive matrix give sub-exponential approximation ratios with high probability.A positive answer to the main conjecture of the paper would lead to polynomial approximation ratios in the above problem.

We consider the following stochastic optimization problem first introduced by Chen et al. in [7]. We are given a vertex set of a random graph where each possible edge is present with probability pe. We do not know which edges are actually present unless we scan/probe an edge. However whenever we probe an edge and find it to be present, we are constrained to picking the edge and both its end points are deleted from the graph. We wish to find the maximum matching in this model. We compare our results against the optimal omniscient algorithm that knows the edges of the graph and present a 0.573 factor algorithm using a novel sampling technique. We also prove that no algorithm can attain a factor better than 0.898 in this model.

Abstract. We investigate the rank of random (symmetric) sparse matrices. Our main finding is that with high probability, any dependency that occurs in such a matrix is formed by a set of few rows that contains an overwhelming number of zeros. This allows us to obtain an exact estimate for the co-rank.

Given fixed 0 = q0 < q1 < q2 < • • • < q k = 1, a constellation in [n] is a scaled translated realization of the qi with all elements in [n], i.e.,We consider the problem of minimizing the number of monochromatic constellations in a two-coloring of [n]. We show how, given a coloring based on a block pattern, to find the number of monochromatic solutions to a lower-order term, and also how experimentally we might find an optimal block pattern. We also show for the case k = 2 that there is always a block pattern that beats random coloring.

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