We show that with high probability a random subset of {1, . . . , n} of size Î˜(n) contains two elements a and a + d, where d is a positive integer. As a consequence, we prove an analogue of theâ€¦ (More)

Let An be an n by n random matrix whose entries are independent real random variables with mean zero and variance one. We show that the logarithm of | det An| satisfies a central limit theorem. Moreâ€¦ (More)

Let X be a matrix sampled uniformly from the set of doubly stochastic matrices of size nÃ—n. We show that the empirical spectral distribution of the normalized matrix âˆš n(X âˆ’ EX) converges almostâ€¦ (More)

Let Zp be the finite field of prime order p and A be a subsequence of Zp. We prove several classification results about the following questions: (1) When can one represent zero as a sum of someâ€¦ (More)

Condition numbers of random polynomial systems have been widely studied in the literature under certain coefficient ensembles of invariant type. In this note we introduce a method that allows us toâ€¦ (More)

Let Î¾ be a real random variable with mean zero and variance one and A = {a1, . . . , an} be a multi-set in R. The random sum SA := a1Î¾1 + Â· Â· Â·+ anÎ¾n where Î¾i are iid copies of Î¾ is of fundamentalâ€¦ (More)

Let Mn denote a random symmetric n by n matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value âˆ’1 and 1 with probability 1/2). Improving the earlier result byâ€¦ (More)

A fundamental problem in random matrix theory is to determine the limiting distribution of the ESD as the size of the matrix tends to infinity. In certain cases when the entries have specialâ€¦ (More)

Let Zp be the finite field of prime order p and A be a subset of Zp. We prove several sharp results about the following two basic questions: (1) When can one represent zero as a sum of distinctâ€¦ (More)

Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models.â€¦ (More)