Amir Dembo

TL;DR: For a polynomial of large degree n whose coefficients are independent, identically distributed, non-degenerate random variables having zero mean and finite moments of all orders, it was shown in this article that the probability that such polynomials have exactly k real zeros with probability n −b+o(1)$ as n --> infinity through integers of the same parity as the fixed integer k >= 0.

TL;DR: In this paper, it was shown that the Hausdorff dimension of the set of "thick points" of a set of small balls of radius σ = σ 2 is at most 2-a-pi^2 /8.

TL;DR: In this paper, the authors considered a simple random walk on a Galton-Watson tree and showed that it is possible to obtain an almost sure equality of the large deviation rate under the quenched measure and the annealed measure.

TL;DR: It is proved that the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure, $${{\hat{\mu}}_{a_N^{-1} {\bf Y}_N^\sigma + {\varvec {D}}_N}}$$ converges in mean towards a limiting probability measure which the authors characterize.

TL;DR: In this article, it was shown that the size of the set of α, β, n-late points in a disc of radius (n) centered at a non-random point in the torus can be computed by a simple random walk, and the expected number of pairs of α and β points within a given distance of each other is smaller than what one might predict by multiplying the total number of late points by the number of points in the disc.