The upper tail problem in the Erdős–Rényi random graph G ∼ Gn,p asks to estimate the probability that the number of copies of a graph H in G exceeds its expectation by a factor 1 + δ. Chatterjee and… (More)

We consider high dimensional Wishart matrices XX⊤ where the entries of X ∈ Rn×d are i.i.d. from a log-concave distribution. We prove an information theoretic phase transition: such matrices are close… (More)

Rotor walk is a deterministic analogue of random walk. We study its recurrence and transience properties on Z for the initial configuration of all rotors aligned. If n particles in turn perform rotor… (More)

We introduce a two-type internal DLA model which is an example of a non-unary abelian network. Starting with n “oil” and n “water” particles at the origin, the particles diffuse in Z according to the… (More)

The prevalent technique for DNA sequencing consists of two main steps: shotgun sequencing, where many randomly located fragments, called reads, are extracted from the overall sequence, followed by an… (More)

Let (G, ρ) be a stationary random graph, and use Bρ (r) to denote the ball of radius r about ρ in G. Suppose that (G, ρ) has annealed polynomial growth, in the sense that E[|Bρ (r)|] 6 O(rk) for some… (More)

Consider “Frozen Random Walk” on Z: n particles start at the origin. At any discrete time, the leftmost and rightmost bn4 c particles are “frozen” and do not move. The rest of the particles in the… (More)

We prove tight (up to small constant factors) results on how localized an eigenvector of a high girth regular graph can be (the girth is the length of the shortest cycle). This study was initiated by… (More)