Shirshendu Ganguly

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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 in total variation distance to the corresponding Gaussian ensemble if and only if d is much larger than n3. Our proof is entropy-based, making use of the chain(More)
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 Dembo (2014) showed that in the sparse regime of p→ 0 as n→∞ with p ≥ n−α for an explicit α = αH > 0, this problem reduces to a natural variational problem on(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 following rule: whenever some site x ∈ Z has at least 1 oil and at least 1 water particle present, it fires by sending 1 oil particle and 1 water particle each(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 k > 0 and every r > 1. Then there is an infinite sequence of times {tn} at which the random walk {Xt} on (G, ρ) is at most diffusive: Almost surely (over the(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 “bulk” independently jump to the left and right uniformly. The goal of this note is to understand the limit of this process under scaling of mass and time. To(More)