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Rotor walk is a deterministic analogue of random walk. We study its recurrence and transience properties on Z d for the initial configuration of all rotors aligned. If n particles in turn perform rotor walks starting from the origin, we show that the number that escape (i.e., never return to the origin) is of order n in dimensions d ≥ 3, and of order n/ log(More)
We consider high dimensional Wishart matrices XX ⊤ where the entries of X ∈ R n×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 n 3. Our proof is entropy-based, making use of the(More)
J o u r n a l o f P r o b a b i l i t y Electron. Abstract We study the joint convergence of independent copies of several patterned matrices in the non-commutative probability setup. In particular, joint convergence holds for the well known Wigner, Toeplitz, Hankel, Reverse Circulant and Symmetric Circulant matrices. We also study some properties of the(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(More)
Let (G, ρ) be a stationary random graph, and use B G ρ (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[|B G ρ (r)|] O(r k) for some k > 0 and every r 1. Then there is an infinite sequence of times {t n } at which the random walk {X t } on (G, ρ) is at most diffusive: Almost surely(More)
In 2006, the fourth author of this paper proposed a graph-theoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle that can be either red or blue. New red and blue particles alternately get emitted from their respective bases and perform random walk. On encountering a particle of the opposite color(More)