
Rankings in directed configuration models with heavy tailed indegrees
We consider the extremal values of the stationary distribution of sparse...
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Time Dependent Biased Random Walks
We study the biased random walk where at each step of a random walk a "c...
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On the cover time of dense graphs
We consider arbitrary graphs G with n vertices and minimum degree at lea...
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The diameter of the directed configuration model
We show that the diameter of the directed configuration model with n ver...
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Learning lowdegree functions from a logarithmic number of random queries
We prove that for any integer n∈ℕ, d∈{1,…,n} and any ε,δ∈(0,1), a bounde...
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Iterated PiecewiseStationary Random Functions
Within the study of uncertain dynamical systems, iterated random functio...
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The complexity of approximating the matching polynomial in the complex plane
We study the problem of approximating the value of the matching polynomi...
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Minimum stationary values of sparse random directed graphs
We consider the stationary distribution of the simple random walk on the directed configuration model with bounded degrees. Provided that the minimum outdegree is at least 2, with high probability (whp) there is a unique stationary distribution. We show that the minimum positive stationary value is whp n^(1+C+o(1)) for some constant C ≥ 0 determined by the degree distribution. In particular, C is the competing combination of two factors: (1) the contribution of atypically "thin" inneighbourhoods, controlled by subcritical branching processes; and (2) the contribution of atypically "light" trajectories, controlled by large deviation rate functions. Additionally, our proof implies that whp the hitting and the cover time are both n^1+C+o(1). Our results complement those of Caputo and Quattropani who showed that if the minimum indegree is at least 2, stationary values have logarithmic fluctuations around n^1.
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