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We study the k-core of a random (multi)graph on n ver-tices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability the k-core is empty, and other conditions that imply that with high probability the(More)
We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high(More)
A. We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For β < 1, we prove that the dynamics exhibits a cutoff: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1 − β)] −1 n log n. For β = 1, we prove that the mixing time is of order n 3/2. For β >(More)
Talagrand (Publ. Math. Inst. Hautes Etudes Sci. 81 (1995) 73) gave a concentration inequality concerning permutations picked uniformly at random from a symmetric group, and this was extended in McDiarmid (Combin. Probab. Comput. 11 (2002) 163) to handle permutations picked uniformly at random from a direct product of symmetric groups. Here we extend these(More)
We study the susceptible-infective-recovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In this model infective vertices infect each of their susceptible neighbours, and recover, at a constant rate. Suppose that initially there are only a few infective vertices. We prove there is a threshold for a parameter(More)
In the supermarket model there are n queues, each with a unit rate server. Customers arrive in a Poisson process at rate λn, where 0 < λ < 1. Each customer chooses d ≥ 2 queues uniformly at random, and joins a shortest one. It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as n → ∞.(More)