Abstract. 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 probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy and Reed [24;25] on the size of the largest component in a random graph with a given degree sequence.We further obtain a new sharp result for the giant component just above the threshold, generalizing the case of G(n, p) with np = 1 + ω(n)n −1/3 , where ω(n) → ∞ arbitrarily slowly. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs.

Abstract. 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 involving the rates and vertex degrees below which only a small number of infections occur. Above the threshold a large outbreak occurs with probability bounded away from zero. Our main result is that, conditional on a large outbreak, the evolutions of certain quantities of interest, such as the fraction of infective vertices, converge to deterministic functions of time.We also consider more general initial conditions for the epidemic, and derive criteria for a simple vaccination strategy to be successful.In contrast to earlier results for this model, our approach only requires basic regularity conditions and a uniformly bounded second moment of the degree of a random vertex.En route, we prove analogous results for the epidemic on the configuration model multigraph under much weaker conditions. Essentially, our main result requires only that the initial values for our processes converge, i.e. it is the best possible.

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 cut-off: 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 > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).1. I 1.1. Ising model and Glauber dynamics. Let G = (V, E) be a finite graph. Elements of the state space Ω := {−1, 1} V will be called configurations, and for σ ∈ Ω, the value σ(v) will be called the spin at v. The nearest-neighbor energy H(σ) of a configuration σ ∈ {−1, 1} V is defined bywhere w ∼ v means that {w, v} ∈ E. The parameters J(v, w) measure the interaction strength between vertices; we will always take J(v, w) ≡ J, where J is a positive constant. For β ≥ 0, the Ising model on the graph G with parameter β is the probability measure µ on Ω given by µ(σ) = e −βH(σ) Z(β) ,where Z(β) = σ∈Ω e −βH(σ) is a normalizing constant.

ABSTRACT:We study the k-core 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 the k-core is empty and other conditions that imply that with high probability the k-core is non-empty and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the result by Pittel, Spencer, and Wormald (J Combinator Theory 67 (1996), 111-151) on the existence and size of a k-core in G(n, p) and G(n, m), see also Molloy (Random Struct Algor 27 (2005), 124-135) and Cooper (Random Struct Algor 25 (2004), 353-375).Our method is based on the properties of empirical distributions of independent random variables and leads to simple proofs.

There are $n$ queues, each with a single server. Customers arrive in a Poisson process at rate $\lambda n$, where $0<\lambda<1$. Upon arrival each customer selects $d\geq2$ servers uniformly at random, and joins the queue at a least-loaded server among those chosen. Service times are independent exponentially distributed random variables with mean 1. We show that the system is rapidly mixing, and then investigate the maximum length of a queue in the equilibrium distribution. We prove that with probability tending to 1 as $n\to\infty$ the maximum queue length takes at most two values, which are $\ln\ln n/\ln d+O(1)$.Comment: Published at http://dx.doi.org/10.1214/00911790500000710 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org