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We consider Markovian models on graphs with local dynamics. We show that, under suitable conditions, such Markov chains exhibit both rapid convergence to equilibrium and strong concentration of measure in the stationary distribution. We illustrate our results with applications to some known chains from computer science and statistical mechanics.

There are n queues, each with a single server. Customers arrive in a Poisson process at rate λn, where 0< λ < 1. Upon arrival each customer selects d ≥ 2 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… (More)

- Svante Janson, Malwina J. Luczak
- Random Struct. Algorithms
- 2007

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… (More)

- Svante Janson, Malwina J. Luczak
- Random Struct. Algorithms
- 2009

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)

Abstract 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… (More)

- Svante Janson, Malwina J. Luczak, Peter Windridge
- Random Struct. Algorithms
- 2014

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)

- Malwina J. Luczak, Colin McDiarmid
- Discrete Mathematics
- 2003

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)

Suppose that there are n bins, and balls arrive in a Poisson process at rate λn, where λ > 0 is a constant. Upon arrival, each ball chooses a fixed number d of random bins, and is placed into one with least load. Balls have independent exponential lifetimes with unit mean. We show that the system converges rapidly to its equilibrium distribution; and when… (More)

- Malwina J. Luczak, Peter Winkler
- Random Struct. Algorithms
- 2004

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)]−1n log n. For β = 1, we prove that the mixing time is of order n3/2. For β >… (More)