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- Malwina J Luczak
- 2008

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.

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)

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)

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 define the decision problem data arrangement, which involves arranging the vertices of a graph G at the leaves of a d-ary tree so that a weighted sum of the distances between pairs of vertices measured with respect to the tree topology is at most a given value. We show that data arrangement is strongly NP-complete for any fixed d ≥ 2 and explain the… (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)

Consider partitions of the vertex set of a graph G into two sets with sizes differing by at most 1: the bisection width of G is the minimum over all such partitions of the number of ''cross edges'' between the parts. We are interested in sparse random graphs Ž. G with edge probability crn. We show that, if c) ln 4, then the bisection width is ⍀ n n, c r n… (More)

- Remco Van Der Hofstad, Malwina J Luczak
- 2006

We study random subgraphs of the 2-dimensional Hamming graph H(2, n), which is the Cartesian product of two complete graphs on n vertices. Let p be the edge probability, and write p = 1+ε 2(n−1) for some ε ∈ R. In [4, 5], the size of the largest connected component was estimated precisely for a large class of graphs including H(2, n) for ε ≤ ΛV −1/3 , where… (More)