The probability that a random multigraph is simple

Consider a random multigraph G* with given vertex degrees d_1,...,d_n, contructed by the configuration model. We show that, asymptotically for a sequence of such multigraphs with the number of edges (d_1+...+d_n)/2 tending to infinity, the probability that the multigraph is simple stays away from 0 if and only if \sum d_i^2=O(\sum d_i). This was previously known only under extra assumtions on the maximum degree. We also give an asymptotic formula for this probability, extending previous results by several authors.Comment: 24 page

Overcoming the critical slowing down of flat-histogram Monte Carlo simulations: Cluster updates and optimized broad-histogram ensembles

We study the performance of Monte Carlo simulations that sample a broad histogram in energy by determining the mean first-passage time to span the entire energy space of d-dimensional ferromagnetic Ising/Potts models. We first show that flat-histogram Monte Carlo methods with single-spin flip updates such as the Wang-Landau algorithm or the multicanonical method perform sub-optimally in comparison to an unbiased Markovian random walk in energy space. For the d=1,2,3 Ising model, the mean first-passage time \tau scales with the number of spins N=L^d as \tau \propto N^2L^z. The critical exponent z is found to decrease as the dimensionality d is increased. In the mean-field limit of infinite dimensions we find that z vanishes up to logarithmic corrections. We then demonstrate how the slowdown characterized by z>0 for finite d can be overcome by two complementary approaches - cluster dynamics in connection with Wang-Landau sampling and the recently developed ensemble optimization technique. Both approaches are found to improve the random walk in energy space so that \tau \propto N^2 up to logarithmic corrections for the d=1 and d=2 Ising model

Preferential attachment without vertex growth: emergence of the giant component

We study the following preferential attachment variant of the classical Erdos-Renyi random graph process. Starting with an empty graph on n vertices, new edges are added one-by-one, and each time an edge is chosen with probability roughly proportional to the product of the current degrees of its endpoints (note that the vertex set is fixed). We determine the asymptotic size of the giant component in the supercritical phase, confirming a conjecture of Pittel from 2010. Our proof uses a simple method: we condition on the vertex degrees (of a multigraph variant), and use known results for the configuration model.Comment: 20 page