Analysis and optimization of weighted ensemble sampling

We give a mathematical framework for weighted ensemble (WE) sampling, a binning and resampling technique for efficiently computing probabilities in molecular dynamics. We prove that WE sampling is unbiased in a very general setting that includes adaptive binning. We show that when WE is used for stationary calculations in tandem with a coarse model, the coarse model can be used to optimize the allocation of replicas in the bins.Comment: 22 pages, 3 figure

Percolation of hard disks

Random arrangements of points in the plane, interacting only through a simple hard core exclusion, are considered. An intensity parameter controls the average density of arrangements, in analogy with the Poisson point process. It is proved that at high intensity, an infinite connected cluster of excluded volume appears with positive probability.Comment: 16 pages, 6 figure

On the phase transition curve in a directed exponential random graph model

We consider a family of directed exponential random graph models parametrized by edges and outward stars. Much of the important statistical content of such models is given by the normalization constant of the models, and in particular, an appropriately scaled limit of the normalization, which is called the free energy. We derive precise asymptotics for the normalization constant for finite graphs. We use this to derive a formula for the free energy. The limit is analytic everywhere except along a curve corresponding to a first order phase transition. We examine unusual behavior of the model along the phase transition curve.Comment: 31 pages, 2 figure