We give a mathematical framework for temperature accelerated dynamics (TAD), an algorithm proposed by M.R. S{\o}rensen and A.F. Voter to efficiently generate metastable stochastic dynamics. Using the notion of quasistationary distributions, we propose some modifications to TAD. Then considering the modified algorithm in an idealized setting, we show how TAD can be made mathematically rigorous.Comment: 28 pages, 2 figure

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.

We study the asymptotics of large directed graphs, constrained to have certain densities of edges and/or outward $p$-stars. Our models are close cousins of exponential random graph models (ERGMs), in which edges and certain other subgraph densities are controlled by parameters. The idea of directly constraining edge and other subgraph densities comes from Radin and Sadun. Such modeling circumvents a phenomenon first made precise by Chatterjee and Diaconis: that in ERGMs it is often impossible to independently constrain edge and other subgraph densities. In all our models, we find that large graphs have either uniform or bipodal structure. When edge density (resp. $p$-star density) is fixed and $p$-star density (resp. edge density) is controlled by a parameter, we find phase transitions corresponding to a change from uniform to bipodal structure. When both edge and $p$-star density are fixed, we find only bipodal structures and no phase transition.Comment: 24 pages, 2 figure

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.

The parallel replica dynamics, originally developed by A.F. Voter, efficiently simulates very long trajectories of metastable Langevin dynamics. We present an analogous algorithm for discrete time Markov processes. Such Markov processes naturally arise, for example, from the time discretization of a continuous time stochastic dynamics. Appealing to properties of quasistationary distributions, we show that our algorithm reproduces exactly (in some limiting regime) the law of the original trajectory, coarsened over the metastable states.KEY WORDS Markov chain, parallel computing, parallel replica dynamics, quasistationary distributions, metastabilityReceived XXX