arXiv subjects

David Aristoff

Publications and source records attributed to David Aristoff.

31 records · Page 2Linked to original sources

arXiv2015

Asymptotic structure and singularities in constrained directed graphs

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.

math.PR↗

arXiv2015

The parallel replica method for computing equilibrium averages of Markov chains

An algorithm is proposed for computing equilibrium averages of Markov chains which suffer from metastability -- the tendency to remain in one or more subsets of state space for long time intervals. The algorithm, called the parallel replica method (or ParRep), uses many parallel processors to explore these subsets more efficiently. Numerical simulations on a simple model demonstrate consistency of the method. A proof of consistency is given in an idealized setting. The parallel replica method can be considered a generalization of A.F. Voter's parallel replica dynamics, originally developed to efficiently simulate metastable Langevin stochastic dynamics.

math.NA↗

arXiv2014

The parallel replica method for simulating long trajectories of Markov chains

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.

math.NA↗

arXiv2014

Mathematical Analysis of Temperature Accelerated Dynamics

We give a mathematical framework for temperature accelerated dynamics (TAD), an algorithm proposed by M.R. Sørensen 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.

math-ph↗

arXiv2013

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.

math-ph↗

arXiv2012

Estimating Small Probabilities for Langevin Dynamics

The problem of estimating small transition probabilities for overdamped Langevin dynamics is considered. A simplification of Girsanov's formula is obtained in which the relationship between the infinitesimal generator of the underlying diffusion and the change of probability measure corresponding to a change in the potential energy is made explicit. From this formula an asymptotic expression for transition probability densities is derived. Separately the problem of estimating the probability that a small noise Langevin process excapes a potential well is discussed.

math-ph↗

arXiv2011

Emergent structures in large networks

We consider a large class of exponential random graph models and prove the existence of a region of parameter space corresponding to multipartite structure, separated by a phase transition from a region of disordered graphs.

math.PR↗

arXiv2011

Rigidity in Solids

We address the question of whether solids may be distinguished from fluids by their response to shear stress

cond-mat.stat-mech↗

arXiv2011

First Order Phase Transition in a Model of Quasicrystals

We introduce a family of two-dimensional lattice models of quasicrystals, using a range of square hard cores together with a soft interaction based on an aperiodic tiling set. Along a low temperature isotherm we find, by Monte Carlo simulation, a first order phase transition between disordered and quasicrystalline phases.

cond-mat.stat-mech↗

arXiv2010

First Order Phase Transition of a Long Polymer Chain

We consider a model consisting of a self-avoiding polygon occupying a variable density of the sites of a square lattice. A fixed energy is associated with each $90^\circ$-bend of the polygon. We use a grand canonical ensemble, introducing parameters $μ$ and $β$ to control average density and average (total) energy of the polygon, and show by Monte Carlo simulation that the model has a first order, nematic phase transition across a curve in the $β$-$μ$ plane.

cond-mat.soft↗

arXiv2010

Dilatancy transition in a granular model

We introduce a model of granular matter and use a stress ensemble to analyze shearing. Monte Carlo simulation shows the model to exhibit a second order phase transition, associated with the onset of dilatancy.

cond-mat.soft↗

arXiv2010

Layering in Crumpled Sheets

We introduce a toy model of crumpled sheets. We use simulation to show there is a first order phase transition in the model, from a disordered dilute phase to a mixture with a layered phase.

cond-mat.soft↗

arXiv2008

Random Loose Packing in Granular Matter

We introduce and simulate a two dimensional probabilistic model of granular matter at vanishing pressure. The model exhibits a perfectly sharp random loose packing density, a phenomenon that should be verifiable for real granular matter.

cond-mat.soft↗