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J. Theodore Cox

Publications and source records attributed to J. Theodore Cox.

13 recordsLinked to original sources
arXiv2019

Rescaling the spatial lambda Fleming-Viot process and convergence to super-Brownian motion

We show that a space-time rescaling of the spatial Lamba-Fleming-Viot process of Barton and Etheridge converges to super-Brownian motion. This can be viewed as an extension of a result of Chetwynd-Diggle and Etheridge (2018). In that work the scaled impact factors (which govern the event based dynamics) vanish in the limit, here we drop that requirement. The analysis is particularly interesting in the biologically relevant two-dimensional case.

math.PR↗
arXiv2016

Weak atomic convergence of finite voter models toward Fleming-Viot processes

We consider the empirical measures of multi-type voter models with mutation on large finite sets, and prove their weak atomic convergence in the sense of Ethier and Kurtz (1994) toward a Fleming-Viot process. Convergence in the weak atomic topology is strong enough to answer a line of inquiry raised by Aldous (2013) concerning the distributions of the corresponding entropy processes and diversity processes for types.

math.PR↗
arXiv2016

Cutoff for the noisy voter model

Given a continuous time Markov Chain $\{q(x,y)\}$ on a finite set $S$, the associated noisy voter model is the continuous time Markov chain on $\{0,1\}^S$, which evolves in the following way: (1) for each two sites $x$ and $y$ in $S$, the state at site $x$ changes to the value of the state at site $y$ at rate $q(x,y)$; (2) each site rerandomizes its state at rate 1. We show that if there is a uniform bound on the rates $\{q(x,y)\}$ and the corresponding stationary distributions are almost uniform, then the mixing time has a sharp cutoff at time $\log|S|/2$ with a window of order 1. Lubetzky and Sly proved cutoff with a window of order 1 for the stochastic Ising model on toroids; we obtain the special case of their result for the cycle as a consequence of our result. Finally, we consider the model on a star and demonstrate the surprising phenomenon that the time it takes for the chain started at all ones to become close in total variation to the chain started at all zeros is of smaller order than the mixing time.

math.PR↗
arXiv2014

A complete convergence theorem for voter model perturbations

We prove a complete convergence theorem for a class of symmetric voter model perturbations with annihilating duals. A special case of interest covered by our results is the stochastic spatial Lotka-Volterra model introduced by Neuhauser and Pacala [Ann. Appl. Probab. 9 (1999) 1226-1259]. We also treat two additional models, the "affine" and "geometric" voter models.

math.PR↗
arXiv2013

On the convergence of densities of finite voter models to the Wright-Fisher diffusion

We study voter models defined on large sets. Through a perspective emphasizing the martingale property of voter density processes, we prove that in general, their convergence to the Wright-Fisher diffusion only involves certain averages of the voter models over a small number of spatial locations. This enables us to identify suitable mixing conditions on the underlying voting kernels, one of which may just depend on their eigenvalues in some contexts, to obtain the convergence of density processes. Our examples show that these conditions are satisfied by a large class of voter models on growing finite graphs.

math.PR↗
arXiv2011

A Branching Process for Virus Survival

Quasispecies theory predicts that there is a critical mutation probability above which a viral population will go extinct. Above this threshold the virus loses the ability to replicate the best adapted genotype, leading to a population composed of low replicating mutants that is eventually doomed. We propose a new branching model that shows that this is not necessarily so. That is, a population composed of ever changing mutants may survive.

math.PR↗
arXiv2011

Voter Model Perturbations and Reaction Diffusion Equations

We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions $d \ge 3$. Combining this result with properties of the PDE, some methods arising from a low density super-Brownian limit theorem, and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are close to the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, and (iii) a continuous time version of the non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first application confirms a conjecture of Cox and Perkins and the second confirms a conjecture of Ohtsuki et al in the context of certain infinite graphs. An important feature of our general results is that they do not require the process to be attractive.

math.PR↗
arXiv2010

Intermediate range migration in the two-dimensional stepping stone model

We consider the stepping stone model on the torus of side $L$ in $\mathbb{Z}^2$ in the limit $L\to\infty$, and study the time it takes two lineages tracing backward in time to coalesce. Our work fills a gap between the finite range migration case of [Ann. Appl. Probab. 15 (2005) 671--699] and the long range case of [Genetics 172 (2006) 701--708], where the migration range is a positive fraction of $L$. We obtain limit theorems for the intermediate case, and verify a conjecture in [Probability Models for DNA Sequence Evolution (2008) Springer] that the model is homogeneously mixing if and only if the migration range is of larger order than $(\log L)^{1/2}$.

math.PR↗
arXiv2009

Contact process in a wedge

We prove that the supercritical one-dimensional contact process survives in certain wedge-like space-time regions, and that when it survives it couples with the unrestricted contact process started from its upper invariant measure. As an application we show that a type of weak coexistence is possible in the nearest-neighbor ``grass-bushes-trees'' successional model introduced in Durrett and Swindle (1991).

math.PR↗
arXiv2009

Survival and coexistence for a multitype contact process

We study the ergodic theory of a multitype contact process with equal death rates and unequal birth rates on the $d$-dimensional integer lattice and regular trees. We prove that for birth rates in a certain interval there is coexistence on the tree, which by a result of Neuhauser is not possible on the lattice. We also prove a complete convergence result when the larger birth rate falls outside of this interval.

math.PR↗
arXiv2005

Rescaled Lotka-Volterra models converge to super-Brownian motion

We show that a sequence of stochastic spatial Lotka-Volterra models, suitably rescaled in space and time, converges weakly to super-Brownian motion with drift. The result includes both long range and nearest neighbor models, the latter for dimensions three and above. These theorems are special cases of a general convergence theorem for perturbations of the voter model.

math.PR↗
arXiv2005

The stepping stone model. II: Genealogies and the infinite sites model

This paper extends earlier work by Cox and Durrett, who studied the coalescence times for two lineages in the stepping stone model on the two-dimensional torus. We show that the genealogy of a sample of size n is given by a time change of Kingman's coalescent. With DNA sequence data in mind, we investigate mutation patterns under the infinite sites model, which assumes that each mutation occurs at a new site. Our results suggest that the spatial structure of the human population contributes to the haplotype structure and a slower than expected decay of genetic correlation with distance revealed by recent studies of the human genome.

math.PR↗