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Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
The arcsine law as a universal aging scheme for trap models
 COMM. PURE APPL. MATH
, 2006
"... We give a general proof of aging for trap models using the arcsine law for stable subordinators. This proof is based on abstract conditions on the potential theory of the underlying graph and on the randomness of the trapping landscape. We apply this proof to aging for trap models on large twodime ..."
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We give a general proof of aging for trap models using the arcsine law for stable subordinators. This proof is based on abstract conditions on the potential theory of the underlying graph and on the randomness of the trapping landscape. We apply this proof to aging for trap models on large twodimensional tori and for trap dynamics of the Random Energy Model on a broad range of time scales.
A new model for evolution in a spatial continuum
"... o b a b i l i t y Vol. 15 (2010), Paper no. 7, pages 162–216. Journal URL ..."
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o b a b i l i t y Vol. 15 (2010), Paper no. 7, pages 162–216. Journal URL
The stepping stone model, II: Genealogies and the infinite sites model, submitted
, 2005
"... This paper extends earlier work by Cox and Durrett, who studied the coalescence times for two lineages in the stepping stone model on the twodimensional 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 inves ..."
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Cited by 17 (3 self)
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This paper extends earlier work by Cox and Durrett, who studied the coalescence times for two lineages in the stepping stone model on the twodimensional 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. 1. Introduction. Sequencing
The spread of a rumor or infection in a moving population
 2005) Shape and Propagation of Fronts 27
"... We consider the following interacting particle system: There is a “gas ” of particles, each of which performs a continuoustime simple random walk on Z d, with jump rate DA. These particles are called Aparticles and move independently of each other. They are regarded as individuals who are ignorant ..."
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We consider the following interacting particle system: There is a “gas ” of particles, each of which performs a continuoustime simple random walk on Z d, with jump rate DA. These particles are called Aparticles and move independently of each other. They are regarded as individuals who are ignorant of a rumor or are healthy. We assume that we start the system with NA(x,0−) Aparticles at x, and that the NA(x,0−),x ∈ Z d, are i.i.d., meanµA Poisson random variables. In addition, there are Bparticles which perform continuoustime simple random walks with jump rate DB. We start with a finite number of Bparticles in the system at time 0. Bparticles are interpreted as individuals who have heard a certain rumor or who are infected. The Bparticles move independently of each other. The only interaction is that when a Bparticle and an Aparticle coincide, the latter instantaneously turns into a Bparticle. We investigate how fast the rumor, or infection, spreads. Specifically, if ˜B(t): = {x ∈ Z d: a Bparticle visits x during [0,t]} and B(t) = ˜B(t)+[−1/2,1/2] d, then we investigate the asymptotic behavior of B(t). Our principal result states that if DA = DB (so that the A and Bparticles perform the same random walk), then there exist constants 0 < Ci < ∞ such that almost surely C(C2t) ⊂ B(t) ⊂ C(C1t) for all large t, where C(r) = [−r,r] d. In a further paper we shall use the results presented here to prove a full “shape theorem, ” saying that t −1 B(t) converges almost surely to a nonrandom set B0, with the origin as an interior point, so that the true growth rate for B(t) is linear in t. If DA ̸ = DB, then we can only prove the upper bound B(t) ⊂ C(C1t) eventually.
Coalescent processes in subdivided populations subject to recurrent mass extinctions
 Electr. J. Probab
, 2009
"... E l e c t r o n J o u r n a l ..."
Evolutionary Sound Synthesis: Rendering Spectrograms from Cellular Automata Histograms
"... Abstract. In this paper we report on the synthesis of sounds using cellular automata, specifically the multitype voter model. The mapping process adopted is based on digital signal processing analysis of automata evolutions and consists in mapping histograms onto spectrograms. The main problem of ce ..."
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Abstract. In this paper we report on the synthesis of sounds using cellular automata, specifically the multitype voter model. The mapping process adopted is based on digital signal processing analysis of automata evolutions and consists in mapping histograms onto spectrograms. The main problem of cellular automata is the difficulty of control and, consequently, sound synthesis methods based on these computational models normally present a high factor of randomness in the output. We have achieved a significant degree of control as to predict the type of sounds that we can obtain. We are able to develop a flexible sound design process with emphasis on the possibility of controlling over time the spectrum complexity.
ONEDIMENSIONAL STEPPING STONE MODELS, SARDINE GENETICS AND BROWNIAN LOCAL TIME
, 801
"... Consider a onedimensional stepping stone model with colonies of size M and pergeneration migration probability ν, or a voter model on Z in which interactions occur over a distance of order K. Sample one individual at the origin and one at L. We show that if Mν/L and L/K 2 converge to positive fini ..."
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Consider a onedimensional stepping stone model with colonies of size M and pergeneration migration probability ν, or a voter model on Z in which interactions occur over a distance of order K. Sample one individual at the origin and one at L. We show that if Mν/L and L/K 2 converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a onedimensional parabolic differential equation with an interesting boundary condition at 0. 1. Introduction. Cox and Durrett [6] and Zähle, Cox and Durrett [15] have recently studied the twodimensional stepping stone model. Space is represented as a torus Λ(L) = (ZmodL) 2. To avoid a factor of 2 and to make the dynamics easier to describe, we suppose that at each point x ∈ Λ(L) there is a colony of M haploid individuals labeled 1,2,...,M. Each
Cellular Automata Sound Synthesis with an Extended Version of the Multitype Voter Model
, 2010
"... The papers at this Convention have been selected on the basis of a submitted abstract and extended precis that have been peer reviewed by at least two qualified anonymous reviewers. This convention paper has been reproduced from the author's advance manuscript, without editing, corrections, or ..."
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The papers at this Convention have been selected on the basis of a submitted abstract and extended precis that have been peer reviewed by at least two qualified anonymous reviewers. This convention paper has been reproduced from the author's advance manuscript, without editing, corrections, or consideration by the Review Board. The AES takes no responsibility for the contents.
The Distribution of Fst and other Genetic Statistics for a Class of Population Structure Models
, 2008
"... We examine genetic statistics used in the study of structured populations. In a 1999 paper, Wakeley observed that the coalescent process associated with the finite island model can be decomposed into a scattering phase and a collecting phase. In this paper, we introduce a class of population structu ..."
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We examine genetic statistics used in the study of structured populations. In a 1999 paper, Wakeley observed that the coalescent process associated with the finite island model can be decomposed into a scattering phase and a collecting phase. In this paper, we introduce a class of population structure models, which we refer to as G/KC models, that obey such a decomposition. In a large population, large sample limit we derive the distribution of the statistic Fst for all G/KC models under the assumptions of strong or weak mutation. We show that in the large population, large sample limit the island and two dimensional stepping stone models are members of the G/KC class of models, thereby deriving the distributions of Fst for these two well known models as a special case of a general formula. We show that our analysis of Fst can be extended to an entire class of genetic statistics, and we use our approach to examine homozygosity measures. Our analysis uses coalescent based methods. 1