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Vlada Limic, University of British Columbia
Anja Sturm, University of Delaware
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Abstract
This paper extends the notion of the Λ-coalescent of Pitman (1999)
to the spatial setting. The partition elements of the spatial
Λ-coalescent migrate in a (finite) geographical space and may only coalesce if
located at the same site of the space. We characterize the Λ-coalescents
that come down from infinity, in an analogous way to Schweinsberg (2000).
Surprisingly, all spatial coalescents that come down from infinity, also come
down from infinity in a uniform way. This enables us to study space-time
asymptotics of spatial Λ-coalescents on large tori in d ≥ 3
dimensions.
Some of our results generalize and strengthen the corresponding results in
Greven et al.~(2005) concerning the spatial Kingman coalescent.
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Full text: PDF
Pages: 363-393
Published on: May 19, 2006
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Bibliography
D.J. Aldous.
Deterministic and stochastic models for coalescence (aggregation,
coagulation): A review of the mean-field theory for Probabilists.
Bernoulli 5 (1999), 3-48.
Math. Review 2001c:60153
R. Arratia.
Coalescing Brownian motion and the whole voter model on Z.
Ph.D.disertation. Univ. Wisconsin-Madison (1979).
N.H. Barton, A.M. Etheridge and A.K. Sturm.
Coalescence in a random background.
Ann. Appl. Probab. 14 (2) (2004), 754-785.
Math. Review 2005e:60190
M. Bramson, D. Griffeath.
Asymptotics for interacting particle systems on Z^d.
Z. Wahrscheinlichkeit verw. Gebiete
53 (1980), 183-196.
Math. Review 82a:60147
J. Berestycki
Exchangeable fragmentation-coalescence processes and their equilibrium
measures.
Electronic J. Probab. 9 (2004), 770-824.
Math. Review 2005k:60232
J. Berestycki, N. Berestycki and J. Schweinsberg.
Continuous-state branching processes and the properties of
beta coalescents.
Preprint.
J. Bertoin, J.-F. Le Gall.
The Bolthausen-Sznitman coalescent and the genealogy of
continuous-state branching processes.
Probab. Theory Relat. Fields
117 (2000), 249-266.
Math. Review 2001h:60150
J. Bertoin, J.-F. Le Gall. Stochastic flows associated to coalescent processes.
Probab. Theory Relat. Fields 126 (2003), 261-288.
Math. Review 2004f:60080
J. Bertoin, J.-F. Le Gall.
Stochastic flows associated to coalescent processes (2005) III: Limit theorems.
preprint at arXiv.org: math.PR/0506092
M. Birkner, J. Blath, M. Capaldo, A. Etheridge, M. Möhle, J.
Schweinsberg, A. Wakolbinger.
Alpha-stable branching and Beta coalescents.
Electronic J. Probab 10 (9) (2005), 303-325.
Math. Review 2006c:60100
E. Bolthausen, A. Sznitman.
On Ruelle's probability cascades and an abstract cavity method.
Comm. Math. Phys. 197 (1998), 247-276.
Math. Review 99k:60244
J.T. Cox. Coalescing random walks and voter model consensus
times on the torus in Z^d.
Ann. Probab. 17 (1989), 1333-1366.
Math. Review 91d:60250
R. Durrett, J. Schweinsberg.
A coalescent model for the effect of advantageous mutations on the
genealogy of a population.
Stochastic Process. Appl. 115 (2005), 1628-1657.
Math. Review 2165337
S.N. Ethier, T.G. Kurtz.
Markov processes: characterization and convergence (1986),
Wiley Series in Probability and Mathematical Statistics.
Math. Review 88a:60130
S.N. Evans, J. Pitman.
Construction of Markovian coalescents.
Ann. Inst. Henri Poincarè 34 (3) (1998), 339-383.
Math. Review 99k:60184
A. Greven, V. Limic and A. Winter.
Representation theorems for interacting Moran models, interacting
Fisher-Wright diffusions and applications.
Electronic J. Probab. 10 (39) (2005), 1286-1358.
Math. Review 2176385
A. Greven, V. Limic and A. Winter.
Coalescent processes arising in
a study of diffusive clustering.
In preparation (2006).
A. Greven, V. Limic and A. Winter.
Large Cluster formation in spatial Moran models in critical dimension via
particle representations.
In preparation (2006).
H.M. Herbots. The structured coalescent.
In P.Donnelly and S.Tavare, editors, Progress of Population
Genetics and Human Evolution 231-255 (1997), Springer.
Math. Review 99h:92016
J.F.C. Kingman. The coalescent.
Stochastic Process. Appl. 13 (1982), 235-248.
Math. Review 84a:60079
M. Möhle, S. Sagitov. A Classification of coalescent
processes for haploid exchangeable population models. Ann. Probab.
29 (2001), 1547-1562.
Math. Review 2003b:60134
M. Notohara.
The coalescent and the genealogical process in geographically
structured populations.
Journal of Mathematical Biology 31 (1990), 841-852.
Math. Review 92b:92043
J. Pitman. Coalescents with multiple collisions.
Ann. Probab. 27 (4) (1999), 1870-1902.
Math. Review 2001h:60016
L.C.G. Rogers, D. Williams.
Diffusion, Markov Processes and Martingales, Volume I.
Cambridge University Press (1979).
Math. Review 2001g:60188
S. Sagitov.
The general coalescent with asynchronous mergers of ancestral lines.
J. Appl. Prob. 36 (4)(1999), 1116-1125.
Math. Review 2001f:92019
J. Schweinsberg.
A necessary and sufficient condition for the Lambda-coalescent
to come down from infinity.
Electr. Comm. Probab. 5 (2000), 1-11.
Math. Review 2001g:60025
J. Schweinsberg.
Coalescent processes obtained from supercritical Galton-Watson processes.
Stochastic Process. Appl. 106 (2005), 107-139.
Math. Review 2004d:60222
I. Zähle, J.T. Cox and R. Durrett.
The stepping stone model. II: Genealogies and the infinite sites model.
Ann. Appl. Probab. 15 (1B) (2005), 671-699.
Math. Review 2006d:60157
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Electronic Journal of Probability. ISSN: 1083-6489 |
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