Abstract
We study several fundamental properties of a class of stochastic processes called spatial Λ-coalescents. In these models, a number of particles perform independent random walks on some underlying graph G. In addition, particles on the same vertex merge randomly according to a given coalescing mechanism. A remarkable property of mean-field coalescent processes is that they may come down from infinity, meaning that, starting with an infinite number of particles, only a finite number remains after any positive amount of time, almost surely. We show here however that, in the spatial setting, on any infinite and bounded-degree graph, the total number of particles will always remain infinite at all times, almost surely. Moreover, if \({G\,=\,\mathbb{Z}^d}\), and the coalescing mechanism is Kingman’s coalescent, then starting with N particles at the origin, the total number of particles remaining is of order (log* N)d at any fixed positive time (where log* is the inverse tower function). At sufficiently large times the total number of particles is of order (log* N)d-2, when d > 2. We provide parallel results in the recurrent case d = 2. The spatial Beta-coalescents behave similarly, where log log N is replacing log* N.
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Aldous D.J.: Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5(1), 3–48 (1999)
Arratia R.: Limiting point processes for rescalings of coalescing and annihilating random walks on Z d. Ann. Probab. 9(6), 909–936 (1981)
Berestycki, J., Berestycki, N., Limic, V.: The Λ-coalescent speed of coming down from infinity. Ann. Probab. (2010)
Berestycki J., Berestycki N., Schweinsberg J.: Beta-coalescents and continuous stable random trees. Ann. Probab. 35(5), 1835–1887 (2007)
Berestycki J., Berestycki N., Schweinsberg J.: Small-time behavior of beta-coalescents. Ann. Inst. H. Poincaré Probab. Statist. 44(2), 214–238 (2008)
Berestycki N.: Recent progress in coalescent theory. Ensaios Matematicos 16, 1–193 (2010)
Berg J.v.d., Kesten H.: Asymptotic density in a coalescing random walk model. Ann. Probab. 28(1), 303–352 (2000)
Berg J.v.d., Kesten H.: Randomly coalescing random walks in dimension ≥ 3. In: Sidoravicius, V. (eds) In and Out of Equilibrium, Volume 51 of Progress in Probability, pp. 1–45. Birkhauser, Boston (2002)
Bertoin J.: Random Fragmentation and Coagulation Processes, Volume 102 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2006)
Bertoin J., Le Gall J.-F.: Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50(1–4), 147–181 (2006) (electronic)
Birkner M., Blath J., Capaldo M., Etheridge A., Möhle M., Schweinsberg J., Wakolbinger A.: Alpha-stable branching and beta-coalescents. Electron. J. Probab. 10(9), 303–325 (2005) (electronic)
Bolthausen E., Sznitman A.-S.: On Ruelle’s probability cascades and an abstract cavity method. Commun. Math. Phys. 197(2), 247–276 (1998)
Bovier, A., Kurkova, I.: Much ado about Derrida’s GREM. In: Bolthausen, E., Bovier, A. (eds.) Spin Glasses, vol. 1900 of Lecture Notes in Mathematics, pp. 81–113. Springer, Berlin (2007)
Bramson M., Griffeath D.: Asymptotics for interacting particle systems on Z d. Z. Wahrsch. Verw. Gebiete 53(2), 183–196 (1980)
Brunet E., Derrida B., Simon D.: Universal tree structures in directed polymers and models of evolving populations. Phys. Rev. E 78, 061102 (2008)
Cox J., Durrett R.: The stepping stone model: new formulae expose old myths. Ann. Appl. Probab. 12, 1348–1377 (2002)
Cox J.T., Griffeath D.: Diffusive clustering in the two dimensional voter model. Ann. Probab. 14, 347–370 (1986)
Durrett R.: Probability Models for DNA Sequence Evolution, 2nd edn. Springer, New York (2008)
Erdös P., Taylor S.J.: Some problems concerning the structure of random walk paths. Acta Math. Acad. Sci. Hungar. 11, 137–162 (1960)
Greven, A., Limic, V., Winter, A.: Coalescent processes arising in the study of diffusive clustering. http://arxiv.org/abs/math/0703875
Greven A., Limic V., Winter A.: Representation theorems for interacting moran models, interacting fisher-wrighter diffusions and applications. Electron. J. Probab. 7, 74 (2005)
Hammond A., Rezakhanlou F.: Kinetic limit for a system of coagulating planar Brownian particles. J. Stat. Phys. 124(2–4), 997–1040 (2006)
Hammond A., Rezakhanlou F.: The kinetic limit of a system of coagulating Brownian particles. Arch. Ration. Mech. Anal. 185(1), 1–67 (2007)
Kesten H.: The number of alleles in electrophoretic experiments. Theoret. Pop. Biol. 18(2), 290–294 (1980)
Kesten H.: The number of distinguishable alleles according to the ohta-kimura model of neutral mutation. J. Math. Biol. 10, 167–187 (1980)
Kimura M.: “Stepping stone” model of population genetics. Ann. Rept. Nat. Inst. Genetics. Jpn 3, 62–63 (1953)
Kimura M., Weiss G.: The stepping stone model of population structure and the decrease of genetic correlations with distance. Genetics 49, 561–576 (1964)
Kingman J.F.C.: The coalescent. Stochastic Process. Appl. 13(3), 235–248 (1982)
Kingman J.F.C.: On the genealogy of large populations. J. Appl. Probab., Special vol. 19A, 27–43 (1982) (Essays in statistical science)
Lawler, G.F., Limic, V.: Random Walk: a modern introduction. Book in preparation. http://www.math.uchicago.edu/~lawler/books.html
Liggett, T.M.: Interacting particle systems, volume 276 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, New York (1985)
Limic V., Sturm A.: The spatial Λ-coalescent. Electron. J. Probab. 11(15), 363–393 (2006) (electronic)
Merle M.: Hitting probability of a distant point for the voter model started with a single one. Ann. Probab. 36(3), 807–861 (2008)
Pitman J.: Coalescents with multiple collisions. Ann. Probab. 27(4), 1870–1902 (1999)
Pitman, J.: Combinatorial stochastic processes, volume 1875 of Lecture Notes in Mathematics. Springer, Berlin, 2006. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24 (2002)
Schweinsberg J.: Coalescent processes obtained from supercritical Galton-Watson processes. Stochastic Process. Appl. 106(1), 107–139 (2003)
Spitzer F.: Principles of Random Walk, volume 34 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1976)
Talagrand M.: The Parisi formula. Ann. Math. 16(1), 221–263 (2006)
Zähle I., Cox J.T., Durrett R.: The stepping stone model II: Genealogies and the infinite sites model. Ann. Appl. Probab. 15, 671–699 (2005)
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O. Angel was supported in part by NSERC; N. Berestycki was supported in part by EPSRC grant EP/G055068/1; and V. Limic was supported in part by Alfred P. Sloan Research Fellowship, and in part by ANR MAEV grant.
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Angel, O., Berestycki, N. & Limic, V. Global divergence of spatial coalescents. Probab. Theory Relat. Fields 152, 625–679 (2012). https://doi.org/10.1007/s00440-010-0332-5
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DOI: https://doi.org/10.1007/s00440-010-0332-5