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 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.

[1]  William E. Pruitt,et al.  The Growth of Random Walks and Levy Processes , 1981 .

[2]  Bernard Derrida,et al.  Effect of Microscopic Noise on Front Propagation , 2001 .

[3]  J. L. Gall,et al.  Branching processes in Lévy processes: the exploration process , 1998 .

[4]  F. Guerra,et al.  Ju l 1 99 8 General properties of overlap probability distributions in disordered spin systems , 1998 .

[5]  S. Evans.  Probability and Real Trees , 2008 .

[6]  Thomas Duquesne,et al.  Random Trees, Levy Processes and Spatial Branching Processes , 2002 .

[7]  Martin Möhle,et al.  Alpha-Stable Branching and Beta-Coalescents , 2005 .

[8]  Jean-François Delmas,et al.  Asymptotic results on the length of coalescent trees , 2007, 0706.0204.

[9]  D. Aldous.  Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists , 1999 .

[10]  Jason Schweinsberg.  ELECTRONIC COMMUNICATIONS in PROBABILITY A NECESSARY AND SUFFICIENT CONDITION FOR THE Λ-COALESCENT TO COME DOWN FROM IN- FINITY. , 2022 .

[11]  Stefan Grosskinsky Warwick,et al.  Interacting Particle Systems , 2016 .

[12]  J. Kingman.  The Representation of Partition Structures , 1978 .

[13]  J. Bertoin,et al.  Stochastic flows associated to coalescent processes II: Stochastic differential equations , 2005 .

[14]  D. Applebaum.  Lévy Processes—From Probability to Finance and Quantum Groups , 2004 .

[15]  G. Grimmett,et al.  On the asymptotic distribution of large prime factors , 1993 .

[16]  Christina Goldschmidt,et al.  Random Recursive Trees and the Bolthausen-Sznitman Coalesent , 2005, math/0502263.

[17]  D. Aldous.  Exchangeability and related topics , 1985 .

[18]  T. E. Harris,et al.  Coalescing and noncoalescing stochastic flows in R1 , 1984 .

[19]  Yueyun Hu,et al.  Asymptotics for the survival probability in a killed branching random walk , 2008 .

[20]  J. Pitman.  Combinatorial Stochastic Processes , 2006 .

[21]  N. Berestycki,et al.  Random paths with bounded local time , 2008, 0805.3326.

[22]  R. Fisher.  THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[23]  C. J-F,et al.  THE COALESCENT , 1980 .

[24]  Martin Möhle,et al.  Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models , 2000, Advances in Applied Probability.

[25]  J. L. Gall,et al.  Spatial Branching Processes, Random Snakes, and Partial Differential Equations , 1999 .

[26]  John Lamperti,et al.  Continuous state branching processes , 1967 .

[27]  M. Kimura,et al.  'Stepping stone' model of population , 1953 .

[28]  J. Coyne.  Lack of genic similarity between two sibling species of drosophila as revealed by varied techniques. , 1976, Genetics.

[29]  Miloslav Jiřina,et al.  Stochastic branching processes with continuous state space , 1958 .

[30]  Alison Etheridge,et al.  Drift, draft and structure: some mathematical models of evolution , 2008 .

[31]  Serik Sagitov,et al.  The general coalescent with asynchronous mergers of ancestral lines , 1999 .

[32]  J. Pitman.  Coalescents with multiple collisions , 1999 .

[33]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[34]  W. Ewens.  Mathematical Population Genetics , 1980 .

[35]  Ruelle's probability cascades seen as a fragmentation process , 2005, math/0501088.

[36]  Dudley Stark.  LOGARITHMIC COMBINATORIAL STRUCTURES: A PROBABILISTIC APPROACH (EMS Monographs in Mathematics) By R ICHARD A RRATIA , A. D. B ARBOUR and S IMON T AVARÉ : 363 pp., €69.00, ISBN 3-03719-000-0 (European Mathematical Society, 2003) , 2005 .

[37]  V. M. Tikhomirov,et al.  A Study of the Diffusion Equation with Increase in the Amount of Substance, and its Application to a Biological Problem , 1991 .

[38]  Small-time behavior of beta coalescents , 2006, math/0601032.

[39]  G. Parisi.  A sequence of approximated solutions to the S-K model for spin glasses , 1980 .

[40]  J. Bertoin,et al.  The Bolthausen–Sznitman coalescent and the genealogy of continuous-state branching processes , 2000 .

[41]  A. Bovier,et al.  Statistical Mechanics of Disordered Systems , 2006 .

[42]  M. Möhle.  On sampling distributions for coalescent processes with simultaneous multiple collisions , 2006 .

[43]  Richard Durrett,et al.  Approximating selective sweeps. , 2004, Theoretical population biology.

[44]  E. Brunet,et al.  Noisy traveling waves: Effect of selection on genealogies , 2006 .

[45]  J. T. Cox,et al.  Coalescing Random Walks and Voter Model Consensus Times on the Torus in $\mathbb{Z}^d$ , 1989 .

[46]  Amaury Lambert,et al.  Proof(s) of the Lamperti representation of Continuous-State Branching Processes , 2008, 0802.2693.

[47]  Jason Schweinsberg.  Coalescents with Simultaneous Multiple Collisions , 2000 .

[48]  Matthias Birkner,et al.  Computing likelihoods for coalescents with multiple collisions in the infinitely many sites model , 2008, Journal of mathematical biology.

[49]  A. Meir,et al.  Cutting down random trees , 1970, Journal of the Australian Mathematical Society.

[50]  R. Griffiths,et al.  Ewens' sampling formula and related formulae: combinatorial proofs, extensions to variable population size and applications to ages of alleles. , 2005, Theoretical population biology.

[51]  Patrick Billingsley,et al.  On the distribution of large prime divisors , 1972 .

[52]  Chris Cannings,et al.  The latent roots of certain Markov chains arising in genetics: A new approach, II. Further haploid models , 1974, Advances in Applied Probability.

[53]  E. A. Perkins.  Conditional Dawson—Watanabe Processes and Fleming—Viot Processes , 1992 .

[54]  J. Gillespie.  Genetic drift in an infinite population. The pseudohitchhiking model. , 2000, Genetics.

[55]  D. Ruelle.  A mathematical reformulation of Derrida's REM and GREM , 1987 .

[56]  E. Perkins,et al.  An application of the voter model-super-Brownian motion invariance principle , 2004 .

[57]  Alois Panholzer.  Destruction of Recursive Trees , 2004 .

[58]  N. Berestycki,et al.  BETA-COALESCENTS AND CONTINUOUS STABLE RANDOM TREES , 2006, math/0602113.

[59]  J. Berg,et al.  Randomly Coalescing Random Walk in Dimension ≥3 , 2002 .

[60]  Ken-iti Sato.  Lévy Processes and Infinitely Divisible Distributions , 1999 .

[61]  W. Ewens.  Mathematical Population Genetics : I. Theoretical Introduction , 2004 .

[62]  Richard Arratia,et al.  Limiting Point Processes for Rescalings of Coalescing and Annihilating Random Walks on $Z^d$ , 1981 .

[63]  T. Rolski.  On random discrete distributions , 1980 .

[64]  A. Etheridge,et al.  An introduction to superprocesses , 2000 .

[65]  R. Durrett,et al.  Random partitions approximating the coalescence of lineages during a selective sweep , 2004, math/0411069.

[66]  Coalescing Markov labelled partitions and a continuous sites genetics model with infinitely many types , 1997 .

[67]  E. Bolthausen,et al.  On Ruelle's Probability Cascades and an Abstract Cavity Method , 1998 .

[68]  Erwin Bolthausen,et al.  Ten Lectures on Random Media , 2002 .

[69]  Bernard Derrida,et al.  Shift in the velocity of a front due to a cutoff , 1997 .

[70]  A. Hammond,et al.  The Kinetic Limit of a System of Coagulating Brownian Particles , 2004, math/0408395.

[71]  Kinetic Limit for a System of Coagulating Planar Brownian Particles , 2005, math/0507522.

[72]  J.-B. Gouéré,et al.  Brunet-Derrida Behavior of Branching-Selection Particle Systems on the Line , 2008, 0811.2782.

[73]  R. Wolpert.  Lévy Processes , 2000 .

[74]  S. Ramaswami.  Semigroups of Measures on Lie Groups , 1974 .

[75]  Alexander Gnedin,et al.  On the number of collisions in ¤-coalescents ∗ , 2007 .

[76]  A. Wakolbinger,et al.  An approximate sampling formula under genetic hitchhiking , 2005, math/0503485.

[77]  J. T. Cox,et al.  Rescaled voter models converge to super-Brownian motion , 2000 .

[78]  N. Berestycki,et al.  Kingman's coalescent and Brownian motion. , 2009, 0904.1526.

[79]  C. Goldschmidt,et al.  Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent , 2007, 0706.2808.

[80]  Maury Bramson,et al.  Asymptotics for interacting particle systems onZd , 1980 .

[81]  M. Kimura,et al.  The Stepping Stone Model of Population Structure and the Decrease of Genetic Correlation with Distance. , 1964, Genetics.

[82]  W. Ewens.  The sampling theory of selectively neutral alleles. , 1972, Theoretical population biology.

[83]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[84]  N. Berestycki,et al.  Gibbs Distributions for Random Partitions Generated by a Fragmentation Process , 2005, math/0512378.

[85]  A. Hammond,et al.  Coagulation, diffusion and the continuous Smoluchowski equation , 2008, 0811.4601.

[86]  Jean Bertoin,et al.  Random fragmentation and coagulation processes , 2006 .

[87]  A. Meir,et al.  Cutting down recursive trees , 1974 .

[88]  T. F. Móri.  On random trees , 2002 .

[89]  Jean-François Marckert,et al.  The depth first processes of Galton--Watson trees converge to the same Brownian excursion , 2003 .

[90]  Jason Schweinsberg.  Coalescent processes obtained from supercritical Galton-Watson processes , 2003 .

[91]  B. Derrida.  Random-Energy Model: Limit of a Family of Disordered Models , 1980 .

[92]  R. Arratia,et al.  Logarithmic Combinatorial Structures: A Probabilistic Approach , 2003 .

[93]  M. Aizenman,et al.  On the structure of quasi-stationary competing particle systems , 2007, 0709.2901.

[94]  B. Derrida,et al.  Universal tree structures in directed polymers and models of evolving populations. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[95]  J. L. Gall,et al.  Random trees and applications , 2005 .

[96]  J. T. Cox,et al.  Diffusive Clustering in the Two Dimensional Voter Model , 1986 .

[97]  A mathematical analysis of the stepping stone model of genetic correlation. , 1965 .

[98]  John Lamperti,et al.  The Limit of a Sequence of Branching Processes , 1967 .

[99]  S. Sawyer.  A limit theorem for patch sizes in a selectively-neutral migration model , 1979, Journal of Applied Probability.

[100]  J. Pitman,et al.  The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator , 1997 .

[101]  N. Berestycki,et al.  Survival, extinction and ergodicity in a spatially continuous population model , 2009 .

[102]  M. Talagrand.  Mean Field Models for Spin Glasses: Some Obnoxious Problems , 2007 .

[103]  Z. Yang,et al.  Probability models for DNA sequence evolution , 2004, Heredity.

[104]  M. F.,et al.  Bibliography , 1985, Experimental Gerontology.

[105]  M. Birkner,et al.  A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks , 2008, 0808.0412.

[106]  École d'été de probabilités de Saint-Flour,et al.  École d'été de probabilités de Saint-Flour XIII - 1983 , 1985 .

[107]  W. König,et al.  A Survey of One-Dimensional Random Polymers , 2001 .

[108]  D. R. Grey.  Asymptotic behaviour of continuous time, continuous state-space branching processes , 1974 .

[109]  Peter Bajorski,et al.  Wiley Series in Probability and Statistics , 2010 .

[110]  G. Parisi.  Brownian motion , 2005, Nature.

[111]  I. Kurkova,et al.  Much Ado about Derrida's GREM , 2007 .

[112]  J. Pitman,et al.  Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws ∗ , 2007, math/0701718.

[113]  Robustness results for the coalescent , 1998 .

[114]  J. Bertoin,et al.  Stochastic flows associated to coalescent processes , 2003 .

[115]  R. Durrett.  Stepping Stone Model , 2008 .

[116]  William Feller,et al.  Diffusion Processes in Genetics , 1951 .

[117]  Frank Spitzer,et al.  The Galton-Watson Process with Mean One and Finite Variance , 1966 .

[118]  J. Quastel,et al.  Effect of noise on front propagation in reaction-diffusion equations of KPP type , 2009, 0902.3423.

[119]  J. Pitman.  Brownian Motion, Bridge, Excursion, and Meander Characterized by Sampling at Independent Uniform Times , 1999 .

[120]  F. Guerra,et al.  General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity , 1998, cond-mat/9807333.

[121]  R. W. Davis,et al.  Detection of numerous Y chromosome biallelic polymorphisms by denaturing high-performance liquid chromatography. , 1997, Genome research.

[122]  C. Cannings.  The latent roots of certain Markov chains arising in genetics: A new approach, II. Further haploid models , 1975, Advances in Applied Probability.

[123]  R. Durrett.  Probability: Theory and Examples , 1993 .

[124]  J. Pitman,et al.  Size-biased sampling of Poisson point processes and excursions , 1992 .

[125]  Asymptotics for the survival probability in a supercritical branching random walk , 2008 .

[126]  N. Berestycki,et al.  Global divergence of spatial coalescents , 2009, 0909.4859.

[127]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[128]  B. Derrida,et al.  Quasi-Stationary Regime of a Branching Random Walk in Presence of an Absorbing Wall , 2007, 0710.3689.

[129]  David Aldous,et al.  The Continuum Random Tree III , 1991 .

[130]  M. Möhle.  On the number of segregating sites for populations with large family sizes , 2006, Advances in Applied Probability.

[131]  J. T. Cox,et al.  Super-Brownian Limits of Voter Model Clusters , 2001 .

[132]  B. Derrida,et al.  Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[133]  N. Barton,et al.  A New Model for Evolution in a Spatial Continuum , 2009, 0904.0210.

[134]  Rick Durrett,et al.  A coalescent model for the effect of advantageous mutations on the genealogy of a population , 2004, math/0411071.

[135]  A. Véber,et al.  Coalescent processes in subdivided populations subject to recurrent mass extinctions , 2008, 0805.1010.

[136]  Alex Gamburd,et al.  Poisson–Dirichlet distribution for random Belyi surfaces , 2006 .

[137]  M. Möhle,et al.  On the number of allelic types for samples taken from exchangeable coalescents with mutation , 2008, Advances in Applied Probability.

[138]  J. Norris,et al.  Differential equation approximations for Markov chains , 2007, 0710.3269.

[139]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[140]  N. Berestycki,et al.  The Λ-coalescent speed of coming down from infinity , 2008, 0807.4278.

[141]  Peter Donnelly,et al.  Particle Representations for Measure-Valued Population Models , 1999 .

[142]  J. Berg,et al.  Asymptotic density in a coalescing random walk model , 1998 .

[143]  B. Derrida.  A generalization of the Random Energy Model which includes correlations between energies , 1985 .

[144]  J. Wakeley,et al.  Coalescence Times and FST Under a Skewed Offspring Distribution Among Individuals in a Population , 2009, Genetics.

[145]  John Wakeley,et al.  Coalescent Processes When the Distribution of Offspring Number Among Individuals Is Highly Skewed , 2006, Genetics.

[146]  An example of Brunet-Derrida behavior for a branching-selection particle system on $\Z$ , 2008, 0810.5567.

[147]  M. Drmota,et al.  Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent , 2007 .