DOCSLIB.ORG

The Brownian Web, the Brownian Net, and Their Universality

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Brownian Web, the Brownian Net, and Their Universality 6 The Brownian Web, the Brownian Net, and their Universality EMMANUEL SCHERTZER, RONGFENG SUN AND JAN M. SWART 6.1 Introduction The Brownian web originated from the work of Arratia’s Ph.D. thesis [1], where he studied diffusive scaling limits of coalescing random walk paths starting from everywhere on Z, which can be seen as the spatial genealogies of the population in the dual voter model on Z. Arratia showed that the collection of coalescing random walks converge to a collection of coalescing Brownian motions on R, starting from every point on R at time 0. Subsequently, Arratia [2] attempted to generalize his result by constructing a system of coalescing Brownian motions starting from everywhere in the space-time plane R2, which would be the scaling limit of coalescing random walk paths starting from everywhere on Z at every time t ∈ R. However, the manuscript [2] was never completed, even though fundamental ideas have been laid down. This topic remained dormant until Toth´ and Werner [99] discovered a surprising connection between the one-dimensional space-time coalescing Brownian motions that Arratia tried to construct, and an unusual process called the true self-repelling motion, which is repelled by its own local time profile. Building on ideas from [2], Toth´ and Werner [99] gave a construction of the system of space-time coalescing Brownian motions, and then used it to construct the true self-repelling motion. On the other hand, Fontes, Isopi, Newman and Stein [37] discovered that this system of space-time coalescing Brownian motions also arises in the study of aging and scaling limits of one-dimensional spin systems. To establish weak convergence of discrete models to the system of coalescing Brownian motions, Fontes et al. [38, 40] introduced a topology that the system of coalescing Brownian motions starting from every space-time point can be realized as a random variable taking values in a Polish space, and they named this random variable the Brownian web.An extension to the Brownian web was later introduced by the authors in [86], and independently by Newman, Ravishankar and Schertzer in [73]. This object was named the Brownian net in [86], where the coalescing paths in the Brownian web 270 Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 23 Apr 2018 at 02:40:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781316403877.007 The Brownian Web and the Brownian Net 271 are also allowed to branch. To counter the effect of instantaneous coalescence, the branching occurs at an effectively “infinite” rate. The Brownian web and net have very interesting properties. Their construction is nontrivial due to the uncountable number of starting points in space-time. Coalescence allows one to reduce the system to a countable number of starting points. In fact, the collection of coalescing paths starting from every point on R at time 0 immediately becomes locally finite when time becomes positive, similar to the phenomenon of coming down from infinity in Kingman’s coalescent (see e.g., [10]). In fact, the Brownian web can be regarded as the spatial analogue of Kingman’s coalescent, with the former arising as the limit of genealogies of the voter model on Z, and the latter arising as the limit of genealogies of the voter model on the complete graph. The key tool in the analysis of the Brownian web, as well as the Brownian net, is its self-duality, similar to the self-duality of critical bond percolation on Z2. Duality allows one to show that there exist random space-time points where multiple paths originate, and one can give a complete classification of these points. The Brownian web and net also admit a coupling, where the web can be constructed by sampling paths in the net, and conversely, the net can be constructed from the web by Poisson marking a set of “pivotal” points in the web and turning these into points where paths can branch. The latter construction is similar to the construction of scaling limits of near-critical planar percolation from that of critical percolation [19, 48, 49]. The Brownian web and net give rise to a new universality class. In particular, they are expected to arise as the universal scaling limits of one-dimensional interacting particle systems with coalescence, respectively branching-coalescence. One such class of models are population genetic models with resampling and selection, whose spatial genealogies undergo branching and coalescence. Establishing weak convergence to the Brownian web or net can also help in the study of the discrete particle systems themselves. Related models which have been shown to converge to the Brownian web include coalescing random walks [72], succession lines in Poisson trees [36, 23, 46] and drainage network type models [17, 22, 80]. Interesting connections with the Brownian web and net have also emerged from many unexpected sources, including supercritical oriented percolation on Z1+1 [6], planar aggregation models [76, 77], true self-avoiding random walks on Z [94, 99], random matrix theory [101, 100], and also one-dimensional random walks in i.i.d. space-time random environments [92]. There are also close parallels between the Brownian web and the scaling limit of critical planar percolation, which are the only known examples of two-dimensional black noise [95, 96, 87, 33]. The goal of this article is to give an introduction to the Brownian web and net, their basic properties, and how they arise in the scaling limits of one-dimensional interacting particle systems with branching and coalescence. We will focus on the Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 23 Apr 2018 at 02:40:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781316403877.007 272 E. Schertzer et al. key ideas, while referring many details to the literature. Our emphasis is naturally biased toward our own research. However, we will also briefly survey related work, including the many interesting connections mentioned above. We have left out many other closely related studies, including diffusion-limited reactions [28, 8] where a dynamic phase transition is observed for branching-coalescing random walks, the propagation of cracks in a sheet [27], rill erosion [31] and the directed Abelian Sandpile Model [25], quantum spin chains [60], etc., which all lie within the general framework of non-equilibrium critical phenomena discussed in the physics surveys [79, 53]. The rest of this article is organized as follows. In Section 6.2, we will construct and give a characterization of the Brownian web and study its properties. In Section 6.3, we do the same for the Brownian net. In Section 6.4, we introduce a coupling between the Brownian web and net and show how one can be constructed from the other. In Section 6.5, we will explain how the Brownian web and net can be used to construct the scaling limits of one-dimensional random walks in i.i.d. space-time random environments. In Section 6.6, we formulate convergence criteria for the Brownian web, which are then applied to coalescing random walks. We will also discuss strategies for proving convergence to the Brownian net. In Section 6.7, we survey other interesting models and results connected to the Brownian web and net. Lastly, in Section 6.8, we conclude with some interesting open questions. 6.2 The Brownian Web The Brownian web is best motivated by its discrete analogue, the collection of discrete time coalescing simple symmetric random walks on Z, with one Z2 ={ ∈ Z2 walker starting from every site in the space-time lattice even : (x,n) : + } Z2 x n is even . The restriction to the sublattice even is necessary due to parity. ∈ Z2 Figure 6.1 illustrates a graphical construction, where from each (x,n) even an independent arrow is drawn from (x,n) to either (x − 1,n + 1) or (x + 1,n + 1) with probability 1/2 each, determining whether the walk starting at x at time n should move to x−1orx+1 at time n+1. The objects of interest for us are the collection of upward random walk paths (obtained by following the arrows) starting from every space-time lattice point. The question is: Q.1 What is the diffusive scaling limit of this collection√ of coalescing random walk paths if space and time are scaled by 1/ n and 1/n respectively? Intuitively, it is not difficult to see that the limit should be a collection of coalescing Brownian motions, starting from everywhere in the space-time plane R2.This is what we will call the Brownian web. However, a conceptual difficulty arises, namely that we need to construct the joint realization of uncountably many Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 23 Apr 2018 at 02:40:07, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781316403877.007 The Brownian Web and the Brownian Net 273 Z2 Figure 6.1. Discrete space-time coalescing random walks on even, and its dual Z2 on odd. Brownian motions. Fortunately it turns out that coalescence allows us to reduce the construction to only a countable collection of coalescing Brownian motions. Note that in Figure 6.1, we have also drawn a collection of downward arrows Z2 ={ ∈ Z2 + } connecting points in the odd space-time lattice odd : (x,n) : x n is odd , which are dual to the upward arrows by the constraint that the upward and backward arrows do not cross each other.
Load more

Recommended publications
  • Collision Times of Random Walks and Applications to the Brownian
    Collision times of random walks and applications to the Brownian web David Coupier1, Kumarjit Saha2, Anish Sarkar3, and Viet Chi Tran4 1Univ. de Valenciennes, CNRS 2Ashoka University 3Indian Statistical Institute, Delhi 4Univ. Lille, CNRS February 12, 2019 Abstract Convergence of directed forests, spanning on random subsets of lattices or on point processes, towards the Brownian web has made the subject of an abundant literature, a large part of which relies on a criterion proposed by Fontes, Isopi, Newman and Ravishankar (2004). One of their conver- gence condition, called (B2), states that the probability of the event that there exists three distinct paths for a time interval of length t(> 0), all starting within a segment of length ε, is of small order of ε. This condition is often verified by applying an FKG type correlation inequality together with a coalescing time tail estimate for two paths. For many models where paths have complex interactions, it is hard to establish FKG type inequalities. In this article, we show that for a non-crossing path model, with certain assumptions, a suitable upper bound on expected first col- lision time among three paths can be obtained directly using Lyapunov functions. This, in turn, provides an alternate verification of Condition (B2). We further show that in case of independent simple symmetric one dimensional random walks or in case of independent Brownian motions, the expected value can be computed explicitly. We apply this alternate method of verification of (B2) to several models in the basin of attraction of the Brownian web studied earlier in the literature ([22], [14], [11]).
    [Show full text]
  • The Brownian Web
    The Annals of Probability 2008, Vol. 36, No. 3, 1153–1208 DOI: 10.1214/07-AOP357 c Institute of Mathematical Statistics, 2008 THE BROWNIAN NET By Rongfeng Sun and Jan M. Swart1 TU Berlin and UTIA´ Prague The (standard) Brownian web is a collection of coalescing one- dimensional Brownian motions, starting from each point in space and time. It arises as the diffusive scaling limit of a collection of coalescing random walks. We show that it is possible to obtain a nontrivial limiting object if the random walks in addition branch with a small probability. We call the limiting object the Brownian net, and study some of its elementary properties. Contents 1. Introduction and main results....................... .....................1154 1.1. Arrow configurations and branching-coalescing random walks.......1154 1.2. Topology and convergence......................... .................1155 1.3. The Brownian web................................. ................1158 1.4. Characterization of the Brownian net using hopping . .............1159 1.5. The left-right Brownian web....................... .................1160 1.6. Characterization of the Brownian net using meshes . .............1161 1.7. The dual Brownian web............................. ...............1162 1.8. Dual characterization of the Brownian net........... ...............1164 1.9. The branching-coalescing point set ................ ................. 1164 1.10. The backbone................................... ...................1166 1.11. Discussion, applications and open problems . .................1167
    [Show full text]
  • The Brownian Web
    The Brownian Web Linus Mattauch University of Oxford, Honour School of Mathematics and Philosophy, Part C Dissertation - March 2008 1 Acknowledgements I am extremely grateful to my supervisors, Alison Etheridge and Pierre Tarr`es,for guiding me through the challenging material on which this dissertation is based and for their patient explanations of mathematical details. I wish to thank R. Arratia for making a seminal unpublished manuscript [1] available to me. I am very grateful to R. Sun for helpful comments (in particular regarding [30, 31]) and explanation of a detail in [12]. I am, moreover, very grateful to Arend Janssen for reading a draft of this disser- tation and to my parents, Angela and Hans Mattauch, for proof-reading and encour- agement. Special mention goes to David Yadin for constant support and to Glenys Luke for giving wise advice whenever needed. Finally I wish to thank Robert Goudie and Tsubasa Itani for some help with LATEX and everyone else who contributed to a fantastic time as an undergraduate at St. Hugh's College. Linus Mattauch March 2008 Note concerning the proposal I wish to note that my study of the Brownian web turned out to be somewhat less connected to the theory of interacting particle systems than I anticipated at the time of submitting the proposal. So the connection between the voter model and coalescing random walks is included in this dissertation (as Appendix A), but the focus of the present work is exclusively on the Brownian web itself. An alternative for the end of the project was stated in the proposal: I decided to include the topic of the \Poisson tree" (not that of self-interacting random walks) into this dissertation and also tailored my account of the Brownian web accordingly.
    [Show full text]
  • The Brownian Fan
    The Brownian fan April 9, 2014 Martin Hairer1 and Jonathan Weare2 1 Mathematics Department, the University of Warwick 2 Statistics Department and the James Frank Institute, the University of Chicago Email: [email protected], Email: [email protected] Abstract We provide a mathematical study of the modified Diffusion Monte Carlo (DMC) algorithm introduced in the companion article [HW14]. DMC is a simulation technique that uses branching particle systems to represent expectations associated with Feynman- Kac formulae. We provide a detailed heuristic explanation of why, in cases in which a stochastic integral appears in the Feynman-Kac formula (e.g. in rare event simulation, continuous time filtering, and other settings), the new algorithm is expected to converge in a suitable sense to a limiting process as the time interval between branching steps goes to 0. The situation studied here stands in stark contrast to the “na¨ıve” generalisation of the DMC algorithm which would lead to an exponential explosion of the number of particles, thus precluding the existence of any finite limiting object. Convergence is shown rigorously in the simplest possible situation of a random walk, biased by a linear potential. The resulting limiting object, which we call the “Brownian fan”, is a very natural new mathematical object of independent interest. Keywords: Diffusion Monte Carlo, quantum Monte Carlo, rare event simulation, sequential Monte Carlo, particle filtering, Brownian fan, branching process Contents 1 Introduction 2 1.1 Notations . .5 2 The continuous-time limit and the Brownian fan 5 2.1 Heuristic derivation of the continuous-time limit . .6 2.2 Some properties of the limiting process .
    [Show full text]
  • A Course in Interacting Particle Systems
    A Course in Interacting Particle Systems J.M. Swart January 14, 2020 arXiv:1703.10007v2 [math.PR] 13 Jan 2020 2 Contents 1 Introduction 7 1.1 General set-up . .7 1.2 The voter model . .9 1.3 The contact process . 11 1.4 Ising and Potts models . 14 1.5 Phase transitions . 17 1.6 Variations on the voter model . 20 1.7 Further models . 22 2 Continuous-time Markov chains 27 2.1 Poisson point sets . 27 2.2 Transition probabilities and generators . 30 2.3 Poisson construction of Markov processes . 31 2.4 Examples of Poisson representations . 33 3 The mean-field limit 35 3.1 Processes on the complete graph . 35 3.2 The mean-field limit of the Ising model . 36 3.3 Analysis of the mean-field model . 38 3.4 Functions of Markov processes . 42 3.5 The mean-field contact process . 47 3.6 The mean-field voter model . 49 3.7 Exercises . 51 4 Construction and ergodicity 53 4.1 Introduction . 53 4.2 Feller processes . 54 4.3 Poisson construction . 63 4.4 Generator construction . 72 4.5 Ergodicity . 79 4.6 Application to the Ising model . 81 4.7 Further results . 85 5 Monotonicity 89 5.1 The stochastic order . 89 5.2 The upper and lower invariant laws . 94 5.3 The contact process . 97 5.4 Other examples . 100 3 4 CONTENTS 5.5 Exercises . 101 6 Duality 105 6.1 Introduction . 105 6.2 Additive systems duality . 106 6.3 Cancellative systems duality . 113 6.4 Other dualities .
    [Show full text]
  • Mathematics of the Gateway Arch Page 220
    ISSN 0002-9920 Notices of the American Mathematical Society ABCD springer.com Highlights in Springer’s eBook of the American Mathematical Society Collection February 2010 Volume 57, Number 2 An Invitation to Cauchy-Riemann NEW 4TH NEW NEW EDITION and Sub-Riemannian Geometries 2010. XIX, 294 p. 25 illus. 4th ed. 2010. VIII, 274 p. 250 2010. XII, 475 p. 79 illus., 76 in 2010. XII, 376 p. 8 illus. (Copernicus) Dustjacket illus., 6 in color. Hardcover color. (Undergraduate Texts in (Problem Books in Mathematics) page 208 ISBN 978-1-84882-538-3 ISBN 978-3-642-00855-9 Mathematics) Hardcover Hardcover $27.50 $49.95 ISBN 978-1-4419-1620-4 ISBN 978-0-387-87861-4 $69.95 $69.95 Mathematics of the Gateway Arch page 220 Model Theory and Complex Geometry 2ND page 230 JOURNAL JOURNAL EDITION NEW 2nd ed. 1993. Corr. 3rd printing 2010. XVIII, 326 p. 49 illus. ISSN 1139-1138 (print version) ISSN 0019-5588 (print version) St. Paul Meeting 2010. XVI, 528 p. (Springer Series (Universitext) Softcover ISSN 1988-2807 (electronic Journal No. 13226 in Computational Mathematics, ISBN 978-0-387-09638-4 version) page 315 Volume 8) Softcover $59.95 Journal No. 13163 ISBN 978-3-642-05163-0 Volume 57, Number 2, Pages 201–328, February 2010 $79.95 Albuquerque Meeting page 318 For access check with your librarian Easy Ways to Order for the Americas Write: Springer Order Department, PO Box 2485, Secaucus, NJ 07096-2485, USA Call: (toll free) 1-800-SPRINGER Fax: 1-201-348-4505 Email: [email protected] or for outside the Americas Write: Springer Customer Service Center GmbH, Haberstrasse 7, 69126 Heidelberg, Germany Call: +49 (0) 6221-345-4301 Fax : +49 (0) 6221-345-4229 Email: [email protected] Prices are subject to change without notice.
    [Show full text]
  • Recent Progress in Coalescent Theory
    SOCIEDADEBRASILEIRADEMATEMÁTICA ENSAIOS MATEMATICOS´ 200X, Volume XX, X–XX Recent Progress in Coalescent Theory Nathana¨el Berestycki Abstract. 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 mathe- matics 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. arXiv:0909.3985v1 [math.PR] 22 Sep 2009 2000 Mathematics Subject Classification: 60J25, 60K35, 60J80. Introduction The probabilistic theory of coalescence, which is the primary subject of these notes, has expanded at a quick pace over the last decade or so. I can think of three factors which have essentially contributed to this growth. On the one hand, there has been a rising demand from population geneticists to develop and analyse models which incorporate more realistic features than what Kingman’s coalescent allows for. Simultaneously, the field has matured enough that a wide range of techniques from modern probability theory may be success- fully applied to these questions. These tools include for instance martingale methods, renormalization and random walk arguments, combinatorial embeddings, sample path analysis of Brownian motion and L´evy processes, and, last but not least, continuum random trees and measure-valued processes. Finally, coalescent processes arise in a natural way from spin glass models of statistical physics. The identification of the Bolthausen-Sznitman coalescent as a universal scaling limit in those models, and the connection made by Brunet and Derrida to models of population genetics, is a very exciting re- cent development.
    [Show full text]
  • Lecture Notes in Mathematics 1840 Editors: J.--M.Morel, Cachan F
    Lecture Notes in Mathematics 1840 Editors: J.--M.Morel, Cachan F. Takens, Groningen B. Teissier, Paris 3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo Boris Tsirelson Wendelin Werner Lectures on Probability Theory and Statistics Ecole d’EtedeProbabilit´ es´ de Saint-Flour XXXII - 2002 Editor: Jean Picard 13 Authors Editor Boris Tsirelson Jean Picard School of Mathematics Laboratoire de Mathematiques´ Appliquees´ TelAvivUniversity UMR CNRS 6620 Tel Av iv 69978 Universite´ Blaise Pascal Clermont-Ferrand Israel 63177 Aubiere` Cedex, France e-mail: [email protected] e-mail: [email protected] Wendelin Werner Laboratoire de Mathematiques´ Universite´ Paris-Sud Batˆ 425, 91405 Orsay Cedex France e-mail: [email protected] Cover picture: Blaise Pascal (1623-1662) Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de Mathematics Subject Classification (2001): 60-01, 60Gxx, 60J65, 60K35, 82B20, 82b27, 82B41 ISSN 0075-8434 Lecture Notes in Mathematics ISSN 0721-5363 Ecole d’Ete´ des Probabilites´ de St. Flour ISBN 3-540-21316-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
    [Show full text]
  • Convergence to the Brownian Web for a Generalization of the Drainage Network Model
    CONVERGENCE TO THE BROWNIAN WEB FOR A GENERALIZATION OF THE DRAINAGE NETWORK MODEL CRISTIAN COLETTI AND GLAUCO VALLE Abstract. We introduce a system of one-dimensional coalescing nonsimple random walks with long range jumps allowing paths that can cross each other and are dependent even before coalescence. We show that under diffusive scaling this system converges in distribution to the Brownian Web. Resum´ e.´ Nous introduisons un syst`eme des marches al´eatoires nonsimple uni- dimensionnelles coalescentes avec des sauts `alongue port´ee.Le syst`eme permet des chemins qui se croisent et qui sont d´ependants avant mˆeme d’avoir coa- lescence. Nous montrons que, sous ´echelle diffusive, ce syst`eme converge en distribution vers le r´eseau Brownien. 1. Introduction The paper is devoted to the analysis of convergence in distribution for a diffu- sively rescaled system of one-dimensional coalescing random walks starting at each point in the space and time lattice Z Z. To not get repetitive, when we make mention to the convergence of a system× of random walks, we always consider one- dimensional random walks and diffusive space time scaling. In this introduction, we aim at describing informally the system and explain why its study is relevant. First, the limit is the Brownian Web (BW) which is a system of coalescing Brownian motions starting at each point in the space and time plane R R. It is the natural scaling limit of a system of simple coalescing random walks.× Here, the BW is a proper random element of a metric space whose points are compact sets of paths.
    [Show full text]
  • The Symbiotic Branching Model: Duality and Interfaces 3
    THE SYMBIOTIC BRANCHING MODEL: DUALITY AND INTERFACES JOCHEN BLATH AND MARCEL ORTGIESE Abstract. The symbiotic branching model describes the dynamics of a spatial two-type population, where locally particles branch at a rate given by the frequency of the other type combined with nearest-neighbour migration. This model generalizes various classic models in population dynamics, such as the stepping stone model and the mutually catalytic branching model. We are particularly interested in understanding the region of coexistence, i.e. the interface between the two types. In this chapter, we give an overview over our results that describe the dynamics of these interfaces at large scales. One of the reasons that this system is tractable is that it exhibits a rich duality theory. So at the same time, we take the opportunity to provide an introduction to the strength of duality methods in the context of spatial population models. 1. Introduction Over recent years spatial stochastic models have become increasingly im- portant in population dynamics. Of particular interest are the spatial pat- terns that emerge through the interaction of different types via competition, spatial colonization, predation and (symbiotic) branching. The classic model in this field is the stepping stone model of Kimura [26]. More recent devel- opments include [46, 9, 6, 3], see also the contributions by Birkner/Gantert and Greven/den Hollander in this volume. A particularly useful technique in this context is duality. This technique allows to relate two (typically Markov) processes in such a way that infor- mation e.g. about the long-term behaviour of one process can be translated to the other one.
    [Show full text]
  • A Lecture on the Averaging Processt1
    Probability Surveys Vol. 9 (2012) 90–102 ISSN: 1549-5787 DOI: 10.1214/11-PS184 A lecture on the averaging process∗ David Aldous† Department of Statistics, 367 Evans Hall # 3860, U.C. Berkeley CA 94720 e-mail: [email protected] url: www.stat.berkeley.edu and Daniel Lanoue Department of Mathematics, 970 Evans Hall # 3840, U.C. Berkeley, CA 94720 e-mail: [email protected] Abstract: To interpret interacting particle system style models as social dynamics, suppose each pair {i,j} of individuals in a finite population meet at random times of arbitrary specified rates νij , and update their states according to some specified rule. The averaging process has real- valued states and the rule: upon meeting, the values Xi(t−),Xj (t−) are 1 1 replaced by 2 (Xi(t−) + Xj (t−)), 2 (Xi(t−) + Xj (t−)). It is curious this simple process has not been studied very systematically. We provide an expository account of basic facts and open problems. AMS 2000 subject classifications: Primary 60K35; secondary 60K99. Keywords and phrases: Duality, interacting particle systems, rate of convergence, spectral gap, voter model. Received September 2011. 1. Introduction The models in the field known to mathematical probabilists as interacting parti- cle systems (IPS) are often exemplified, under the continuing influence of Liggett [9], by the voter model, contact process, exclusion process, and Glauber dynam- ics for the Ising model, all on the infinite d-dimensional lattice, and a major motivation for the original development of the field was as rigorous study of phase transitions in the discipline of statistical physics.
    [Show full text]
  • Probabilistic Structures in Evolution Chapter 1
    EURANDOM PREPRINT SERIES 2019-008 July 1, 2019 Probabilistic Structures in Evolution Chapter 1 A. Greven, F. den Hollander ISSN 1389-2355 1 Probabilistic Structures in Evolution Ellen Baake Anton Wakolbinger (Editors) Contents Chapter 1. From high to low volatility: spatial Cannings with block resampling and spatial Fleming-Viot with seed-bank Andreas Greven and Frank den Hollander 1 1.1. Background 4 1.2. Two models 5 1.3. Equilibrium 12 1.4. Random environment 17 1.5. Extensions 19 1.6. Perspectives 20 References 21 iii CHAPTER 1 From high to low volatility: spatial Cannings with block resampling and spatial Fleming-Viot with seed-bank Andreas Greven and Frank den Hollander In neutrally evolving populations subject to resampling and migration on ge- ographic spaces, the longtime behaviour exhibits a classical dichotomy between clustering of types versus coexistence of types (= between convergence to locally monotype equilibria versus locally multitype equilibria). Which of the two scenar- ios prevails depends on whether the difference between the lineages of two typical individuals is a recurrent random walk or a transient random walk. In the present contribution, for two classes of models we present results on how this dichotomy is affected when the resampling has either high volatility or low volatility: (I) The spatial Cannings model with block resampling, where a positive fraction of the individuals in the next generation may inherit the type of a single individual in the previous generation. (II) The spatial Fleming-Viot model with seed-bank, where individuals may become dormant for awhile, suspending resampling and migration, until they become active again.
    [Show full text]