Recent Progress in Coalescent Theory
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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]). -
Science Fiction and Astronomy
Sci Fi Science Fiction and Astronomy Many science fiction books include subjects of astronomical interest. Here is a list of some that have been recommended to me or I’ve read. I expect that most are not in the University library but many are available in Kindle and other e-formats. At the top of my list is a URL to a much longer list by Andrew Fraknoi of the Astronomical Society of the Pacific. Each title in his list has a very brief summary indicating the kind of story it is. Andrew Fraknoi’s list (http://www.astrosociety.org/education/resources/scifi.html). Gregory Benford is a plasma physicist who has been rated by some as one of the finest observes and interpreters of science in modern fiction. Note: Timescape [Vista, ISBN 0575600500] In the Ocean of Night [Vista, ISBN 0575600357] Across the Sea of Suns [Vista, ISBN 0575600551] Great Sky River [Gallancy, ISBN 0575058315] Eater and over 2 dozen more available as e-books. Stephen Baxter Titan [Voyager, 1997, ISBN 0002254247] has been strongly recommended by New Scientist as a tense, near future, thriller you shouldn’t miss. David Brin has a PhD in astrophysics with which he brings real understanding of the Universe to his stories. The Crystal Spheres won an award. Fred Hoyle is probably the most famous astronomer to have written science fiction. The Back Cloud [Macmillan, ISBN 0333556011] is his classic, followed by A for Andromeda. Try also October the First is too late. Larry Niven’s stories include plenty of ideas inspired by modern astronomy. -
Science Fiction Stories with Good Astronomy & Physics
Science Fiction Stories with Good Astronomy & Physics: A Topical Index Compiled by Andrew Fraknoi (U. of San Francisco, Fromm Institute) Version 7 (2019) © copyright 2019 by Andrew Fraknoi. All rights reserved. Permission to use for any non-profit educational purpose, such as distribution in a classroom, is hereby granted. For any other use, please contact the author. (e-mail: fraknoi {at} fhda {dot} edu) This is a selective list of some short stories and novels that use reasonably accurate science and can be used for teaching or reinforcing astronomy or physics concepts. The titles of short stories are given in quotation marks; only short stories that have been published in book form or are available free on the Web are included. While one book source is given for each short story, note that some of the stories can be found in other collections as well. (See the Internet Speculative Fiction Database, cited at the end, for an easy way to find all the places a particular story has been published.) The author welcomes suggestions for additions to this list, especially if your favorite story with good science is left out. Gregory Benford Octavia Butler Geoff Landis J. Craig Wheeler TOPICS COVERED: Anti-matter Light & Radiation Solar System Archaeoastronomy Mars Space Flight Asteroids Mercury Space Travel Astronomers Meteorites Star Clusters Black Holes Moon Stars Comets Neptune Sun Cosmology Neutrinos Supernovae Dark Matter Neutron Stars Telescopes Exoplanets Physics, Particle Thermodynamics Galaxies Pluto Time Galaxy, The Quantum Mechanics Uranus Gravitational Lenses Quasars Venus Impacts Relativity, Special Interstellar Matter Saturn (and its Moons) Story Collections Jupiter (and its Moons) Science (in general) Life Elsewhere SETI Useful Websites 1 Anti-matter Davies, Paul Fireball. -
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 -
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. -
The Drink Tank Sixth Annual Giant Sized [email protected]: James Bacon & Chris Garcia
The Drink Tank Sixth Annual Giant Sized Annual [email protected] Editors: James Bacon & Chris Garcia A Noise from the Wind Stephen Baxter had got me through the what he’ll be doing. I first heard of Stephen Baxter from Jay night. So, this is the least Giant Giant Sized Crasdan. It was a night like any other, sitting in I remember reading Ring that next Annual of The Drink Tank, but still, I love it! a room with a mostly naked former ballerina afternoon when I should have been at class. I Dedicated to Mr. Stephen Baxter. It won’t cover who was in the middle of what was probably finished it in less than 24 hours and it was such everything, but it’s a look at Baxter’s oevre and her fifth overdose in as many months. This was a blast. I wasn’t the big fan at that moment, the effect he’s had on his readers. I want to what we were dealing with on a daily basis back though I loved the novel. I had to reread it, thank Claire Brialey, M Crasdan, Jay Crasdan, then. SaBean had been at it again, and this time, and then grabbed a copy of Anti-Ice a couple Liam Proven, James Bacon, Rick and Elsa for it was up to me and Jay to clean up the mess. of days later. Perhaps difficult times made Ring everything! I had a blast with this one! Luckily, we were practiced by this point. Bottles into an excellent escape from the moment, and of water, damp washcloths, the 9 and the first something like a month later I got into it again, 1 dialed just in case things took a turn for the and then it hit. -
Frontiers in Coalescent Theory: Pedigrees, Identity-By-Descent, and Sequentially Markov Coalescent Models
Frontiers in Coalescent Theory: Pedigrees, Identity-by-Descent, and Sequentially Markov Coalescent Models The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Wilton, Peter R. 2016. Frontiers in Coalescent Theory: Pedigrees, Identity-by-Descent, and Sequentially Markov Coalescent Models. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:33493608 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Frontiers in Coalescent Theory: Pedigrees, Identity-by-descent, and Sequentially Markov Coalescent Models a dissertation presented by Peter Richard Wilton to The Department of Organismic and Evolutionary Biology in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Biology Harvard University Cambridge, Massachusetts May 2016 ©2016 – Peter Richard Wilton all rights reserved. Thesis advisor: Professor John Wakeley Peter Richard Wilton Frontiers in Coalescent Theory: Pedigrees, Identity-by-descent, and Sequentially Markov Coalescent Models Abstract The coalescent is a stochastic process that describes the genetic ancestry of individuals sampled from a population. It is one of the main tools of theoretical population genetics and has been used as the basis of many sophisticated methods of inferring the demo- graphic history of a population from a genetic sample. This dissertation is presented in four chapters, each developing coalescent theory to some degree. -
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 . -
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. -
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. -
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. -
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.