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This page intentionally left blank ELEMENTS OF THE RANDOM WALK AnIntroductionforAdvancedStudentsandResearchers Randomwalkshaveproventobeausefulmodelinunderstandingprocessesacross awidespectrumofscientificdisciplines.ElementsoftheRandomWalkisanintro- ductiontosomeofthemostpowerfulandgeneraltechniquesusedintheapplication oftheseideas. The mathematical construct that runs through the analysis of each of the topics coveredinthisbook,andwhichthereforeunifiesthemathematicaltreatment,isthe generatingfunction.Althoughthereaderisintroducedtomodernanalyticaltools, suchaspathintegralsandfield-theoreticalformalism,thebookisself-containedin thatbasicconceptsaredevelopedandrelevantfundamentalfindingsfullydiscussed. Thebookalsoprovidesanexcellentintroductiontofrontiertopicssuchasfractals, scalingandcriticalexponents,pathintegrals,applicationoftheGLWHamiltonian formalism, and renormalization group theory as they relate to the random walk problem.Mathematicalbackgroundisprovidedinsupplementsattheendofeach chapter,whenappropriate. Thisself-containedtextwillappealtograduatestudentsacrossscience,engineer- ing, and mathematics who need to understand the application of random walk techniques,aswellastoestablishedresearchers. Joseph Rudnick earned his Ph.D. in 1970. He has held faculty positions at TuftsUniversityandtheUniversityofCalifornia,SantaCruz,aswellasavisiting position at Harvard University. He is currently a Professor in the Department of PhysicsandAstronomyattheUniversityofCalifornia,LosAngeles. George GaspariiscurrentlyEmeritusProfessorattheUniversityofCalifornia, SantaCruz.HehasheldvisitingpositionsattheUniversityofBristol,UK;Stanford University; and the University of California Los Angeles. He has been a Sloan FoundationFellow. ELEMENTS OF THE RANDOM WALK An Introduction for Advanced Students and Researchers JOSEPH RUDNICK DepartmentofPhysicsandAstronomyUniversityofCalifornia, LosAngeles GEORGE GASPARI DepartmentofPhysicsUniversityofCalifornia,SantaCruz cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521828918 © J. Rudnick and G. Gaspari 2004 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2004 isbn-13 978-0-511-18657-8 eBook (EBL) isbn-10 0-511-18657-6 eBook (EBL) isbn-13 978-0-521-82891-8 hardback isbn-10 0-521-82891-0 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. ForAliceandNancy Contents Preface pagexi 1 Introductiontotechniques 1 1.1 Thesimplestwalk 1 1.2 Someveryelementarycalculationsonthesimplestwalk 5 1.3 Backtotheprobabilitydistribution 10 1.4 Recursionrelationfortheone-dimensionalwalk 13 1.5 Backing into the generating function for a random walk 15 1.6 Supplement:methodofsteepestdescents 20 2 GeneratingfunctionsI 25 2.1 Generalintroductiontogeneratingfunctions 25 2.2 Supplement1:Gaussianintegrals 41 2.3 Supplement2:Fourierexpansionsonalattice 42 2.4 Supplement3:asymptoticcoefficientsofpowerseries 47 3 GeneratingfunctionsII:recurrence,sitesvisited,andtheroleof dimensionality 51 3.1 Recurrence 51 3.2 Anewgeneratingfunction 51 3.3 Derivationofthenewgeneratingfunction 52 3.4 Dimensionalityandtheprobabilityofrecurrence 55 3.5 Recurrenceintwodimensions 58 3.6 Recurrencewhenthedimensionality,d,liesbetween2and4 60 3.7 Theprobabilityofnon-recurrenceinwalksondifferentcubic latticesinthreedimensions 62 3.8 Thenumberofsitesvisitedbyarandomwalk 63 4 Boundaryconditions,steadystate,andtheelectrostaticanalogy 69 4.1 Theeffectsofspatialconstraintsonrandomwalkstatistics 70 4.2 Randomwalkinthesteadystate 82 4.3 Supplement:boundaryconditionsatanabsorbingboundary 93 vii viii Contents 5 Variationsontherandomwalk 95 5.1 Thebiasedrandomwalk 95 5.2 Thepersistentrandomwalk 98 5.3 Thecontinuoustimerandomwalk 118 6 Theshapeofarandomwalk 127 6.1 Thenotionandquantificationofshape 127 6.2 Walksind (cid:2) 3dimensions 135 6.3 Finalcommentary 154 6.4 Supplement1:principalradiiofgyrationandrotationalmotion 154 6.5 Supplement2:calculationsforthemeanasphericity 159 6.6 Supplement3:derivationof(6.21)fortheradiusofgyration ↔ tensor,T,andtheeigenvaluesoftheoperator 165 7 Pathintegralsandself-avoidance 167 7.1 Theunrestrictedrandomwalkasapathintegral 168 7.2 Self-avoidingwalks 171 8 Propertiesoftherandomwalk:introductiontoscaling 193 8.1 Universality 193 9 Scalingofwalksandcriticalphenomena 203 9.1 Scalingandtherandomwalk 203 9.2 Criticalpoints,scaling,andbrokensymmetries 204 9.3 Ginzburg–Landau–WilsoneffectiveHamiltonian 215 9.4 Scalingandthemeanend-to-enddistance;(cid:4)R2(cid:5) 218 9.5 Connectionbetweenthe O(n)modelandtheself-avoidingwalk 219 9.6 Supplement:evaluationofGaussianintegrals 227 10 Walksandthe O(n)model:meanfieldtheoryandspinwaves 233 10.1 Meanfieldtheoryandspinwavescontributions 233 10.2 Themeanfieldtheoryofthe O(n)model 236 10.3 Fluctuations:loworderspinwavetheory 239 10.4 Thecorrelationhole 251 11 Scaling,fractals,andrenormalization 255 11.1 Scaleinvarianceinmathematicsandnature 255 11.2 Moreontherenormalizationgroup:therealspacemethod 264 11.3 Recursionrelations:fixedpointsandcriticalexponents 277 12 Moreontherenormalizationgroup 285 12.1 Themomentum-shellmethod 285 12.2 TheeffectiveHamiltonianwhenthereisfourthorder interactionbetweenthespindegreesoffreedom 286