Table Of Content

Random Walks and Geometry Random Walks and Geometry Proceedings of a Workshop at the Erwin Schrödinger Institute Vienna (cid:18) June 18 July 13, 2001 Editor Vadim A. Kaimanovich in collaboration with Klaus Schmidt and Wolfgang Woess ≥ Walter de Gruyter · Berlin · New York Editor VadimA.Kaimanovich IRMAR UMR6625duC.N.R.S. Universite´deRennes-1 CampusdeBeaulieu 35042RennesCedex France e-mail:[email protected] MathematicsSubjectClassification2000: 22D40,37H15,43A05,58J65,60B99,60J45,82B20 Keywords: randomwalksongroupsandgraphs,Markovprocesses,randommatrices,Lyapunovexponents,harmonic functions,stochasticLoewnerevolution,expandergraphs,quantumchaos,spectraltheory,cellularautomata, randomnumbergenerators EPPrintedonacid-freepaperwhichfallswithintheguidelinesofthe ANSItoensurepermanenceanddurability. LibraryofCongressCataloging-in-PublicationData Random walks and geometry : proceedings of a workshop at the Erwin Schrödinger Institute, Vienna, June 18(cid:18)July 13, 2001 / editor, Vadim A. Kaimanovich, in collaboration with KlausSchmidtandWolfgangWoess. p. cm. English,withonecontributioninFrench. ISBN3-11-017237-2(acid-freepaper) 1. Random walks(Mathematics)(cid:18)Congresses. 2. Geom- etry(cid:18)Congresses. I.Kaimanovich,VadimA. II.Schmidt, Klaus,1943(cid:18) III.Woess,Wolfgang,1954(cid:18) QA274.73.R37 2004 519.2(cid:25)82(cid:18)dc22 2004043902 ISBN 3-11-017237-2 BibliographicinformationpublishedbyDieDeutscheBibliothek DieDeutscheBibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataisavailableintheInternetat(cid:12)http://dnb.ddb.de(cid:14). ”Copyright2004byWalterdeGruyterGmbH&Co.KG,10785Berlin,Germany. All rights reserved, including those of translationinto foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, includingphotocopy,recordingoranyinformationstorageandretrievalsystem,withoutpermis- sioninwritingfromthepublisher. PrintedinGermany. Coverdesign:ThomasBonnie,Hamburg. Typesetusingtheauthors’T Xfiles:I.Zimmermann,Freiburg. E Printingandbinding:Hubert&Co.GmbH&Co.KG,Göttingen. DedicatedtothememoryofMartineBabillot Preface Thisvolumeisanoutcomeofthespecialsemester2001-RandomWalksheldatthe SchrödingerInstituteinVienna,Austria,fromFebruaryuntilJuly2001. Itwasdedi- catedtovariousproblemsrelatedtostochasticprocessesongeometricandalgebraic structures,withanemphasisontheirinterplay,andalsoontheirinteractionwiththe- oretical physics. Some of the focal points were: probability on groups; products of randommatricesandtheLyapunovspectrum;boundarybehaviour,harmonicfunctions andotherpotentialtheoreticaspects; Brownianmotiononmanifolds; combinatorial and spectral properties of random walks on graphs; random walks and diffusion on fractals. Thereweretwoseparatemainperiodsofactivityinthefirst(February/March) andinthesecond(May/June/July)halvesofthesemester. Thefirstperiodstartedwith atwo-weekworkshopwiththegeneralthemeRandomWalksandStatisticalPhysics (February19–March2). Towardstheendofthesecondperiodtherewasanotherwork- shopwiththegeneralthemeRandomWalksandGeometrywhichlastedforalmosta month(June18–July13). Thepaperscollectedinthisvolumeare(withacoupleofexceptions)contributed by the participants of the second workshop and show how the ideas connected with Markovchainsongeometricandalgebraicstructurespermeatesuchdifferentsubjects ashyperbolicgeometry,Liegroups,geometricgrouptheory,cellularautomata,graph theory,randomnumbergenerators,percolation,andstatisticalphysics. Amongthese papersarebothsurveysandoriginalresearcharticles.Allofthemhavebeenthoroughly refereedandproofread. Fruitful complementary interaction between the geometry and randomness is a common feature and unifying link between all the contributions, and we are glad to presentthispanoramaofrecentworkintherapidlygrowingareaatthecrossroadsof severalmathematicaldisciplines. WearegratefultotheErwinSchrödingerInternationalInstituteforMathematical Physics inVienna for generous financial support and for creating excellent working atmosphereduringourspecialsemester. Whentheworkonthisvolumewasalmostfinishedwelearnedabouttheuntimely death of Martine Babillot caused by a foudroyant and devastating disease. Mar- tine’s bright personality, with her ability to synthesize different points of view and approaches, was very close to the spirit of our program, of which she was an active participant. ShefinishedproofreadinghercontributiontotheProceedingsjustseveral weeksbeforepassingaway. Wededicatethisvolumetohermemory. March2004 VadimA.Kaimanovich,KlausSchmidtandWolfgangWoess Table of contents Preface vii Surveysandlongerarticles AbbasAlhakimandStanislavMolchanov SomeMarkovchainsonabeliangroupswithapplications 3 RaffaellaBurioni,DavideCassiandAlessandroVezzani Randomwalksandphysicalmodelsoninfinitegraphs: anintroduction 35 TullioCeccherini-Silberstein,FrancescaFiorenziandFabioScarabotti TheGardenofEdenTheoremforcellularautomataandforsymbolicdynamical systems 73 AlexGamburd Expandergraphs,randommatricesandquantumchaos 109 RostislavI.GrigorchukandAndrzejZ˙uk TheIharazetafunctionofinfinitegraphs,theKNSspectralmeasure andintegrablemaps 141 YvesGuivarc’hetÉmileLePage SimplicitédespectresdeLyapounovetpropriétéd’isolationspectrale pourunefamilled’opérateursdetransfertsurl’espaceprojectif 181 GregoryF.Lawler AnintroductiontotheStochasticLoewnerEvolution 261 GeorgeA.Willis Acanonicalformforautomorphismsoftotallydisconnected locallycompactgroups 295 Researchcommunications MartineBabillot Ontheclassificationofinvariantmeasuresforhorospherefoliations onnilpotentcoversofnegativelycurvedmanifolds 319 MartinT.BarlowandStevenN.Evans Markovprocessesonvermiculatedspaces 337 x Tableofcontents LaurentBartholdi Cactustreesandlowerboundsonthespectralradiusofvertex-transitivegraphs 349 EnriqueBendito,ÁngelesCarmonaandAndrésM.Encinas Equilibriummeasure,Poissonkernelandeffectiveresistanceonnetworks 363 SébastienBlachère Internaldiffusionlimitedaggregationondiscretegroupsofpolynomialgrowth 377 MassimoCampaninoandDimitriPetritis Onthephysicalrelevanceofrandomwalks: anexampleofrandomwalks onarandomlyorientedlattice 393 TullioCeccherini-SilbersteinandFabioScarabotti Randomwalks,entropyandhopfianityoffreegroups 413 AnnaErschler Growthratesofsmallcancellationgroups 421 AlexEskinandGregoryMargulis Recurrencepropertiesofrandomwalksonfinitevolume homogeneousmanifolds 431 AlessandraIozzi Onthecohomologyoffoliationswithamenablegroupoid 445 AndersKarlsson Linearrateofescapeandconvergenceindirection 459 AnnaMariaManteroandAnnaZappa Remarksonharmonicfunctionsonaffinebuildings 473 TatianaNagnibeda Randomwalks,spectralradii,andRamanujangraphs 487 SamNorthshield Cogrowthofarbitrarygraphs 501 LaurentSaloff-Coste TotalvariationlowerboundsforfiniteMarkovchains: Wilson’slemma 515