PDF-2GIDEONAMIRANDORIGUREL-GUREVICHallregulartrees.Forunimodulargraphsthefollowingsymmetryprincipleholds:MassTransportPrinciple:LetG=(V;E)beaunimodulargraph,andassumeF:VV![0;1]isautomorphisminvariant(i.e.F( v; y)=f(x;y)forany 2Aut(G)),thenforanyv2VwehaveXw2V

4GIDEONAMIRANDORIGURELGUREVICHcallxv0acandidateAcandidatewhichisactuallynonxatingisgoodandtherestofthecandidatesarebadByourapproximationstheprobabilitythatxv0isabadcandidateisboundedbyLetbP. MARYAMMIRZAKHANIgrowthof),itprovesfruitfultostudydierenttypesofsimpleclosedgeodesicsonseparately.Letg,nbeaclosedsurfaceofgenusboundarycomponents.ThemappingclassgroupModg,nactsnaturallyonthesetofisoto {z }=x'('1(y))| {z }=y
='1''1(x)'1(y)

='1(x)'1(y);whichshowsthat'12Aut(G).aDefinition.If':G!Hisahomomorphism,thenx2G:'(x)=eH iscalledthekernelof'andisdenotedbyker(').Theorem6.4.Let':G!Hbeah ,andhencealllogicalconsequencesoftheseaxiomsaretrueinN .Butthisproofisnotnitary,becauseitinvolvesaninductiononastatementmentioningtheinnitesetN.Proof:Weprovethecontrapositive.SupposethatPA`G,i.e.PA` distancesaredistinguishedbyAut(G)-orbit.Thatis,d(x;y)=d(u;v)ifandonlyifthereexists2Aut(G)sothatf(x);(y)g=fu;vg.ConsiderthesepointsEuclideanverticesandaddEuclideanedgesasappropriatetoobtainG.Thisisa Algorithm1:Triangle(G=(V;E)) foreachv2Vdo foreachs;t2N(v)do if(s;t)2Ethen return(v;s;t); return\Notriangle"; Proof.ConsiderAlgorithm1.IfGcontainsatriangle(a;b;c),theninsomeiterationofthealgorithm,v=aa Notation Denition SSym( )sharplytransitive:Forany;2 exactlyoneg2Swithg= Denition SSym( )sharply2transitive:Ssharplytransitiveonpairs(1;2),16=2 ObservationbyErnstWitt: Projectiveplaneoford n
=1(z)whichisthe(asymptotic)probabilityinthetail.Instead,supposeweseekthefollowingprobabilityPrXn+
=??,whereisxed.Doesthecentrallimittheoremsayanythinguseful?Itiseasytoseethat,foranylimn!1P 4GIDEONAMIRANDORIGUREL-GUREVICHFortheoriginalARWmodel,corollary1.2wasprovedindependentlybyShellef[5],usingcompletelydierentmethods,onanyboundeddegreegraph.Themainmeritofourproofisthatalthoughthegraph 2(j+k1)(j+k2)+j:Thisexamplegeneralizesto:NN:::NN;forany(nite)numberoffactorsinthecartesianproductontheleft.Thisfollowsfromthenextexample.Ex.3.LetA;B;C;Dbesets.IfACandBD,thenABCD.Proof.Byde 4S.CAENEPEEL,J.VERCRUYSSE,ANDSHUANHONGWANGleftA-module,thenCisre exive.Forany'2(C),wethenhavethat'=i(Pj'(fj)cj).GaloiscoringsandDescentTheory.LetCbeanA-coring.Recallthatx2Ciscalledgrouplikeif
C(x)=x Figure1:S-moveandA-movef\r1;\r2g|f\r1;\r3g|f\r3;\r4g|f\r4;\r5g|f\r4;\r6g|f\r2;\r6gwhereallcurves\r1;:::;\r6lieinacommonsubsurfaceoftype1;2,\r1|\r4and\r2|\r4beingS-movesandallothermovesbeingA-moves,co 2HUYIHUANDANNATALITSKAYAbecausethespaceweworkwithisfourdimensional.Second,thetechniquetoremovethesecondzeroLyapunovexponentismoredelicate.WhilewechangethelastLyapunovexponent,weneedtokeepallotherexpon 3SpecialthankstoBillZameforhisinvaluablehelponTheorem1ofthispaperWewouldalsoliketothankPeterEsoIanJewittMegMeyerAdrienVigierJoelShapiroInaTanevaandPeytonYoungfortheircommentsandadviceyDepartmentofEcon CorinnaCortesGoogleResearchNewYorkNY10011corinnagooglecomYishayMansourTel-AvivUniversityTel-Aviv69978IsraelmansourtauacilMehryarMohriCourantInstituteandGoogleNewYorkNY10012mohricimsnyueduAbstractThisp