RandomWalksonGraphs
Thetheoryofelectricalnetworksisafundamentaltoolforstudying therecurrenceofreversibleMarkovchains.TheKirchhofflawsand ThomsonprinciplepermitaneatproofofP ´ olya’stheoremforrandom walkona d -dimensionalgrid.
1.1RandomWalksandReversibleMarkovChains
Abasicknowledgeofprobabilitytheoryisassumedinthisvolume.Readers keentoacquirethisarereferredto[150]foranelementaryintroduction,and to[148]forasomewhatmoreadvancedaccount.Weshallgenerallyusethe letter P todenoteagenericprobabilitymeasure,withmorespecificnotation whenhelpful.Theexpectationofarandomvariable f willbewrittenas either P( f ) or E( f ).
Onlyalittleknowledgeis assumedaboutgraphs,andmanyreaderswill havesufficientacquaintancealready.OthersareadvisedtoconsultSection 1.6.Ofthemanybooksongraphtheory,wemention[50].
Let G = ( V , E ) beafiniteorcountablyinfinitegraph,whichwegenerally assume,forsimplicity,tohaveneitherloopsnormultipleedges.If G is infinite,weshallusuallyassumeinadditionthateveryvertex-degreeis finite.Aparticlemovesaroundthevertex-set V .Havingarrivedatthe vertex Sn attime n ,itsnextposition Sn +1 ischosenuniformlyatrandom fromthesetofneighboursof Sn .Thetrajectoryoftheparticleiscalleda symmetricrandomwalk (SRW)on G .
Twoofthebasicquestionsconcerningsymmetricrandomwalkare:
1.Underwhatconditionsisthewalk recurrent ,inthatitreturns(almost surely)toitsstartingpoint?
2.Howdoesthedistancebetween S0 and Sn behaveas n →∞?
TheaboveSRWissymmetricinthatthejumpsarechosen uniformly fromthesetofavailableneighbours.Inamoregeneralprocess,wetakea function w : E → (0 , ∞),andwejumpalongtheedge e withprobability proportionalto we .
AnyreversibleMarkovchain 1 ontheset V givesrisetosuchawalkas follows.Let Z = ( Z n : n ≥ 0 ) beaMarkovchainon V withtransition matrix P ,andassumethat Z isreversiblewithrespecttosomepositive function π : V → (0 , ∞),whichistosaythat
(1.1) πu pu ,v = πv pv,u , u ,v ∈ V .
Witheachdistinctpair u ,v ∈ V ,weassociatetheweight
(1.2) wu ,v = πu pu ,v , notingby(1.1)that wu ,v = wv,u .Then
(1.3) pu ,v = wu ,v Wu , u ,v ∈ V , where
Thatis,giventhat Z n = u ,thechainjumpstoanewvertex v withprobability proportionalto wu ,v .Thismaybesetinthecontextofarandomwalkon thegraphwithvertex-set V andedge-set E containingall e = u ,v such that pu ,v > 0.Withedge e ∈ E weassociatetheweight we = wu ,v .
Inthischapter,wedeveloptherelationshipbetweenrandomwalkson G andelectricalnetworkson G .Therearesomeexcellentaccountsofthis subjectarea,andthereaderisreferredtothebooksofDoyleandSnell [83],LyonsandPeres[221],andAldous andFill[19],amongstothers.The connectionbetweenthesetwotopicsismadeviatheso-called‘harmonic functions’oftherandomwalk.
1.4Definition Let U ⊆ V ,andlet Z beaMarkovchainon V withtransitionmatrix P ,thatisreversiblewithrespecttothepositivefunction π .The function f : V → R is harmonic on U (withrespectto P )if f (u ) = v ∈ V pu ,v f (v), u ∈ U , or,equivalently,if f (u ) = E( f ( Z 1 ) | Z 0 = u ) for u ∈ U .
Fromthepair ( P ,π),wecanconstructthegraph G asabove,andthe weightfunction w asin(1.2).Werefertothepair ( G ,w) astheweighted graphassociatedwith ( P ,π).Weshallspeakof f asbeingharmonic(for ( G ,w))ifitisharmonicwithrespectto P .
1 AccountsofMarkovchaintheoryarefoundin[148,Chap.6]and[150,Chap.12]. .002 17:53:41
Theso-calledhittingprobabilitiesarebasicexamplesofharmonicfunctionsforthechain Z .Let U ⊆ V , W = V \ U ,and s ∈ U .For u ∈ V ,let g (u ) betheprobabilitythatthechain,startedat u ,hits s before W .Thatis,
g (u ) = Pu ( Z n = s forsome n < TW ), where
TW = inf {n ≥ 0: Z n ∈ W } isthefirst-passagetimeto W ,and Pu (·) = P(·| Z 0 = u ) denotesthe conditionalprobabilitymeasuregiventhatthechainstartsat u .
1.5Theorem ThefunctiongisharmoniconU \{s }.
Evidently, g (s ) = 1,and g (v) = 0for v ∈ W .Wespeakofthesevalues of g asbeingthe‘boundaryconditions’oftheharmonicfunction g .See Exercise1.13fortheuniquenessofharmonicfunctionswithgivenboundary conditions.
Proof. ThisisanelementaryexerciseusingtheMarkovproperty.For u / ∈ W ∪{s },
g (u ) = v ∈ V pu ,v Pu Z n = s forsome n < TW Z 1 = v = v ∈ V pu ,v g (v), asrequired.
1.2ElectricalNetworks
Throughoutthissection, G = ( V , E ) isafinitegraphwithneitherloops normultipleedges,and w : E → (0 , ∞) isaweightfunctionontheedges. Weshallassumefurtherthat G isconnected.
Wemaybuildanelectricalnetworkwithdiagram G ,inwhichtheedge e hasconductance we (or,equivalently,resistance1/we ).Let s , t ∈ V bedistinctverticestermed sources ,andwrite S ={s , t } forthe source-set . Supposeweconnectabatteryacrossthepair s , t .Itisaphysicalobservation thatelectronsflowalongthewiresinthenetwork.Theflowisdescribedby theso-calledKirchhofflaws,asfollows.
Toeachedge e = u ,v ,thereareassociated(directed)quantities φu ,v and i u ,v ,calledthe potentialdifference from u to v ,andthe current from u to v ,respectively.Theseareantisymmetric,
1.6Kirchhoff’spotentiallaw Thecumulativepotentialdifferencearound anycycle v1 ,v2 ,...,vn ,vn +1 = v1 of G iszero,thatis, (1.7) n j =1
j ,v j +1 = 0 .
1.8Kirchhoff’scurrentlaw Thetotalcurrentflowingoutofanyvertex u ∈ V otherthanthesource-setiszero,thatis, (1.9)
Therelationshipbetweenresistance/conductance,potentialdifference, andcurrentisgivenbyOhm’slaw.
1.10Ohm’slaw Foranyedge e = u ,v , i u ,v = w
u ,v .
Kirchhoff’spotentiallawisequivalenttothestatementthatthereexists afunction φ : V → R,calleda potentialfunction,suchthat
Since φ isdetermineduptoanadditiveconstant,wearefreetopickthe potentialofanysinglevertex.Noteourconventionthat currentflowsuphill : i u ,v hasthesamesignas φu ,v = φ(v) φ(u ).
1.11Theorem Apotentialfunctionisharmoniconthesetofallvertices otherthanthesource-set.
Proof. Let U = V \{s , t }.ByKirchhoff’scurrentlawandOhm’slaw,
whichistosaythat
Thatis, φ isharmonicon U .
WecanuseOhm’slawtoexpresspotentialdifferencesintermsofcurrents,andthusthetwoKirchhofflawsmaybeviewedasconcerningcurrents only.Equation(1.7)becomes
(1.12)
0 ,
validforanycycle v1 ,v2 ,...,vn ,vn +1 = v1 .With(1.7)writtenthus,each lawislinearinthecurrents,andthesuperpositionprinciplefollows.
1.13Theorem(Superpositionprinciple) Ifi 1 andi 2 aresolutionsofthe twoKirchhofflawswiththesamesource-setthensoisthesumi 1 + i 2 .
Nextweintroducetheconceptofa‘flow’onagraph.
1.14Definition Let s , t ∈ V , s = t .An s / t-flowj isavector j = ( ju ,v : u ,v ∈ V , u = v),suchthat: (a) ju ,v =− jv,u , (b) ju ,v = 0whenever u v , (c)forany u = s , t ,wehavethat v ∈ V ju ,v = 0.
Thevertices s and t arecalledthe‘source’and‘sink’ofan s / t flow,and weusuallyabbreviate‘s / t flow’to‘flow’.Foranyflow j ,wewrite Ju = v ∈ V ju ,v , u ∈ V ,
notingby(c)abovethat Ju = 0for u = s , t .Thus, Js + Jt =
=
V
,v
ju ,v = 1 2 u ,v
V ( ju ,v + jv,u ) = 0 .
Therefore, Js =− Jt ,andwecall | Js | the size oftheflow j ,denoted | j |.If | Js |= 1,wecall j a unitflow .Weshallnormallytake Js > 0,inwhich case s isthe source and t isthe sink oftheflow,andwesaythat j isaflow from s to t
Notethatanysolution i totheKirchhofflawswithsource-set {s , t } isan s / t flow.
1.15Theorem Leti 1 andi 2 betwosolutionsoftheKirchhofflawswith thesamesourceandsinkandequalsize.Theni 1 = i 2 .
Proof. Bythesuperpositionprinciple, j = i 1 i 2 satisfiesthetwoKirchhofflaws.Furthermore,undertheflow j ,nocurrententersorleaves thesystem.Therefore, Jv = 0forall v ∈ V .Suppose ju 1 ,u 2 > 0for someedge u 1 , u 2 .BytheKirchhoffcurrentlaw,thereexists u 3 suchthat
ju 2 ,u 3
> 0.Since | V | < ∞,thereexistsbyiterationacycle u l , u l +1 ,..., u m , u m +1 = u l suchthat ju k ,u k +1 > 0for k = l , l + 1,..., m .ByOhm’s law,thecorrespondingpotentialfunctionsatisfies
φ(u l )<φ(u l +1 )< ··· <φ(u m +1 ) = φ(u l ), acontradiction.Therefore, ju ,v = 0forall u , v .
Foragivensizeofinputcurrent,andgivensource s andsink t ,therecan benomorethanonesolutiontothetwoKirchhofflaws,butisthereasolution atall?Theanswerisofcourseaffirmative,andtheuniquesolutioncanbe expressedexplicitlyintermsofcountsofspanningtrees.2 Considerfirstthe specialcasewhen we = 1forall e ∈ E .Let N bethenumberofspanning treesof G .Foranyedge a , b ,let (s , a , b , t ) bethepropertyofspanning treesthat:theunique s / t pathinthetreepassesalongtheedge a , b inthe directionfrom a to b .Let N (s , a , b , t ) bethesetofspanningtreesof G withtheproperty (s , a , b , t ),andlet N (s , a , b , t ) =|N (s , a , b , t )| 1.16Theorem Thefunction (1.17) i a ,b = 1 N N (s , a , b , t ) N (s , b , a ,
E , definesaunitflowfromstotsatisfyingtheKirchhofflaws.
Let T beaspanningtreeof G chosenuniformlyatrandomfromtheset T ofallsuchspanningtrees.ByTheorem1.16andthepreviousdiscussion, theuniquesolutiontotheKirchhofflawswithsource s ,sink t ,andsize1is givenby i a ,b = P T has (s , a , b , t ) P T has (s , b , a , t ) .
WeshallreturntouniformspanningtreesinChapter2. WeproveTheorem1.16next.Exactlythesameproofisvalidinthecase ofgeneralconductances we .Inthatcase,wedefinetheweightofaspanning tree T as w( T ) =
, andweset (1.18) N ∗ = T
(s , a , b , t ) = T with (s ,a ,b ,t ) w( T ).
TheconclusionofTheorem1.16holdsinthissettingwith i a ,b = 1 N
, b , t )
2 ThiswasdiscoveredinanequivalentformbyKirchhoffin1847,[188].
ProofofTheorem1.16. WefirstchecktheKirchhoffcurrentlaw.Inevery spanningtree T ,thereexistsauniquevertex b suchthatthe s / t pathof T containstheedge s , b ,andthepathtraversesthisedgefrom s to b . Therefore,
By(1.17),
and,byasimilarargument,
Let T beaspanningtreeof G .Thecontributiontowardsthequantity i a ,b ,madeby T ,dependsonthe s / t path π of T andequals
N 1 if π passesalong a , b from a to b , N 1 if π passesalong a , b from b to a , (1.19) 0if π doesnotcontaintheedge a , b .
Let v ∈ V , v = s , t ,andwrite Iv = w ∈ V i v,w .If v ∈ π ,thecontribution of T towards Iv is N 1 N 1 = 0since π arrivesat v alongsomeedgeof theform a ,v anddepartsfrom v alongsomeedgeoftheform v, b .If v/ ∈ π ,then T contributes0to Iv .Summingover T ,weobtainthat Iv = 0 forall v = s , t ,asrequiredfortheKirchhoffcurrentlaw.
WenextchecktheKirchhoffpotentiallaw.Let v1 ,v2 ,...,vn ,vn +1 = v1 beacycle C of G .Weshallshowthat (1.20) n j =1 i v j ,v j +1 = 0 ,
andthiswillconfirm(1.12),onrecallingthat we = 1forall e ∈ E .Itis moreconvenientinthiscontexttoworkwith‘bushes’thanspanningtrees. A bush (or,moreprecisely,an s / tbush)isdefinedtobeaforeston V containingexactlytwotrees,onedenoted Ts andcontaining s ,andtheother denoted Tt andcontaining t .Wewrite ( Ts , Tt ) forthisbush.Let e = a , b , andlet B (s , a , b , t ) bethesetofbusheswith a ∈ Ts and b ∈ Tt .The sets B (s , a , b , t ) and N (s , a , b , t ) areinone–onecorrespondence,since theadditionof e to B ∈ B (s , a , b , t ) createsauniquemember T = T ( B ) of N (s , a , b , t ),andviceversa.
By(1.19)andtheabove,abush B = ( Ts , Tt ) makesacontributionto i a ,b of
N 1 if B ∈ B (s , a , b , t ),
N 1 if B ∈ B (s , b , a , t ), 0otherwise
Therefore, B makesacontributiontowardsthesumin(1.20)thatisequalto N 1 ( F+ F ),where F+ (respectively, F )isthenumberofpairs v j ,v j +1 of C ,1 ≤ j ≤ n ,with v j ∈ Ts , v j +1 ∈ Tt (respectively, v j +1 ∈ Ts , v j ∈ Tt ). Since C isacycle,wehave F+ = F ,whenceeachbushcontributes0to thesumand(1.20)isproved.
1.3FlowsandEnergy
Let G = ( V , E ) beaconnectedgraphasbefore.Let s , t ∈ V bedistinct vertices,andlet j bean s / t flow.With we theconductanceoftheedge e , the(dissipated) energy of j isdefinedas E ( j ) = e= u ,v ∈
Thefollowingpieceoflinearalgebrawillbeuseful.
1.21Proposition Let ψ : V → R,andletjbeans / tflow.Then
[ψ(t ) ψ(s )] Js = 1 2 u ,v ∈ V [ψ(v) ψ(u )] ju ,v . Proof. Bythepropertiesofaflow,
[ψ(v) ψ(u )] ju ,v =
Jv )
ψ(u ) Ju =−2[ψ(s ) Js + ψ(t ) Jt ] = 2[ψ(t ) ψ(s )] Js , asrequired.
Let φ and i satisfythetwoKirchhofflaws.WeapplyProposition1.21 with ψ = φ and j = i tofindbyOhm’slawthat (1.22)
E (i ) = [φ(t ) φ(s )] Is .
Thatis,theenergyofthetruecurrent-flow i from s to t equalstheenergy dissipatedina(notional)single s , t edgecarryingthesamepotentialdifferenceandtotalcurrent.Theconductance Weff ofsuchanedgewould satisfyOhm’slaw,thatis,
(1.23) Is = Weff [φ(t ) φ(s )],
andwedefinethe effectiveconductanceWeff bythisequation.Theeffective resistanceis
(1.24)
, which,by(1.22)and(1.23),equals E (i )/ I 2 s .Westatethisasalemma.
1.25Lemma TheeffectiveresistanceReff ofthenetworkbetweenvertices sandtequalsthedissipatedenergywhenaunitflowpassesfromstot.
Itisusefultobeabletodocalculations.Electricalengineershavedevised avarietyofformulaicmethodsforcalculatingtheeffectiveresistanceofa network,ofwhichthesimplestaretheseriesandparallellaws,illustrated inFigure1.1.
Figure1.1 Twoedges e and f inparallelandinseries.
1.26Serieslaw Tworesistorsofsize r 1 and r 2 inseriesmaybereplaced byasingleresistorofsize r 1 + r 2 .
1.27Parallellaw Tworesistorsofsize r 1 and r 2 inparallelmaybereplaced byasingleresistorofsize R ,where R 1 = r 1 1 + r 1 2 .
Athirdsuchrule,theso-called‘star–triangletransformation’,maybe foundatExercise1.5.Thefollowing‘variationalprinciple’hasmanyuses.
1.28Theorem(Thomsonprinciple) LetG = ( V , E ) beaconnected graphand (we : e ∈ E ) strictlypositiveconductances.Lets , t ∈ V, s = t.AmongstallunitflowsthroughGfromstot,theflowthatsatisfies theKirchhofflawsistheuniques / tflowithatminimizesthedissipated energy.Thatis,
E (i ) = inf E ( j ) : j aunitflowfrom s to t .
Proof. Let j beaunitflowfromsource s tosink t ,andset k = j i ,where i isthe(unique)unit-flowsolutiontotheKirchhofflaws.Thus, k isaflow withzerosize.Now,with e = u ,v and r e = 1/we ,
Let φ bethepotentialfunctioncorrespondingto i .ByOhm’slawand Proposition1.21,
whichequalszero.Therefore, E ( j ) ≥ E (i ),withequalityifandonlyif j = i .
TheThomson‘variationalprinciple’leadstoaproofofthe‘obvious’fact thattheeffectiveresistanceofanetworkisanon-decreasingfunctionofthe resistancesofindividualedges.
1.29Theorem(Rayleighprinciple) TheeffectiveresistanceReff ofthe networkisanon-decreasingfunctionoftheedge-resistances (r e : e ∈ E )
Itisleftasanexercisetoshowthat Reff isaconcavefunctionofthevector (r e ).SeeExercise1.6.
Proof. Considertwovectors (r e : e ∈ E ) and (r e : e ∈ E ) ofedgeresistanceswith r e ≤ r e forall e .Let i and i denotethecorrespondingunit flowssatisfyingtheKirchhofflaws.ByLemma1.25,with r e = r u ,v ,
= Reff , asrequired.
1.4RecurrenceandResistance
Let G = ( V , E ) beaninfiniteconnectedgraphwithfinitevertex-degrees, andlet (we : e ∈ E ) be(strictlypositive)conductances.Weshallconsider areversibleMarkovchain Z = ( Z n : n ≥ 0 ) onthestatespace V with transitionprobabilitiesgivenby(1.3).Ourpurposeistoestablishacondition onthepair ( G ,w) thatisequivalenttotherecurrenceof Z
Let0beadistinguishedvertexof G ,calledthe‘origin’,andsuppose that Z 0 = 0.Thegraph-theoreticdistancebetweentwovertices u , v isthe numberofedgesinashortestpathbetween u and v ,denoted δ(u ,v).Let
Wethinkof ∂ n asthe‘boundary’of n .Let G n bethesubgraphof G inducedbythevertex-set n .Welet G n bethegraphobtainedfrom G n by identifyingtheverticesin ∂ n asasinglecompositevertexdenoted In .The resultingfinitegraph G n maybeconsideredasanelectricalnetworkwith sources0and In .Let Reff (n ) betheeffectiveresistanceofthisnetwork.The graph G n maybeobtainedfrom G n +1 byidentifyingallverticeslyingin ∂ n ∪{ In +1 },andthus,bytheRayleighprinciple, Reff (n ) isnon-decreasing in n .Therefore,thelimit Reff = lim n →∞ Reff (n ) exists.
1.30Theorem TheprobabilityofultimatereturnbyZtotheorigin 0 is givenby P0 ( Z n = 0forsome n ≥ 1) = 1 1 W0 Reff , whereW0 = v : v ∼0 w 0,v .
Thereturnprobabilityisnon-decreasingas W0 Reff increases.Bythe Rayleighprinciple,thiscanbeachieved,forexample,byremovinganedge of E thatisnotincidentto0.Theremovalofanedgeincidentto0canhave theoppositeeffect,since W0 decreaseswhile Reff increases(seeFigure1.2).
A0 /∞ flow isavector j = ( ju ,v : u ,v ∈ V , u = v) satisfying(1.14)(a), (b)andalso(c)forall u = 0.Thatis,ithassource0butnosink.
1.31Corollary
(a) ThechainZisrecurrentifandonlyifReff =∞
(b) ThechainZistransientifandonlyifthereexistsanon-zero 0 /∞ flow jonGwhoseenergyE ( j ) = e j 2 e /we satisfiesE ( j )< ∞. .002 17:53:41
Figure1.2 Thisisaninfinitebinarytreewithtwoparalleledgesjoining theorigintotheroot.Wheneachedgehasunitresistance,itisaneasy calculationthat Reff = 3 2 ,sotheprobabilityofreturnto0is 2 3 .Ifthe edge e isremoved,thisprobabilitybecomes 1 2 .
Itisleftasanexercisetoextendthistocountablegraphs G withoutthe assumptionoffinitevertex-degrees.
ProofofTheorem1.30. Let
gn (v) = Pv ( Z hits ∂ n before0),v ∈ n .
ByTheorem1.5andExercise1.13, gn istheuniqueharmonicfunctionon G n withboundaryconditions
gn (0 ) = 0 , gn (v) = 1for v ∈ ∂ n .
Therefore, gn isapotentialfunctionon G n viewedasanelectricalnetwork withsource0andsink In .
Byconditioningonthefirststepofthewalk,andusingOhm’slaw, P0 ( Z returnsto0beforereaching ∂ n ) = 1
v : v ∼0 p0,v gn (v) = 1
v : v ∼0 w0,v W0 [ gn (v) gn (0 )] = 1 |i (n )| W0 , where i (n ) istheflowofcurrentsin G n ,and |i (n )| isitssize.By(1.23)and (1.24), |i (n )|= 1/ Reff (n ).Thetheoremisprovedonnotingthat P0 ( Z returnsto0beforereaching ∂ n ) → P0 ( Z n = 0forsome n ≥ 1)
17:53:41
as n →∞,bythecontinuityofprobabilitymeasures.
ProofofCorollary1.31. Part(a)isanimmediateconsequenceofTheorem 1.30,andweturntopart(b).ByLemma1.25,thereexistsaunitflow i (n ) in G n withsource0andsink In ,andwithenergy E (i (n )) = Reff (n ).Let i beanon-zero0 /∞ flow;bydividingbyitssize,wemaytake i tobeaunit flow.Whenrestrictedtotheedge-set E n of G n , i formsaunitflowfrom0 to In .BytheThomsonprinciple,Theorem1.28,
Therefore,bypart(a), E (i ) =∞ ifthechainisrecurrent.
Suppose,conversely,thatthechainistransient.Bydiagonalselection,3 thereexistsasubsequence (n k ) alongwhich i (n k ) convergestosomelimit j (thatis, i (n k )e → je forevery e ∈ E ).Sinceeach i (n k ) isaunitflow fromtheorigin, j isaunit0 /∞ flow.Now,
(i (n k )) =
Therefore, E ( j ) ≤ lim k
Reff (n k ) = Reff < ∞, and j isaflowwiththerequiredproperties.
3 Diagonalselection:Let ( x m (n ) : m , n ≥ 1) beaboundedcollectionofreals.There existsanincreasingsequence n 1 , n 2 ,... ofpositiveintegerssuchthat,forevery m ,the limitlim k →∞ x m (n k ) exists.
1.5P ´ olya’sTheorem
The d -dimensionalcubiclattice Ld hasvertex-set Zd andedgesbetweenany twoverticesthatareEuclideandistanceoneapart.Thefollowingcelebrated theoremcanbeprovedbyestimatingeffectiveresistances.4
1.32P ´ olya’stheorem[242] Symmetricrandomwalkonthelattice Ld in ddimensionsisrecurrentifd = 1, 2 andtransientifd ≥ 3.
TheadvantageofthefollowingproofofP ´ olya’stheoremovermorestandardargumentsisitsrobustnesswithrespecttotheunderlyinggraph.Similarargumentsarevalidforgraphsthatare,inbroadterms,comparableto Ld whenviewedaselectricalnetworks.
Proof. Forsimplicity,andwithonlylittlelossofgenerality(seeExercise 1.10),weshallconcentrateonthecases d = 2, 3.Let d = 2,forwhichcase weaimtoshowthat Reff =∞.Thisisachievedbyfindinganinfinitelower boundfor Reff ,andlowerboundscanbeobtainedbydecreasingindividual edge-resistances.Theidentificationoftwoverticesofanetworkamounts totheadditionofaresistorwith0resistance,and,bytheRayleighprinciple, theeffectiveresistanceofthenetworkcanonlydecrease.
Figure1.3 Thevertexlabelled i isacompositevertexobtainedby identifyingallverticeswithdistance i from0.Thereare8i 4edgesof L2 joiningvertices i 1and i .
From L2 ,weconstructanewgraphinwhich,foreach k = 1, 2,... , theset ∂ k ={v ∈ Z2 : δ(0 ,v) = k } isidentifiedasasingleton.This transforms L2 intothegraphshowninFigure1.3.Bytheseries/parallel lawsandtheRayleighprinciple, Reff (n ) ≥ n 1 i =1 1 8i 4 ,
whence Reff (n ) ≥ c log n →∞ as n →∞.
Supposenowthat d = 3.Thereareatleasttwowaysofproceeding. Weshallpresentonesuchroute,takenfrom[222],andweshallthensketch
4 Anamusingstoryistoldin[243]aboutP ´ olya’sinspirationforthistheorem. .002 17:53:41
Figure1.4 Theflowalongtheedge u ,v isequaltotheareaofthe projection ( Fu ,v ) ontheunitspherecentredattheorigin,withasuitable conventionforitssign.
thesecond,whichhasitsinspirationin[83].ByCorollary1.31,itsuffices toconstructanon-zero0 /∞ flowwithfiniteenergy.Let S bethesurface oftheunitsphereof R3 withcentreattheorigin0.Take u ∈ Z3 , u = 0, andpositionaunitcube C u in R3 withcentreat u andedgesparalleltothe axes(seeFigure1.4).Foreachneighbour v of u ,thedirectededge[u ,v intersectsauniqueface,denoted Fu ,v ,of C u .
For x ∈ R3 , x = 0,let ( x ) bethepointofintersectionwith S ofthe straightlinesegmentfrom0to x .Let ju ,v beequalinabsolutevaluetothe surfacemeasureof ( Fu ,v ).Thesignof ju ,v istakentobepositiveifand onlyifthescalarproductof 1 2 (u + v) and v u ,viewedasvectorsin R3 ,is positive.Let jv,u =− ju ,v .Weclaimthat j isa0 /∞ flowon L3 .Parts(a) and(b)ofDefinition1.14followbyconstruction,anditremainstocheck (c).
Thesurfaceof C u hasprojection (C u ) on S .Thesum Ju = v ∼u ju ,v istheintegralover x ∈ (C u ),withrespecttosurfacemeasure,ofthe numberofneighbours v of u (countedwithsign)forwhich x ∈ ( Fu ,v ). Almostevery x ∈ (C u ) iscountedtwice,withsigns + and .Thusthe integralequals0,whence Ju = 0forall u = 0.
Itiseasilyseenthat J0 = 0,so j isanon-zeroflow.Next,weestimate itsenergy.Byanelementarygeometricconsideration,thereexist ci < ∞ .002 17:53:41
suchthat:
(i) | ju ,v |≤ c1 /|u |2 for u = 0,where |u |= δ(0 , u ) isthelengthofa shortestpathfrom0to u , (ii)thenumberof u ∈ Z3 with |u |= n issmallerthan c2 n 2 . Itfollowsthat
asrequired.
Anotherwayofshowing Reff < ∞ when d = 3istofindafiniteupper boundfor Reff .Upperboundscanbeobtainedeitherbyincreasingindividual edge-resistancesorbyremovingedges.Theideaistoembedatreewith finiteresistancein L3 .Considerabinarytree Tρ inwhicheachconnection betweengeneration n 1andgeneration n hasresistance ρ n ,where ρ> 0.It isaneasyexerciseusingtheseries/parallellawsthattheeffectiveresistance betweentherootandinfinityis
eff ( Tρ ) =
whichwemakefinitebychoosing ρ< 2.Weproceedtoembed Tρ in Z3 insuchawaythataconnectionbetweengeneration n 1andgeneration n isalattice-pathwithlengthoforder ρ n .Thereare2n verticesof Tρ in generation n ,andtheirlattice-distancefrom0isoforder n k =1 ρ k ,that is,order ρ n .Thesurfaceofthe k -ballin R3 isoforder k 2 ,andthusitis necessarythat c (ρ n )2 ≥ 2n ,
whichistosaythat ρ> √2.
Let √2 <ρ< 2.Itisnowfairlysimpletocheckthat Reff < c Reff ( Tρ ). Thismethodwasusedin[138]toprovethetransienceoftheinfiniteopen clusterofpercolationon L3 .Itisrelatedto,butdifferentfrom,thetree embeddingsof[83].
1.6GraphTheory
Agraph G = ( V , E ) comprisesafiniteorcountablyinfinitevertex-set V andanassociatededge-set E .Eachelementof E isanunorderedpair u , v ofvertices,written u ,v .Twoedgeswiththesamevertex-pairsaresaid tobein parallel ,andedgesoftheform u , u arecalled loops .Thegraphs discussedinthistextwillgenerallycontainneitherparalleledgesnorloops,
andthisisassumedhenceforth.Twovertices u , v aresaidtobejoined(or connected)byanedgeif u ,v ∈ E .Inthiscase, u and v arethe endvertices of e ,andwewrite u ∼ v andsaythat u is adjacent to v .Anedge e issaid tobe incident toitsendvertices.Thenumberofedgesincidenttovertex u iscalledthe degree of u ,denoteddeg(u ).Thenegationoftherelation ∼ is written
Sincetheedgesareunorderedpairs,wecallsuchagraph undirected (or unoriented ).Ifsomeorallofitsedgesare ordered pairs,written[u ,v ,the graphiscalled directed (or oriented ).
A path of G isdefinedasanalternatingsequence v0 , e0 ,v1 , e1 ,..., en 1 , vn ofdistinctvertices vi andedges ei = vi ,vi +1 .Suchapathhas length n ;itissaidtoconnect v0 to vn ,andiscalleda v0 /vn path.A cycle or circuit of G isanalternatingsequence v0 , e0 ,v1 ,..., en 1 ,vn , en ,v0 ofvertices andedgessuchthat v0 , e0 ,..., en 1 ,vn isapathand en = vn ,v0 .Such acyclehaslength n + 1.The(graph-theoretic)distance δ(u ,v) from u to v isdefinedtobethenumberofedgesinashortestpathof G from u to v .
Wewrite u v ifthereexistsapathconnecting u and v .Therelation isanequivalencerelation,anditsequivalenceclassesarecalled components (or clusters )of G .Thecomponentsof G maybeconsideredeitherassetsof verticesorasgraphs.Thegraph G is connected ifithasauniquecomponent. Itisa forest ifitcontainsnocycle,anda tree ifinadditionitisconnected.
A subgraph ofthegraph G = ( V , E ) isagraph H = ( W , F ) with W ⊆ V and F ⊆ E .Thesubgraph H isa spanningtree of G if V = W and H isatree.Asubset U ⊆ V ofthevertex-setof G has boundary ∂ U ={u ∈ U : u ∼ v forsome v ∈ V \ U }.
Lattice-graphsarethemostimportanttypeofgraphforapplicationsin areassuchasstatisticalmechanics.Latticesaresometimestermed‘crystalline’sincetheyareperiodicstructuresofcrystal-likeunits.Ageneral definitionofalatticemayconfusereadersmorethanhelpthem,andinstead wedescribesomeprincipalexamples.
Let d beapositiveinteger.Wewrite Z ={..., 1, 0 , 1,... } forthe setofallintegers,and Zd forthesetofall d -vectors v = (v1 ,v2 ,...,vd ) withintegralcoordinates.For v ∈ Zd ,wegenerallywrite vi forthe i th coordinateof v ,andwedefine
δ(u ,v) = d i =1 |u i vi |
The origin of Zd isdenotedby0.Weturn Zd intoagraph,calledthe ddimensional(hyper)cubiclattice,byaddingedgesbetweenallpairs u , v of pointsof Zd with δ(u ,v) = 1.Thisgraphisdenotedas Ld ,anditsedge-set .002 17:53:41
Figure1.5 Thesquare,triangular,andhexagonal(or‘honeycomb’) lattices.Thesolidanddashedlinesillustratetheconceptof‘planar duality’discussedafter(3.7).
as Ed :thus, Ld = (Zd , Ed ).Weoftenthinkof Ld asagraphembedded in Rd ,theedgesbeingstraightline-segmentsbetweentheirendvertices. The edge-set E V of V ⊆ Zd isthesetofalledgesof Ld bothofwhose endverticesliein V .
Thetwo-dimensionalcubiclattice L2 iscalledthe squarelattice andis illustratedinFigure1.5.Twootherlatticesintwodimensionsthatfeature inthistextaredrawntherealso.
1.7Exercises
1.1 Let G = ( V , E ) beafiniteconnectedgraphwithunitedge-weights.Show thattheeffectiveresistancebetweentwodistinctvertices s , t oftheassociated electricalnetworkmaybeexpressedas B / N ,where B isthenumberof s / t bushes of G ,and N isthenumberofitsspanningtrees.(SeetheproofofTheorem1.16 foranexplanationoftheterm‘bush’.)
Extendthisresulttogeneraledge-weights we > 0.
1.2 Let G = ( V , E ) beafiniteconnectedgraphwithstrictlypositiveedgeweights (we : e ∈ E ),andlet N ∗ begivenby(1.18).Showthat i a ,b = 1 N ∗ N ∗ (s , a , b , t ) N ∗ (s , b , a , t ) constitutesaunitflowthrough G from s to t satisfyingKirchhoff’slaws.
1.3 (continuation)Let G = ( V , E ) befiniteandconnectedwithgivenconductances (we : e ∈ E ),andlet ( x v : v ∈ V ) berealssatisfying v x v = 0.To G weappendanotionalvertexlabelled ∞,andwejoin ∞ toeach v ∈ V .Showthat thereexistsasolution i toKirchhoff’slawsontheexpandedgraph,viewedastwo lawsconcerningcurrentflow,suchthatthecurrentalongtheedge v, ∞ is x v . .002 17:53:41
Another random document with no related content on Scribd:
Inventors of gas apparatus should note that the municipal authorities of Brussels have decided upon holding a competition, with a view to ascertain the best means of using gas for heating and cooking purposes. A large sum is to be offered in prizes to the successful competitors. Apparatus for trial must be forwarded not later than September next, and all particulars regarding the matter may be obtained from the chief engineer, M. Wybauw, Rue de l’Etuve, Brussels.
In the island of Skye, large deposits of the very useful mineral called diatomite have recently been found. Under the German name of kieselguhr, this absorbent earth has been extensively used in the manufacture of dynamite, which consists of nitro-glycerine rendered more safe for handling by admixture with this porous body. It is also used as a non-conducting compound for coating the exterior of steam-pipes and boilers, as a siliceous glaze for pottery, for the manufacture of silicate paints, and for many minor purposes. In this particular deposit the varieties of diatoms are singularly few, only sixteen species of these wonderful microscopic organisms being represented. The deposit is estimated to yield a total of between one and two hundred tons.
At a recent meeting of the Royal Society of Edinburgh, Dr A. B. Griffiths read a most instructive paper on ‘The Effect of Ferrous Sulphate in destroying the Spores of Parasitic Fungi.’ The value of this salt—the common ‘green vitriol’ of commerce—as a plant-food has long ago been established; but Dr Griffiths points out the important antiseptic property it possesses in destroying certain low forms of plant-life. As a preventive of potato disease, it is most effectual, although the spores of that fungus possess such vitality that they may be kept as dry dust for eight months without losing their power for mischief. Dr Griffiths also notes that in damp warm weather, the potato disease is actually encouraged by the use of potash manures. He advocates the treatment of manure with a weak solution of the iron salt before its application to the land. Wheat when treated with the sulphate is rendered proof against mildew.
A clever method of damascening metals by electrolysis is described in a French technical journal. The process consists of two distinct
operations, and is based on the well-known fact, that when two copper plates are hung in a bath of sulphate of copper and connected with the opposite poles of a battery, a transfer of metal from one to the other will take place. In the case before us, a copper plate is covered with a thin layer of insulating material, as in the etching process, and this is drawn upon with an etching needle so as to lay bare the metal beneath. This is now submitted to the action of the electric current, so that the metal is eaten away to a certain depth in the exposed parts. The plate is next washed with acid, to remove all traces of oxide of copper in the bitten-in lines, and is then transferred to another bath by which metallic silver or nickel is deposited in the etched parts, with the result that the sunk lines are ultimately completely filled with the new metal. When the plate is relieved of its waxy coating and is polished, it is impossible to say whether or not the beautiful inlaid appearance has been produced by a mechanical process or by skilled handiwork.
Two remarkable finds of old coins have lately occurred—one at Milverton, a suburb of Leamington; and the other at Aberdeen. In the first case, some labourers were digging foundations, when they found a Roman amphora, which they immediately smashed to ascertain its contents. It contained nearly three hundred coins in silver and copper. These were of very early date, and in a state of excellent preservation. The Aberdeen treasure trove came to light in excavating Ross’s Court, one of the oldest parts of the city. Here the labourers found a bronze urn filled with a large number of silver coins. These coins also are well preserved. They are all English, and are mostly of the reigns of Edward I. and Edward II. Some of these coins are of extreme rarity, and the discovery has great antiquarian interest.
The largest installation of the electric light, worked from a central point, which this country has yet seen has been recently completed at the Paddington terminus of the Great Western Railway. The lights, which are equivalent to thirty thousand ordinary gas jets, are distributed between the Paddington passenger and goods stations, the ‘Royal Oak,’ and Westbourne Park Stations, the terminus hotel, and all the various offices, yards, and approaches to the railway
Company’s premises. The district covers no fewer than sixty-seven acres of ground, and is one mile and a half long. The two Gordon dynamos which are used to generate the current weigh forty-five tons each, and give sufficient power to serve four thousand one hundred and fifteen Swan glow lamps, each of twenty-five candlepower; ninety-eight arc lamps, each of three thousand five hundred candle-power; and two of twelve thousand candle-power each. The current is kept on day and night, except for a few hours on Sunday morning, and each individual lamp is under separate control by a switch, so that it can be turned off and on just like a gas jet. Every detail has been well thought out, and the vast scheme is a success in every way. We understand that the contractors, the Telegraph Maintenance and Construction Company, have undertaken to supply the light at the same price as would have been charged for gas lamps giving the same light-value.
From a paper read by Mr C. Harding before the Royal Meteorological Society on ‘The Severe Weather of the Past Winter,’ we learn that the cold lately experienced has been of the most exceptional character. The persistency with which frost continued for long periods was quite remarkable. In south-west England, there was not a single week from October to the end of March in which the temperature did not fall below the freezing-point; and in one town in Hertfordshire, frost occurred on the grass on seventy-three consecutive nights. Since the formation of the London Skating Club, nearly sixty years ago, the past season has been the only one in which skating has been possible in each of the four months December to March. We therefore must note that we have just passed through an unusually severe season.
Fresh fruit from the antipodes, of which two large consignments have recently reached London, is now being daily sold to eager purchasers in the Australian fruit-market at the Indian and Colonial Exhibition. Grapes, apples, pears, and other fruits, in splendid condition, and with their flavour unaltered by their long separation from their parent stems, can now be conveyed by the shipload, packed in cool chambers, in the same way that meat is imported from the same distant lands. The success of the enterprise opens up
a wide field of promise to those in temperate lands who have been dazzled by the reports of travellers as to the luscious nature of foreign fruits, which hitherto have been quite out of reach of stay-athome Britons. We seem to be fast coming to the time when fairy tales will be considered tame and uninteresting, from being so far eclipsed by current events.
A correspondent of the Times notes a most important means of escape from suffocation by smoke, a fatality by which many lives are lost annually. He points out that if a handkerchief be placed beneath the pillow on retiring to rest so as to be within easy reach of the hand, it can, in case of an alarm of fire, be readily dipped in water and tied over the mouth and nostrils. As an amateur fireman, he has gone through the densest smoke protected in that manner, and he alleges that such a respirator will enable its wearer to breathe freely in an otherwise irrespirable atmosphere.
Professor Dewar lately exhibited at the Royal Institution, London, the apparatus he employs for the production of solid oxygen. If we refer to the physical text-books of only three or four years back, we find oxygen, hydrogen, and nitrogen described as permanent gases, for no one had ever produced either in any other form. At length all three had to give way before scientific research, and they were by special appliances reduced to the liquid state. Professor Dewar is the first experimenter who has taken the further step of producing one of these gases in a solid form. His method consists in allowing liquid oxygen to expand into a partial vacuum, when the great absorption of heat which accompanies the operation causes the liquid to assume a solid state. It is said to resemble snow in appearance, with a temperature greatly below the freezing-point of water It is believed that a means of producing such a degree of cold will be of great service to experimental chemistry.
Mr W. Thomson, F.R.S.E., has devised a new process for determining the calorific power of fuel by direct combustion in oxygen, which promises to supersede, by reason of its greater accuracy, the methods hitherto in use. The process consists in placing a gramme of the coal or fuel to be tested in a platinum crucible covered with an inverted glass vessel. The whole
arrangement is placed under water in a suitable receptacle; and the fuel, burnt in oxygen, burns away in a very few minutes, giving off much heated gas, which escapes through the water. The temperature of the water, compared with its temperature before the operation, gives the data upon which the heating power of the coal can be calculated. The question of heat-value in fuel is of course one of first importance to railway Companies and other large consumers of coal. It is, too, in a minor way of importance to householders, who often find, by painful experience, the little heat-value of the fuel which has been shot into their cellars. If coal-merchants were to furnish some guarantee based on a scientific test as above described, they would find it to their own profit, as well as to the advantage of their customers.
We do not hear very much in these days of mummy wheat and barley, but many people firmly believe that the seeds of both plants found with Egyptian mummies, and supposed to be three or four thousand years old, will sprout if put in the ground. A few years ago, such wheat was commonly sold as a curiosity; and we believe that many purchasers succeeded in raising a small crop from it. Professor Bentley, who has recently commenced a series of lectures on the Physiology of Plants, asserts most emphatically that no grains which with certainty have been identified as contemporaneous with the deposit of the mummified corpse, have ever come to life. In cases where the so-called mummy wheat has germinated, it has been introduced into the coffin shortly before, or at the time of discovery of the body Professor Bentley does not name a limit to the time during which seeds retain their vitality, but he says that very few will germinate after being three years old.
Dr Kosmann of Breslau has designed a safety cartridge for use in fiery mines, but it has not yet passed the ordeal of practical employment. It depends for its efficiency upon the sudden evolution of a large volume of hydrogen gas, which is brought about by the action of dilute acid upon finely divided zinc. The ‘cartridge’ consists of a glass cylinder pinched into a narrow tube at the centre, so that interiorly it is divided into two compartments. One of these contains the powdered zinc, and the other the dilute acid, the passage
between them being closed by a rubber cork. The borehole into which it is inserted is first of all made gas-tight by a lining of clay; then the cartridge is put in position, with an iron rod in connection with it so placed that, when struck with a hammer from the outside of the hole, it will drive in the rubber cork, and so bring the acid into contact with the zinc. We shall be interested to hear how the method answers in practice.
