Discrete random walk with barriers on a locally infinite graph

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1 Drete rdo wl wth rrer o loll fte grh Theo Ue Aterd Shool of Teholog Weeerde 9 97 DZ Aterd The etherld El: te@hl Atrt We ot eeted er of rrl orto rolte d eeted te efore orto for etr drete rdo wl o loll fte grh the reee of ltle fto rrer O eh edge of the grh d eh rrer there re ef rolte defed Ele of ltle fto rrer grh d lto o the teger re ge Keword: Rdo wl orto refleto rrer grh thet Set Clfto: Prr 6G5 Seodr 6J5 Itrodto Rdo wl e ed ro dle: eoo to odel hre re d ther derte ede d olog where org rrer ge trl odel for wde ret of heoe h lfed odel of Brow oto eolog to dere ddl l oeet d olto d ttt to le eetl tet roedre oter ee to ette the e of the World Wde We g rdoed lgorth Bro d C 5 ge reew of rdo wl o grh where the geerlto of the oet of deo to hoogeeo trtre g fte grh odered Drh Joo d Wheter 6 deelo tehe to ot rgoro od o the ehor of rdo wl o o Ug thee od the llte the etrl deo of rdo o wth fte teeth t rdo oto or teeth wth rdo t fte legth Rdo wl he ee tded for dede o reglr trtre h ltte We ow ge ref htorl reew of the e of rrer oe-deol drete rdo wl Weel 96 ded the ll role of rdo wl retrted etwee refletg d org rrer Ug geertg fto he ot elt ereo for the rolt of orto Leher 963 tde oe-deol rdo wl wth rtll refletg rrer g otorl ethod Gt 966 trode the oet of ltle fto rrer f: tte tht or reflet let throgh or hold for oet D Khdlr d Se 976 fd the rte geertg fto of the rolte of rtle rehg ert tte der dfferet odto Per 985 oder etr rdo wl wth oe or two odre o oe-deol ltte At the odre the wler ether ored or refleted to the te Ug geertg fto the rolt dtrto of eg t oto fter te oted well the e er of te efore orto El-Shehwe ot orto rolte t the odre for rdo wl etwee oe or two rtll org odre well the odtol e for the er of te efore tog ge the orto t efed rrer g odtol rolte I th er we ot eeted er of rrl orto rolte d eeted te efore orto for etr drete rdo wl wth ltle fto rrer edg the theor two dreto: A the tte e o loger oe-deol ltte or do loll fte grh B oe or two org or refletg rrer re reled fte et of ltle fto rrer

2 Theo Ue We hooe trtre of or grh tht hdle wth the well ow ltte trtre well trtre le tr grh fte d fte d le grh Or grh ot of ltle fto rrer erte tte o the edge etwee the f d fte et of hlf le eh wth fte er of tte trtg eh rrer Eh hlf le h toologl ed O eh edge of the grh rdo wl wth t ow terl tte d g rolte troded Whe the wler rehe ltle fto rrer rdo wl tted ordg to etr et of rolte or the rtle ored the rrer Eh rrer d eh edge h t ow rolt reter I eto we e geertg fto to fd the eeted er of rrl to tte the rolt of orto d the eeted te efore orto I eto 3 we le oe ele of grh wth ltle fto rrer: two tr grh d le grh I the lt eto we l or theor l tto: the et of teger I ede A d B we ge roe of oe relt eto A grh wth ltle fto rrer Derto of the rdo wl wth ltle fto rrer I grh we he erte rereetg the f Betwee d there rdo wl wth fte er of tte whh we er 3 the dreto fro to whe We wll e the reto for the edge etwee d Eh rdo wl fro to h t ow reter d rr where the oe-te forwrd rolt oe-te wrd d r-- the rolt to t for oet the e oto We ded for eh d It lo ole to oe fro log hlf le wth tte 3 A hlf le trtg lelled d h ed Ω I there rolt to oe oe te the dreto of rolt to t for oet rolt for edte orto d rolt to oe oe te the dreto Ω where ee lo fg Ω tte: 3 oe te rolte: r tte: r Fgre Derto of the rdo wl

3 Drete rdo wl wth rrer 3 We trt ether d o edge d d where e: trt d trt d rereeted d or we trt o edge h where e: trt Eeted er of rrl We re tereted the eeted er of rrl the f well the eeted er of rrl the other tte of or grh We defe: trt fter te tte Pte f ot f for f trt rrl eeted er of rrl trt eeted er of O terl : d O hlf le : We trt o the terl d or o the hlf le h Theore If the: the e olto of Q where f f Strtg o terl d: d d Q δ δ δ Strtg o hlf le h: h f h f Q δ δ O terl we he: If d the:

4 4 Theo Ue the : If d O hlf le we he: the: h If h h h h h the : h If f f If the we ot lr relt lg l Hotl rle Proof See Aed A 3 Prolt of orto We re tereted the rolt of orto f well the ed of the hlf le We defe: trt fter te f Pte The rolt of orto f ge Aother w to detere the rolt of orto f ge the et orollr of Theore Alto of th orollr e fod eto 3 d 4

5 Drete rdo wl wth rrer 5 Corollr Porto f t or wth hlf le trt o f we erl o o hlf le wth we trt f Proof Ug Theore we get: Q f we trt o hlf le wth otherwe Q The e theore ge: o: Q Le Ge r rdo wl o the o-egte teger wth rtl refletg rrer trt d ed Ω we he: rrl eeted er of orto Ω P Proof I we te refletg rolt d orto rolt Solg the relet dfferee eto we fd: Ug th relt we lo get orto P Ω We re ow red for: Corollr If o hlf le wth ed Ω we he the: Ω o Port h f h f h Proof Ue Theore d Le where tte Le ow odered the oe ot otrto of or olete grh or hlf le We wll e th orollr eto 3 d 4 Rer Corollr oeee of orollr

6 6 Theo Ue 4 Eeted te efore orto We re tereted the eeted te efore orto We defe: eeted te efore orto whe trtg o terl eeted te efore orto whe trtg o hlf le eeted te efore orto whe trtg Theore If o eh hlf le wth we he the: the e olto of Λ where: d f d f d f Λ O terl : O hlf le : Proof See ed B

7 Drete rdo wl wth rrer 7 3 Ele of ltle fto rrer grh 3 A fte tr grh We oder tr grh wth the etre d ll edge re of the terl te we trt o edge tte rolte r Fgre A fte tr grh I fte tr grh we he: f { } d otto: r r We he: 3 Eeted er of rrl To ot the eeted er of rrl or fte tr grh we e Theore : Q Ited of the eto wth we wll e: Th eto el dered fro Q orto or fte grh If we fd: ς ς ee lo orollr or oerg tht we del wth the rolt of 3 wth ς where ς 3 9

8 8 Theo Ue If : wth 3 ς ς ς ς I oth e we he : o ς ς ς ζ If we trt the eter of the tr grh we he: ς ς 3 e orto te the reee of org tte the edot We ow oder tr grh wth the etre d org tte the edot : } { f We trt otto: r r We defe for : R W Theore ge ow: Λ R W

9 Drete rdo wl wth rrer 9 33 A el e We get el fte tr grh whe we te d tte rolte r tte rolte r Fgre 3 A el e Ug eto 3 we get the eeted er of rrl: ς ς Iterretto: rdo wl o -AB wth dfferet rolte left d rght fro the trtg ot d three f : -A d B A B Ug eto 3 we ot the e orto te trtg the reee of org tte the edot: Iterretto: rdo wl o -AB wth dfferet rolte left d rght fro the trtg ot f d two org rrer: -A d B A B W R Rer If we get the well ow relt: AB ee Feller A fte tr grh We oder tr grh wth gle f the etre d ll edge re of the hlf le te we trt o the edge otto : r r Ω tte rolte r Ω Ω Fgre 4 A fte tr grh

10 Theo Ue 3 Eeted er of rrl d rolt of rrl To ot the eeted er rrl d rolt of rrl we l Theore : f f where whh led to: f f Ag g Theore we fd: O hlf le : O hlf le 3 : f f We defe f rolt to t whe trtg A well ow relt : f If o hlfle d : f

11 Drete rdo wl wth rrer 3 Aorto rolte Bede orto there the olt of orto Ω whe Whe ge: o: Porto ow we oe tht there t let oe wth: Ug 9 wth org tte d tg we fd for f the ζ d f f We frt oder the e : Th led to: f the: Porto Porto We he: Porto et we le the e : Ω Porto Porto Th d g g 9 wth org tte led to: d f : Ω 3 Porto Ω Porto Ω Here we lo he: Porto Porto 3 Ω All relt th eto lo e erfed lg orollr the f e d orollr the Ω tto

12 Theo Ue 33 e orto te Alg Theore we ot the e orto te f whe trtg If : The to e derget For the e orto te tte o hlf le we fd: 34 Sel e: two hlf le trtg f A oeee we he ow lo led rdo wl o wth gle f d two reter d rght d left fro the org: te two edge reer the tte of the eod edge -- d e reter o the frt d o the eod edge 33 A Pote Oreted Cle Grh We he rrer le grh: We trt d defe rtfl rrer whh the e rrer A ote oreted le grh defed : otto : tte rolte r Fgre 5 A le grh 33 Eeted er of rrl d rolt of rrl Theore ge ow:

13 Drete rdo wl wth rrer 3 Pt for : where We ow he: where I fte grh we he: o: the o f he : we For We lo he: f 33 e orto te Ug Theore we ot: Λ Λ Λ µ µ µ o : µ

14 4 Theo Ue 4 Alto of ltle fto rrer grh 4 Itrodto Theore d lo e ed lg rdo wl wth ltle fto rrer o et of the teger Eg rdo wl o hlf le wth rtll refletg rrer e led g le etwor wth oe f d oe ed Ω I the et eto we wor ot other lto of or theor 4: Rdo wl o the teger wth two f We oder drete rdo wl o the teger of the --r te r wth f d I f we he rolte r r d f we he rolte r r We dere th rdo wl grh wth three ooet: the frt oe fte terl where the tte re ered fro to : wth The eod ooet hlf le trtg tte re ered o th ooet th orreod wth or orgl tte e The lt ooet hlf le trtg tte re ered o th hlf le whh orreod wth - the teger O the terl d the frt hlf le we he reter o the lt hlf le we e reter to get the dered rdo wl Ω r r Ω Fgre 6 Rdo wl o the teger wth two f Frt we hdle wth We e Theore whe trtg Solg the two eto we fd: Ug orollre d : Port o Porto Porto Ω Porto Ω f : Porto f f : Porto f

15 Drete rdo wl wth rrer 5 The eod rt of Theore ge: f f f f the lt for le re dted o ehlf of dfferet erg of the tte e d reter o the egte teger et we e Theore whe trtg : orgl tte e We td the e roeed log the e le d fd: ow we ot the rolt of rrl f If d the: f Th the forl grh lgge for the orgl tte e we eed to hge - ow we roeed wth the e We e Theore whe trtg olg the two eto we fd:

16 6 Theo Ue Porto Porto et we e Theore whe trtg If d the: orgl tte e We get: f Th the forl grh lgge for the orgl tte e we he to hge - Bee of the ft tht ot oth d e le th we he: Ζ 5 Colo Ug geertg fto we oted eeted er of rrl orto rolte d eeted te efore orto for etr drete rdo wl o loll fte grh the reee of ltle fto rrer eto Elt olto were oted for oreted le grh fte d fte tr grh eto 3 We lo got relt the feld of oedeol rdo wl: o the teger wth two rrer eto 4 o terl wth three rrer d dfferet rolte etwee the rrer eto 33 d o the teger wth oe rrer d dfferet rolte left d rght fro the rrer eto 34

17 Drete rdo wl wth rrer 7 Aed A A Proof of theore Ce : or The rdo wl etwee d d d d the rdo wl o hlf le h e dered the dfferee eto: r wth hrtert eto: r Bee of or we he: wth The rdo wl etwee d d d the rdo wl o h e dered the dfferee eto: r δ wth olto : 4 4 r r 3 We frt loo t the terl e B fog o tte d etwee d d d we get: r r 4 Ug d 4 we ere : d d 5 Proeedg log the e le t ow etwee d d ge: 4 r 6 4 r 7 We ow oder hlf le wth h: Ug wth d r we get: 8 After oe llto we ot o hlfle h:

18 8 Theo Ue 4 4 r r 9 We ow fo o d t eghor: Frt we hdle the terl Whe d the e : Stttg the forle we fod for d 5 we get: d: Both forle re ld for t we eed the lt oe wth terhgg d ge: Whe g 36 d 7: 3 Whe d: 4 For the hlf le wth g 8: 5 Hlf le wth g 9: 6 We re ow red for the fl rt For d we he g d 5:

19 Drete rdo wl wth rrer 9 I the terl e we get ddtol ter whe e 3 d whe d e 4 Whe we get the hlf le e ddtol ter e 6 We get the relt of theore tg d otg tht: the If Ce : d Ug the e ethod e t ow wth d ted of d we fd o terl : the : If d the : If d where O hlf le we he: the : If d wth the : If d where For we he: We get the e wer g d l Hotl rle e Whe trtg or d o the terl d we get the e relt wth ddtol ter whe d whe d Strtg o h we get ddtol ter

20 Theo Ue Aed B B Proof of Theore The rdo wl etwee d d d the rdo wl o h e dered the dfferee eto: r 7 Frt we d the terl rt: Iterl e A olto of 7 ge : 8 Ug 8 wth d we ere d d Ug tht ereo we get: We e the lt forl whe o we he to terhge d the forl: Iterl e Followg the e ethod e we get: The e forle re fod lg l Hotl rle twe the terl e et we d hlf le For hlf le wth we get: Fll we e:

21 Drete rdo wl wth rrer Referee Feller W 968 A Itrodto to rolt theor d t lto thrd edto Vol Joh Wle ew or Weel B 96 The rdo wl etwee refletg d org rrer A th Sttt Leher G 963 Oe-deol rdo wl wth rtll refletg rrer A th Stt Gt H C 966 Rdo wl the reee of ltle fto rrer Jor th S D S Khdlr S d Se K 976 A odfed rdo wl the reee of rtll refletg rrer J Al Pro Per O E 985 Phe trto oe-deol rdo wl wth rtll refletg odre Ad Al Pro El-Shehwe A Aorto rolte for rdo wl etwee two rtll org odre I J Ph A: th Ge Bro R d C D 5 Rdo wl o grh: de tehe d relt J Ph A: th Ge 38 R45-R78 9 Drh B Joo T d Wheter J 6 Rdo wl o o J Ph A: th Ge