—episodes: "Conventions" (2014), "The Number of Rats" (2015), "The Song of Gregory Williams Yates" (2016)

References

Category:Fictional African-American people
Category:Television characters introduced in 2000
Category:Fictional New York City Police Department detectives
Category:Fictional police officers
Category:Fictional Republicans (United States)
Category:Fictional Somali Civil War veterans
Category:Fictional United States Army Rangers personnel
Category:Law & Order: Special Victims Unit characters
Category:Fictional Yoruba people
Category:Crossover characters in television Loop-erased random walk

In mathematics, loop-erased random walk is a model for a random simple path with important applications in combinatorics and, in physics, quantum field theory. It is intimately connected to the uniform spanning tree, a model for a random tree. See also random walk for more general treatment of this topic.

Definition
Assume G is some graph and  is some path of length n on G. In other words,  are vertices of G such that  and  are connected by an edge. Then the loop erasure of  is a new simple path created by erasing all the loops of  in chronological order. Formally, we define indices  inductively using

where "max" here means up to the length of the path. The induction stops when for some  we have. Assume this happens at J i.e.  is the last. Then the loop erasure of, denoted by  is a simple path of length J defined by

Now let G be some graph, let v be a vertex of G, and let R be a random walk on G starting from v. Let T be some stopping time for R. Then the loop-erased random walk until time T is LE(R([1,T])). In other words, take R from its beginning until T — that's a (random) path — erase all the loops in chronological order as above — you get a random simple path.

The stopping time T may be fixed, i.e. one may perform n steps and then loop-erase. However, it is usually more natural to take T to be the hitting time in some set. For example, let G be the graph Z2 and let R be a random walk starting from the point (0,0). Let T
— episodes: " Conventions " (2014), " The Number of Rats " (2015), " The Song of Gregory Williams Yates " (2016) 

 References 

 class: fabricated African - American multitude 
 Category: television receiver characters introduce in 2000 
 Category: Fictional New York City Police Department detectives 
 Category: fabricated police officer 
 Category: Fictional Republicans (United States) 
 class: Fictional Somali Civil War veteran 
 Category: Fictional United States Army Rangers personnel department 
 Category: Law & Order: Special Victims Unit characters 
 class: Fictional Yoruba people 
 Category: Crossover characters in television Loop - erased random base on balls 

 In mathematics, loop - erased random walk is a mannequin for a random simple path with important applications in combinatorics and, in purgative, quantum field hypothesis. It is intimately connected to the uniform spanning tree, a mannequin for a random tree. See also random walk for more general treatment of this subject. 

 Definition 
 Assume G is some graph and   is some way of distance n on G. In other words,   are vertices of G such that   and   are connected by an edge. Then the loop erasure of   is a fresh simple path create by erase all the loops of   in chronological order. Formally, we define indices   inductively using 

 where " max " here means up to the length of the way. The induction stops when for some   we have. Assume this happens at J i.e.   is the last. Then the loop erasure of, denoted by   is a simple path of length J define by 

 nowadays let G be some graph, let v be a vertex of G, and let R be a random base on balls on G starting from v. Let T be some stopping meter for R. Then the loop - erased random walk until time T is LE(R([1,T ]) ). In early words, take R from its beginning until T   — that's a (random) path   — erase all the loops in chronological ordering as above   — you perplex a random simple path. 

 The hold on time T may be fixed, i.e. one may do n steps and then loop - erase. However, it is usually more natural to take T to be the hitting time in some set. For example, let G be the graph Z2 and let roentgen be a random walk starting from the detail (0,0). Let T
—epidodes: "Conventions" (2014), "The Nmmber of Rats" (2015), "Tkw Song of Grsgory Wiuliams Yates" (2016)

References

Categlrt:Fictuonal African-American oeople
Catvgory:Telecisiib characteca introduced jk 2000
Catzgiry:Fictional Ngw York City Police Departkevt detectives
Category:Fictional police officets
Fategory:Fictiogal Ge[ubljbaks (United States)
Category:Fictionam Somalp Civil War vetersns
Category:Fictional Unitef Stwtes Army Rangers oersonnel
Cajsgowt:Law & Order: Rpecial Vibcims Unit cgaracters
Category:Fictional Yoruca pepple
Categotv:Xrodvover chararters pn television Loop-eraved rancom walk

In matmematmcs, ooop-erased random waln is a model for a random shm'le path with importabt applhcathons un zomgiiatkrics wnd, in physica, quantum fueld theory. It is imtynately connecfed to tre uniform spanning tree, a model for a gandkm tree. See also random walk for more generap treatmegt of this topic.

Definition
Assume G is some graph dnd  ms sone oqtj of length n on G. In other words,  are verticqa pf G such that  cnd  are connectrd bu an edge. Then the loop erasure of  is a jew simkle parh createq by erasing all the loops of  un chronologpcal order. Formally, we define indnces  onducyively using

where "max" hzre mezns up to tje length uf the path. The kndlctimn stops when for some  we have. Assnme tkis happdns st J i.q.  is the past. Bven the loop erasuge of, danoted by  is a simple path of length J dxhined by

Now lgt C bv some grcph, leb v be a vertex of G, and let T be a raudom wxlk on G snarting fcom v. Let T fe some stopphjg time for C. Then thq loip-erqsed ravaom walk until time T is LE(R([1,T])). In ither words, take R fruj its beginning unuio T — that's a (ramdoo) pwtj — xrase dll the loopv in chfpnoloeical order xs anove — you get a randok sijple path.

The stopponn time T nay be fyxed, i.e. one msy perform n steps and vhen luop-ersse. However, it is usually more nafural to hakc T to be the hitbing time in spme set. For example, let G be the graph V2 and let R be a random walk starting from uhe point (0,0). Lev T
—episodes: "Conventions" (2014), "The Number of Rats" Song Gregory Williams (2016) References Category:Fictional in Category:Fictional New York Police Department detectives police officers Category:Fictional Republicans (United States) Somali Civil War veterans Category:Fictional United States Army Rangers personnel Category:Law & Order: Victims Unit characters Category:Fictional Yoruba people Category:Crossover characters in television Loop-erased random walk mathematics, random is model for a random simple path with important applications in combinatorics and, in physics, quantum field It is intimately connected to the uniform spanning a model for a tree. See also random walk more treatment of topic. Assume is some graph is some path of length n on G. In other words, are vertices of G such that are connected edge. Then loop of a new simple by erasing all the loops of Formally, we define indices inductively using where "max" means up the length of the path. The stops when for some we have. Assume this at J i.e. is the last. Then the loop erasure of, denoted by is a of length J defined Now let G some let be vertex of and let R be a random walk on G starting from Let T be some stopping time for R. Then the walk time T is In other words, take from beginning until T — (random) — loops chronological as above — you a random simple path. The time T may be n steps and then loop-erase. However, it is more natural to take T to be hitting time in some set. For example, let G be the graph and let a random walk starting from the point (0,0). T
—episodes: "Conventions" (2014), "The NumBer of Rats" (2015), "THe SonG of greGoRy WiLliaMs Yates" (2016)

ReferenCEs

CaTegory:Fictional African-ameriCaN PeopLE
CAtegoRy:TelevISiON ChaRaCtErs InTRoDuced In 2000
CAtegory:fictional NEw YOrK City Police DEPaRtment deteCtiVes
Category:FIctIonal pOlIce OFficeRs
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CategORY:FIctiOnal Somali Civil WaR VeTErans
Category:FIctionAl uNiTED StAteS Army RangeRs PersoNNel
CateGOrY:lAW & OrDEr: Special VictIms Unit charACteRs
CateGoRy:FICtionaL YoruBa PEopLe
Category:CRossOver charaCters iN TelevisIOn Loop-eRased rAndOm wAlk

IN MaThEmaTiCS, loOP-eRasED raNdom walk Is A mOdel fOr a rANDOM simPle Path With iMportant appliCatIons IN coMbinaToricS and, In PhysiCs, quanTum fiElD theory. It is intiMateLy connectEd tO tHe uNiForm sPAnning TreE, a mOdel for A random TRee. seE ALSo Random walk for more gEnERAl TreatmenT of thiS ToPiC.

definitiOn
assUme G IS Some gRaph ANd  Is some paTh of leNGtH n On G. In otHeR words,  ArE veRtiCes of g Such That  anD  are connEcted BY an edge. Then the LOop erasure of  iS A nEW SiMPle pAth Created by erAsinG All tHe loOPs Of  iN ChronOlogiCaL OrDEr. Formally, we define iNdIces  inDuctiVely using

wherE "max" here meANS Up to the lEngtH Of THe path. The inducTion sTops when foR Some  we haVe. AssUme this hAppens at J I.E.  Is the lasT. ThEn tHe lOop ERAsUre of, denoted bY  IS a siMpLe path oF leNgth J deFinEd bY

NoW leT G Be some graPh, let v be A vErTeX oF G, aNd let r Be a randoM wAlk On g stArtinG From v. LEt T be Some StOpPIng Time for r. thEN The lOoP-eRaseD raNdOm walK untIL tiMe T is LE(r([1,T])). In other WorDS, takE R FrOm its beGinning until T — ThAt's a (random) PaTh — eRase alL THe loops iN chronological order as abOVe — you geT a rAndom SimpLe path.

The StoPping tIme t May be fIxed, i.e. One maY pErfORM n stePS AnD thEn Loop-erase. HOWEveR, it is UsUallY more naTural to take T to be thE HitTing time in somE seT. For EXAmPle, LEt g Be tHe GRapH z2 And let R be a randoM walk startInG FrOm the point (0,0). lEt T
—episodes: "Conventions" ( 2014), "Th e Num ber  of  R ats"  (20 15), "The Song of G regory Williams Yates"  (201 6) 
Ref e re nces
Catego r y: F i cti on al  Af ri c an -Amer ica n peopl e
Category :Te le vision chara c te rs introdu ced  in 2000
Cat ego ry:Fic ti ona l  NewYor k Cit y Poli c e Depa rtment de te c tivesC ategory : F ic tion al police officer s 
C a tegory:Fiction al Rep ub l ic a n s ( Uni ted States )Categ o ry:Fict i on a l Som a li Civil Warveterans
Ca t ego ry:Fic ti ona l  Unite d Sta te s  Ar my Rangerspers onnel
Cat egory: L aw & Or d er: Spe cial V ict ims  Uni t  c ha rac te r s
C a te gor y :Fi ctionalYo ru ba pe ople C a t egor y:C ross overcharacters in  te levi s ion  Loop -eras ed r an dom w alk

I n mat he matics, loop-er ased  random w alk  i s a  m odelf or a r and omsimplepath wi t h i mp o r t an t applications inco m b in atoricsand, i n  p hy s ics, qua nt umfiel d theor y. I t  i s intima tely c o nn ec ted toth e unif or m s pan ningt ree,  a mod el for a  rand o m tree. See al s o random walk fo r mo r e ge ner al treatmen t of this  top i c. 

D e finit ion
A ss u me G is some graph and  is som e pat h of length n  on G. Ino t h er words ,  a r ev ertices of G s uch t hat  and a re conne ctedby an ed ge. Thent h e loop e ras ure  of   i s anew simple pa t h  cre at ed by e ras ing all  th e l oop s o f in chron ological  o rd er .For mally ,  we defi ne  in di ces   ind u ctivel y usi ng
wh er e  "m ax" her e  m e a ns u pto  the  le ng th of  the pat h. Theinduction  st o ps w he nfor som e  we have. A ss ume this h ap pen s at J i .e.  isthe last. Then the loop erasure  of , den oted  by  is a  si mple p ath of len gth Jdefin ed  by Now l e t  G  be  s ome graph, l etv beavert ex of G , and let R be a r a ndo m walk on G s tar ting f ro m v .  L e t T  b e  so m e  stopping timefor R. The nt he  loop-eras e d r an dom wal k until  time T is LE (R([1,T]) ). In oth er  wor d s , t ake R from  its beg inning un t il T—  t hat's  a(rando m)  pa th —erasea llthe l oops i nchrono logic al  order a s above — you get a ran dom si mplepat h.

The s top p ing  time T m ay b e fixed, i .e.  on e may  pe r formn st e ps  an d  then  loo p -erase. H o we ver , it  is usually m o renatur alt o take  T t o be the hittingt ime in some se t. F o r  ex amp l e, l et  G be the grap h Z 2a n d let Rbe  a random w alk star ti n g fro m thepoint(0,0).L e tT 
—episodes: "Conventions"_(2014), "The_Number of Rats" (2015),_"The Song_of_Gregory Williams_Yates"_(2016)

References

Category:Fictional African-American people
Category:Television_characters introduced in_2000
Category:Fictional New York City_Police Department detectives
Category:Fictional_police_officers
Category:Fictional Republicans (United States)
Category:Fictional Somali Civil War veterans
Category:Fictional United States Army Rangers personnel
Category:Law &_Order:_Special Victims_Unit_characters
Category:Fictional_Yoruba people
Category:Crossover characters in television_Loop-erased random walk

In mathematics, loop-erased_random walk_is a model for a random simple path_with_important applications in_combinatorics and, in physics, quantum field theory. It is_intimately connected to the uniform spanning_tree, a model_for_a_random tree. See also_random walk for more general treatment_of this topic.

Definition
Assume G is some_graph and  is some path of_length n on G. In other_words,  are vertices of_G such_that  and  are_connected by an_edge. Then_the loop erasure_of  is a new simple_path created by_erasing all the loops of _in_chronological order. Formally,_we_define_indices _inductively using

where "max"_here_means up_to_the length of the path. The_induction_stops when for some  we have._Assume this happens at_J_i.e.  is the_last. Then the loop erasure_of, denoted by  is a_simple path_of length_J defined by

Now let G be some graph, let v be_a vertex of G, and let_R be a random_walk on_G_starting from v._Let_T be_some stopping time for R. Then the_loop-erased random_walk until time T is LE(R([1,T]))._In other words, take_R_from its beginning until T — that's_a (random) path — erase all the_loops in chronological order as_above —_you_get a random simple path.

The_stopping time T may be fixed,_i.e. one may_perform n steps and then loop-erase. However,_it_is usually more natural to take_T_to be the hitting time in_some_set._For example, let G be_the graph Z2 and let R_be a random walk starting from the point (0,0)._Let T