Distribution of a Particle s Position in the ASEP with the Alternating Initial Condition

Transcription

1 J Stat Phys 00 40: DOI 0007/s Dstrbuton of a Partcle s Poston n the ASEP wth the Alternatng Intal Condton Eunghyun Lee Receved: Aprl 00 / Accepted: June 00 / Publshed onlne: 9 July 00 The Authors 00 Ths artcle s publshed wth open access at Sprngerlnkcom Abstract In ths paper we gve the dstrbuton of the poston of a partcle n the asymmetrc smple excluson process ASEP wth the alternatng ntal condton That s, we fnd PX m t x X m t s the poston of the partcle at tme t whch was at m k, k Z at t 0 As n the ASEP wth step ntal condton, there arses a new combnatoral dentty for the alternatng ntal condton, ths dentty relates the ntegr of the ntegral formula for PX m t x to a determnantal form together wth an extra product Keywords ASEP TASEP Bethe Ansatz Introducton The excluson process s an nteractng stochastc partcle system on a countable set SA partcle at x S chooses y S wth probablty px,y after a holdng tme exponentally dstrbuted wth parameter If y s empty, the partcle at x jumps to y but f y s already occuped, then the partcle remans at x, the Posson clock resumes The detaled references on the constructon of the model are Lggett s books [8, 9] The asymmetrc smple excluson process ASEP s defned on S Z by takng px,x + p px,x q for all x Z, p + q If p, we call t the totally asymmetrc smple excluson process TASEP Schütz [] consdered the system of N partcles for the TASEP There Schütz obtaned the probablty that the system s n confguraton {x,,x N } at tme t gven the ntal confguraton {y,,y N } at t 0, expressed the probablty as an N N determnant, moreover, for general ASEP, obtaned the condtonal probablty for N, Johansson [7] studed the TASEP wth a specal ntal condton that one half of the system s occuped the other half of the system s empty at t 0, whch s called the step ntal condton Assumng the left half s occuped the rght half s empty at t 0, Ths work was supported n part by Natonal Scence Foundaton through the grant DMS E Lee Department of Mathematcs, Unversty of Calforna, Davs, CA 9566, USA e-mal: ehnlee@mathucdavsedu

2 636 E Lee Johansson derved the probablty that the N th partcle from the rghtmost moves at least M steps before tme t, whch descrbes the tme-ntegrated current By usng the technque used n [] Rákos Schütz [] obtaned the same result as Johansson [7]fortheTASEP wth step ntal condton Rákos Schütz s method orgnates from the Bethe Ansatz whle Johansson used a combnatoral argument Tracy Wdom extended the prevous results on the TASEP to the ASEP Theorem 5 Theorem 5 n [6] provde the probablty that the mth partcle from the leftmost s at x Z at tme t when the system ntally has fntely many partcles Corollary to Theorem 5 s a generalzaton of the Johansson s result the Rákos Schütz s result to the ASEP the subsequent Remark shows that how ther result leads to the Johansson s result or the Rákos Schütz s result The work for the determnstc step ntal condton n [6] was recently generalzed to step Bernoull ntal condton [8] that assumes that at t 0 each ste n Z + s occuped wth probablty ρ wth 0 <ρ ndependently of the others all other stes are empty In the ASEP wth step ntal condton that the rght half of the system s occuped the left half s empty at t 0, the rom varable x m t, themth partcle s poston from the leftmost at tme t can be related to the tme-ntegrated current Assumng p<q, PT x, t m Px m t x T x, t s the number of partcles whose postons are less than equal to x at tme t That the asymptotcs on the fluctuaton of x m t or T x, t are related to the GUE Tracy- Wdom dstrbuton n rom matrx theory that fluctuatons are n the regme of the t /3 scale are well known for varous stuatons [, 7,, 8] Besdes the step ntal condton, the TASEP wth the alternatng ntal condton also has been nvestgated Ths ntal condton assumes that all even stes are occuped all odd stes are empty at t 0 or vce-versa Whle the current fluctuaton of the TASEP more generally ASEP wth step ntal condton n the long tme lmt s governed by the GUE Tracy-Wdom dstrbuton [, 7,, 4, 8], on the other h, the current fluctuaton of the TASEP wth the alternatng ntal condton s related to the GOE Tracy-Wdom dstrbuton [3 6, 0, 5] The appearance of the GOE Tracy-Wdom dstrbuton n growth models goes back to [] by Bak Rans a breakthrough n the TASEP was led by Sasamoto [3] But to the best of the author s knowledge nothng s known on the ASEP wth the alternatng ntal condton although we expect the GOE statstcs n ths case In ths paper, we study the dstrbuton of a partcle s poston n the ASEP wth the alternatng ntal condton, whch gves nformaton on the current of the system The asymptotc behavor of the current remans as a problem for the future We denote by Y Z the set of ntal postons of partcles by P Y the probablty of the ASEP wth the ntal condton Y The man object we are nterested n s a rom varable X m t, the poston of a partcle at tme t whose ntal poston s m Y the man goal n ths paper s to obtan P Y X m t x when Y Z { : Z}Todo so we wll start wth a fnte set Y { : Z, N + N} wll consder the lmtng case that N LetY + {, 3,,N } Y {, 3,, N + } so that Y Y Y+ Let us denote the poston of the th partcle from the leftmost at tme t by x t Then, X m t x Y + m+ t, so we can use some prevous results on x t n the ASEP wth fnte Y In Sect we revew the ntegral formula for P Y x t x when Y s fnte n Sect 3 by usng a new combnatoral dentty that arses n the alternatng ntal condton we derve P Z X m t x

3 Dstrbuton of a Partcle s Poston n the ASEP wth the Alternatng 637 gve the one-sded verson of t The dentty s gven n Lemma our fnal results for P Z X m t x ts one-sded verson are gven n 35, respectvely Some Known Results The ntegral formulas for the probablty of a partcle s poston for the ASEP wth a fnte ntal condton Y were developed [6, 8, 9] These ntegral formulas have dfferent forms dependng on the contours we choose The ntegral formula of Theorem 5 n [6]whchs over large contours can be used to derve the formula for step ntal condton that postve ntegers are occuped as shown n Corollary Alternatvely, f all negatve ntegers are ntally occuped other stes are empty, the ntegral formula of Theorem 5 n [6] whchs over small contours s needed These are because we need geometrc seres whch arse to be convergent Hence, for the alternatng ntal condton that has nfntely many partcles on both sdes of any reference ste, a recently developed ntegral formula over both large small contours s requred [9] In ths secton we revew the formula developed n [9] whch s the startng pont of ths paper Frst, let p/q ξ ξk ξ,,ξ k+,ξ,,ξ k wth k k + + k We defne εξ : p ξ j ξ + qξ, fξ,ξ j : ξ p + qξ ξ j ξ Ix,ξ: <j fξ,ξ j ξ x eεξ t ξ Notce that Ix,ξ depends on t but we omt t n the notaton Gven two sets U V of ntegers σu,v: #{u, v : u U,v V u v} recall the defnton of the -bnomal coeffcent, [ N n ] N N N n+ n We assume that Y Z Y Y Y+ s a fnte set Y Y + are dsjont all members of Y + are greater than all members n Y ForS ± Y ± we set S ± k ± use postve ndces for S + negatve ndces for S In other words, we set S {s,s,,s k } S + {s,s,,s k+ } Then the dstrbuton of x m t,themth partcle s poston from the leftmost partcle at tme t s gven by P Y x m t x Ix,ξ ξ s k C ± 0 R k+ C k dξ r c m,s,s+ S ± Y ±, S ± k ± c m,s,s + m+ Y \S mm k +m+σs +,Y +σy,y \S m Y + k k + [ ] q kk k m Y \ S

4 638 E Lee C R C r s a crcle wth center zero radus R r Here R r s so large small that all the poles of <j fξ,ξ j n Ix,ξle nsde outsde C R C r The product n the ntegr s over all postve negatve ndces, ntegrals over C R are for varables of postve ndces ntegrals over C r are for varables of negatve ndces If we set Y, exactly becomes 3 n [9], from whch we are able to work on the postve one-sded step ntal condton or the postve one-sded alternatng ntal condton Addtonally, n Sect V of [9], a varant of c m,s,s + was computed It s the coeffcent c m+ Y,S,S + m+k mm σs+,y+\s+ σy \S,S + + k +k++ mk + [ ] q kk / k 3 m + k wll be used n the next secton 3 Alternatng Intal Condton on Z 3 Dstrbuton of a Partcle s Poston at Tme t In ths secton we derve P Z X m t x as the lmtng case N of the formula wth Y { N +, N + 3,,N 3, N } wth x Y + m+ t nstead of x m t Then S + S, subsets of Y + Y, respectvely, may be wrtten as S + {, +,, + + k+ },,, k+ N,k + N S { j +, j +j +,, j + +j k +}, j,,j k N,k N Snce we are workng wth x Y + m+ t X m t, the coeffcent for X m t becomes m+ c Y + m+ +k σs +,Y +\S + σy \S,S + m 8 + k +k++ m+ k+,s,s+ [ ] k q kk / m + k by replacng m by m+ n 3 Notcng that σs +,Y + \ S + σy \ S,S n c Y + m+,s,s+ ξ s n the ntegr n depend on S ±, we consder the sum S ± Y ±, S ± k ± σs +,Y +\S + σy \S,S S + Y +, S + k + σs +,Y +\S + >0 ξ s ξ s S Y, S k σy \S,S <0 ξ s

5 Dstrbuton of a Partcle s Poston n the ASEP wth the Alternatng 639 Let us compute the frst sum Frst, ξ s ξ + ξ + ξ k + + k + >0 ξ ξ k+ ξ ξ k+ ξ ξ k+ ξ k+ k + Now, observe that the number of ponts n Y + less than or equal to s l + + l s + + l the number of ponts n S + less than or equal to s l s l Hence, the number of pars s l,ywth s l y s l S + y Y + \ S + s + + l l, thus σs +,Y + \ S k+ k + k + + k k+ k +k + + Denotng the frst sum by ϕ + k +,ξ +, ϕ + k +,ξ + σs +,Y +\S + ξ s >0 S + Y +, S + k + ξ ξ k+ k +k++ S + Y +, S + k + k+ k + ξ ξ k+ ξ ξ k k + ξk + If we assume that Y + { :,,}, the sum mples geometrc seres, whch converge because we choose large contours for varables wth postve ndces as shown n Hence, ϕ + k +,ξ + ξ ξ k+ <0 k +k++,, k + k+ k + k + ξ ξ k+ ξ ξ k+ ξk + ξ ξ k+ ξ ξ k+ k+ ξ ξ k+ k+ ξ k + Lkewse for negatve ndces, ξ s ξ j ξ j +j ξ j + +j k k For σy \ S,S we set then t s easly seen that so ξ ξ k ξ ξ k j ξ ξ k j ξ j k k s l s l, S + S Ỹ + Y, σy \ S,S σ S +, Ỹ + \ S +

6 640 E Lee σ S +, Ỹ + \ S + j + j + j + +j + +j k k k j + k j + +j k k k + Recallng that we choose small contours for negatve ndces so that geometrc seres for negatve ndces converge, one can obtan for Y { : 0,, } σy \S,S ξ s <0 let S Y, S k k k + ξ ξ k j,,j k ξ ξ k j ξ ξ k j ξ k k k k k + ξ ξ 3 k ξ k k ξ ξ k k ξ ξ k ξ k, ϕ k,ξ : ξ ξ 3 k ξ k k ξ ξ k k ξ ξ k ξ k jk Hence we obtaned P Z X m t x c m,k± k ± 0 C R k+ C k r Ix,ξϕ k,ξ ϕ + k +,ξ + dξ 4 [ ] c m,k± m+ +k m 8 + kk+ k +k m+ k+ q kk k m 5 + k Remark In Corollary n [6] the dentty 7 n [6] was used n obtanng P Z +x m t x for step ntal condton We gve the P Z +x m t x to be compared wth the case of the alternatng ntal condton n the later secton k mk m+/ P Z +x m t x m + kk k! k m C R C R j [ ] k k m ξ j ξ p + qξ ξ j ξ ξ x etεξ ξ ξ dξ Moreover, the ntegr n P Z +x m t x could be expressed as a determnantal form by usng another dentty 3 n [7] It states that P Z +x m t x c m,k, detkξ,ξ j,j k dξ 7 k m C R C R 6 Kξ,ξ ξ x e εξt p + qξξ ξ 8

7 Dstrbuton of a Partcle s Poston n the ASEP wth the Alternatng 64 c m,k, s a constant dependng on m, k, Here we ntroduce the dentty for the later use Lemma [7] det p + qξ ξ j ξ,j k k pq kk q k j 3 Symmetrzaton Combnatoral Identty ξ j ξ p + qξ ξ j ξ ξ ξ As mentoned n Remark a combnatoral dentty was found to derve the ntegral formula of the dstrbuton n case of step ntal condton the ntegr of the formula can be expressed as a determnant Ths dentty s assocated wth a specal ntal structure of the system, that s, the step ntal condton So we may expect to have a new dentty assocated wth the alternatng ntal condton In ths subsecton we fnd the new combnatoral dentty obtan an alternate form of 4 by usng the dentty Lemma Let p q p + q For k N σ S k >j p + qξ σ ξ σj ξ σ ξ σj ξ σ ξ σ ξ σ ξ σk k ξ σ ξ σ3 ξ σk k ξ σk + kk / <j + ξ + ξ j ξ ξ j ξ Proof The equalty clearly holds for k Denote the left h sde by L k ξ,,ξ k the rght h sde by R k ξ,,ξ k, assume that the dentty holds for k, e, L k R k Letσ l We change the sum over all permutatons n L k to the double sum over l,,,k σ,,σk S k,thats, σ S k k l Observe that for all σ S k >j,,j,,k k ξ σ ξ σ ξ σk k ξ ξ ξ k k p + qξ σ ξ σj ξ σ ξ σj ξ σ p + qξ σ ξ σ ξ σ ξ σ ξ σ >j,,j,,k p + qξ σ ξ σj ξ σ ξ σj ξ σ σ,,σ k S k Ths dentty was conjectured by Crag A Tracy through prvate communcaton

8 64 E Lee l p + qξ ξ l ξ ξ l ξ >j,,j,,k p + qξ σ ξ σj ξ σ ξ σj ξ σ L k ξ,,ξ l,ξ l+,,ξ k σ,,σ k S k >j,,j,,k p + qξ σ ξ σj ξ σ ξ σj ξ σ ξ σ ξ σk k ξ σ3 ξ σk k ξ σk Hence L k ξ,,ξ k ξ ξ ξ k k ξ ξ ξ k k k l l k l l p + qξ ξ l ξ ξ l ξ L k ξ,,ξ l,ξ l+,,ξ k p + qξ ξ l ξ ξ l ξ R k ξ,,ξ l,ξ l+,,ξ k, the second equalty comes from the nducton hypothess Our goal s to show that ξ ξ ξ k k k l l p + qξ ξ l ξ ξ l ξ R k ξ,,ξ l,ξ l+,,ξ k R k ξ,,ξ k e, k l l p + qξ ξ l ξ R k ξ,,ξ l,ξ l+,,ξ k ξ ξ l ξ R k ξ,,ξ k ξ ξ k k Recallng the form of R k,whatwewanttoshows ξ ξ ξ k k + k k l ξ l l p + qξ ξ l ξ ξ l ξ ξ l ξ + ξ l ξ 9 For some techncal reasons we multply by p + qξl ξ l + ξ l both the numerator the denomnator of the rght h sde, then usng p + qξl ξ l qξ l pξ l, 9 becomes ξ ξ ξ k k + k k l p + qξ ξ l ξ ξ l ξ ξ l l ξ l ξ + ξ l ξ ξ l qξ l p 0 Observe that gz : l p + qξ l z ξ l z ξ l l zξ l z + z ξ l z qz p qk z l ξ l

9 Dstrbuton of a Partcle s Poston n the ASEP wth the Alternatng 643 for z Thus, the ntegral of gz over a crcle wth suffcently large radus R s equal to the ntegral of qk l ξ l over the crcle, so, z gzdz q k ξl Resgz C R l Recall that p/q p + q It s easy to see that Res z gz p k /q Res z p q gz p k /q Usngp + qξ l ξ l qξ l pξ l we obtan Res zξl gz ξl p + qξ ξ l ξ ξ l ξ ξ l l ξ + ξ l ξ Fnally, observng that + /q, wecanseethatres z+ ξl gz Res zξl gz, thus 9 s verfed Ths completes the proof as We use ths new dentty to symmetrze the ntegr n 4 Frst, we wrte the ntegr ξ x eεξ t ξ ϕ + k +,ξ + ϕ k,ξ Ix,ξϕ + k +,ξ + ϕ k,ξ fξ,ξ j <j [ fξ,ξ j fξ,ξ j <0,j>0 0<<j >0 [ fξ,ξ j ξ xeεξ t ] ϕ k,ξ ξ <j<0 <0 The frst bracket s a functon of varables wth postve ndces that s gven by 0<<j fξ,ξ j 0< j 0<j< k + >0 ξ j ξ p + qξ ξ j ξ ξ x eεξ t ξ ϕ + k +,ξ + k + >0 ξ x+ e εξ t ξ ξ xeεξ t ] ϕ + k +,ξ + ξ p + qξ ξ j ξ ξ j ξ ξ ξ ξ k + k+ ξ ξ 3 ξ k + k+ ξ k + ts symmetrzaton by the dentty n Lemma s c k+ 0< j ξ j ξ ξ x+ e εξ t + ξ + ξ j p + qξ ξ j ξ ξ >0 ξ ξ 0<<j ξ j c k+ k +! + k+k+ /

10 644 E Lee Now, s wrtten as ξ j ξ ξ x+ e εξ t ξ c k+ p + qξ ξ j ξ ξ ξ ξ 0< j 0<<j >0 + ξ + ξ j ξ ξ j c k+ det p + qξ ξ j ξ j c k+ det K + ξ,ξ j,j k +,j k + 0<<j >0 >0 ξ x+ e εξ t ξ ξ + ξ + ξ j ξ ξ j 0<<j c k+ c k+ k+ pq k+k+ / q k+ K + ξ, ξ ξ x e εξ t p + qξξ ξ ξ ξ ξ Lkewse, for the second bracket, let ξ ξ Then <j<0 ϕ k,ξ fξ,ξ j 0<j< 0<<j ξ j ξ p + q ξ ξ j ξ ξ j ξ q + p ξ ξ j ξ 0<j< 3 k ξ ξ ξ k + ξ + ξ j ξ ξ j ξ ξ j p ξ ξ j + q ξ j k ξ ξ k k ξ ξ k ξ k k k + k k + 4 k ξ ξ ξ k ξ ξ k k ξ ξ k k ξ ξ k ξ k k k + ξ ξ k ξ ξ k ξ k ξ k ξ k k Usng Lemma agan, the symmetrzaton of s 0<<j ξ j ξ q + p ξ ξ j ξ c k 0< j ξ j ξ q + p ξ ξ j ξ k k + ξ ξ k ξ ξ k k ξ ξ k k ξ k k >0 ξ ξ 0<<j + ξ + ξ j ξ ξ j

11 Dstrbuton of a Partcle s Poston n the ASEP wth the Alternatng 645 c k k k + k! +, k k / recallng that ξ ξ, we obtan the symmetrzaton of the second bracket c k 0> j ξ j ξ p + qξ ξ j ξ k <0 Usng Lemma, s expressed as ξ x e εξ t ξ ξ / j<<0 c k detk ξ,ξ j k,j j<<0 + / ξ + ξ j / ξ ξ j + ξ + ξ j ξ ξ j c k c k k pq k k / q k K ξ, ξ ξ x e εξ t p + qξξ ξ ξ ξ ξ Now, we summarze our result Recall the defnton of fξ,ξ j n Sect Thenwehave P Z X m t x c m,k± c k c k+ k ± 0 C k r C R k+ <0,j>0 fξ,ξ j G ξ G + ξ + dξ 3 G ξ det K ξ,ξ j k,j G + ξ + det K + ξ,ξ j,j k + 0<<j j<<0 + ξ + ξ ξ ξ j + ξ + ξ j ξ ξ j j, c m,k± c k+ c k m+ k+ k +!k! m 8 + k +k++ m+ k+ q kk+ [ ] k p k+k+ / q k k / m + k 33 The One-Sded Alternatng Intal Condton Let us assume the ntal condton Y {n k 0 : n N} for a fxed k 0 Z In ths case we start wth 3 n [9], P Y x m t x c m,k σs,y Ix,k,ξ k C ξ s dξ 4 R C R S Y, S k

12 646 E Lee [ ] k c m,k q kk / m mm / km m for convergence of geometrc seres By usng the same procedure as the subsecton the dentty n Lemma we can obtan P Y x m t x We gve the ntegral formula wthout the detaled dervaton P Y x m t x m k mk m+/ [ ] k + kk k! k m k m C R C R j <j + ξ + ξ j ξ ξ j ξ j ξ p + qξ ξ j ξ ξ x+k 0 e tεξ ξ ξ dξ 5 Ths s to be compared wth 6 wth step ntal condton 3 wth the two-sded alternatng ntal condton 5 has a product term n the ntegr compared wth 6 s descrbed by the ntegral only over large contours wth the ntegr G + ξ when k 0 compared wth 3 Remark One may consder more generalzed perodc ntal condton that stes n kz k N,k are occuped all other stes are empty The dscrete TASEP wth ths ntal condton was studed n [4] We confrmed by usng a computer that we do not have a combnatoral dentty n a smple form when k 3 That s, currently, we have two specal determnstc ntal condtons, that s, the step ntal condton the alternatng ntal condton, whch are assocated wth combnatoral denttes Acknowledgements The author s grateful to Crag A Tracy for suggestng ths problem nvaluable comments on ths work thanks anonymous referees for useful comments Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton Noncommercal Lcense whch permts any noncommercal use, dstrbuton, reproducton n any medum, provded the orgnal authors source are credted References Bak, J, Rans, EM: Lmtng dstrbutons for a polynuclear growth model wth external sources J Stat Phys 00, Arous, BG, Corwn, I: Current fluctuatons for TASEP: a proof of the Prähofer-Spohn conjecture arxv: Borodn, A, Ferrar, PL, Prähofer, M, Sasamoto, T: Fluctuaton propertes of the TASEP wth perodc ntal confguraton J Stat Phys 9, Borodn, A, Ferrar, PL, Prähofer, M: Fluctuatons n the dscrete TASEP wth perodc ntal confguratons the Ary process Int Math Res Papers , rpm 00 5 Borodn, A, Ferrar, PL, Sasamoto, T: Large tme asymptotcs of growth models on space-lke paths II: PNG parallel TASEP Commun Math Phys 83, Borodn, A, Ferrar, PL, Sasamoto, T: Transton between Ary Ary processes TASEP fluctuatons arxv:math-ph/ Johansson, K: Shape fluctuatons rom matrces Commun Math Phys 09, Lggett, TM: Interactng Partcle Systems Sprnger, Berln 985

13 Dstrbuton of a Partcle s Poston n the ASEP wth the Alternatng Lggett, TM: Stochastc Interactng Systems: Contact, Voter Excluson Processes Sprnger, Berln Prähofer, M, Spohn, H: Current fluctuatons for the totally asymmetrc smple excluson process arxv: cond-mat/0000v Rákos, A, Schütz, GM: Current dstrbuton rom matrx ensembles for an ntegrable asymmetrc fragmentaton process J Stat Phys 8, Schütz, GM: Exact soluton of the master equaton for the asymmetrc excluson process J Stat Phys 88, Sasamoto, T: Spatal correlatons of the D KPZ surface on a flat substrate J Phys A 38, L549 L Tracy, CA, Wdom, H: Level-spacng dstrbutons the Ary kernal Commun Math Phys 59, Tracy, CA, Wdom, H: On orthogonal symplectc matrx ensembles Commun Math Phys 77, Tracy, CA, Wdom, H: Integral formulas for the asymmetrc smple excluson process Commun Math Phys 79, Tracy, CA, Wdom, H: A Fredholm determnant representaton n ASEP J Stat Phys 3, Tracy, CA, Wdom, H: On ASEP wth step Bernoull ntal condton J Stat Phys 37, Tracy, CA, Wdom, H: Formulas for ASEP wth two-sded Bernoull ntal condton arxv: