Random turn walk on a half line with creation of particles at the origin

Random turn w alk on a half line with crea ti on of pa rticles at the o rigin 1 J.W . v an de Leur † 2 and A. Y u. Orlo v ⋆ 3 † Mathematic al Institu te,Univer sity of Utr echt , P .O. Box 80010, 3508 T A Utr echt, The Nethe rlands ⋆ Nonlinea r W ave Pr oce sses Labo ratory , Oceanolo gy Ins titute, 36 Nakhimovski i Pr ospect Moscow 117851 , Russia Abstract W e consid er a ve rsion of r andom motion of hard core pa rticles on the s emi-lattice 1 , 2 , 3 , . . . , wher e in each time instant one of three possible eve nts occurs, viz., (a) a randomly chose n particle hops to a free neig hboring sit e, (b) a particle is created at th e origin (namely , at sit e 1 ) provid ed th at site 1 is free and (c) a particle is eli minated at the orig in (p rovided that the si te 1 is occup ied). Relation s to the BKP equatio n are expla ined. Namely , the tau functions of two diff erent BKP hierarchies provide genera ting func tions respecti vely (I) for transitio n weight s be tween differ ent pa rticle configuratio ns and (II) for an impor tant obj ect: a normalization function which plays the role of the statis tical sum for our non -equilibriu m sy stem. As an e xample we s tudy a model where the h opping rate depen ds on two paramete rs ( r a nd β ). For time T → ∞ we obtain th e asymptotic co nfiguration o f particles ob tained from the initial empty state (the state without particles) and find a n an alog o f th e first o rder t ransition at β = 1 . 1 Introduction In the famous paper [1] M.Fisher introduced models of o ne-dimensional random walk o f hard core particles on the lattice. W e shall consider a specific version of the models t hat Fisher called random t urn walk models. They describe a motion of parti cles where at each tick of the clock a random ly chosen walker takes a random step. In both types of models each site may be occupied by only one walker a t the same time. In our case we consider a v ersion of this model where particles move along a semi-l ine, and where also a particle may be created at the origin. The model . Consider a set of n odes labelled by positive integers, where each nearest neigh- bors are li nked by a pair of opposit e arrows. Let us view the nod es 2 , 3 , 4 , . . . as sit uated to the rig ht of the origin, the node 1 . An a rrow which starts at a node i and ends at a node j 1 This work has been partially su pported b y (1) the Europ ean Union through the FP6 Marie Curie R TN ENIGMA (Contr act number MR TN-CT -2 004-5 652) and th e Europ ean Scien ce Foundation Pr ogram MISGAM and by ( 2) the Russian Academy o f Scien ce pr ogram “Fund amental Meth ods in Nonlinear Dy namics”, RFBR grant No 05-0 1-004 98, and joint RFBR -Con sortium E.I.N.S.T .E.IN gran t No 06-01-9 2054 KE-a 2 vdleur@math .uu.n l 3 orlovs@wa ve.sio.rssi.ru 1 1 2 3 . . . exp(-U) 1 exp(-U) 2 exp(-U) 3 Figure 1 : Configuratio n for T = 1 in case λ ( T =0) is th e emp ty configuratio n. Black ball serves for a particle, white balls for free sites (where j = i ± 1 ) is assign ed a weight equal to e − U j + U i . Here U = ( U 1 , U 2 , . . . ) is a set o f real numbers. As an ini tial stat e of our dynamical sys tem re lated to a tim e T = 0 , we place a certain number of hard core particles at nodes (”hard-core” means that each node may be occupied by at most one particle), this initial configuration will be denoted by λ (0) . Consider a random motion of the hard-core parti cles where in each time instant one of three possible e vents occ urs: (a) a ra nd omly chos en par ti cle hop s alo ng an y of arrows attached to the corresponding nod e (i.e. either to the left or to the right) provided t hat t ar get node is free of particles; (b) A particle is created at th e origin (nam ely , at site 1 ) pro vided th is site is free; (c) a particle is elim inated at the origin (provided that site 1 is occupied). W e shall refer t o configurations as neighboring ones if they dif fer by an elementary e vent from the list abov e. For instance, the emp ty configuratio n is an neighborin g one to the configuration shown on figure 1, because the last is obtained by the ev ent (b) from the configuration without particles. The probability o f each element ary (i.e. which occurs in one tim e instant) event is p ropor- tional to a weight (or , a rate ) of the e vent which we assign as follows: (a) For the hop o f a parti cle along an arro w i → j (where j = i ± 1 ) the weight is giv en by e − U j + U i . The weig ht r ( j ) := e − U j + U i , j = 2 , 3 , . . . will be called (right ) hopping rate; (b) For the birth process the weight is 1 √ 2 e − U 1 (this weight will be called birth rate r (1)) ; (c) For the elimination process the weight is 1 √ 2 e U 1 . Along the random process in each eac h t ime step, say , T → T + 1 a configuration λ ( T ) goes to a neighboring configuration λ ( T +1) with the following probabi lity: p λ ( T ) → λ ( T + 1) = W λ ( T ) → λ ( T + 1) P µ W λ ( T ) → µ , (1) where W λ ( T ) → µ is the weight of the elementary e vent which creates (in one time step) a config- uration µ from the configuratio n λ ( T ) , the sum in th e d enominator ranges over all neighboring configurations. For instance, in case the initial configuration is the em pty configuration, th en by rules (a)-(c) with unit probability we obtain the configuration depicted in figure 1. Each set of configuration s λ (0) , λ (1) , . . . , λ ( T ) where each pair λ ( i ) , λ ( i +1) is a pair of neig h- boring configurations wi ll be referred to as a path of duratio n T which starts from the configu- ration λ (0) and ends at the configuration λ ( T ) . The w eight of t he pat h is defined as the product of the weight s of all elementary ev ents along the pat h, W λ (0) → λ (1) W λ (1) → λ (2) · · · W λ ( T − 1) → λ ( T ) . The transition weight betwee n configurations λ (0) and λ ( T ) is defined as the sum of weight s of all paths of duration T starting from λ (0) ending at λ ( T ) and will be denoted by W λ (0) → λ ( T ) ( T ) . 2 Then, i t is ea sy to see that th e probability to come from a configuration λ (0) to a given configuration λ ( T ) in T steps is the ratio p λ (0) → λ ( T ) = W λ (0) → λ ( T ) ( T ) Z λ (0) ( T ) (2) where the denominator is a normalization function Z λ (0) ( T ) = X µ W λ (0) → µ ( T ) (3) In what follo ws we shall omit superscripts for configurations. W e shall not nee d intermedi - ate configurations which were introduced for better explanation of our model. Our letter is arranged as follows. I n an introductory part t o section 2 we introduce our tools: neut ral fermions and the related F ock space, and quadratic operators (11)-(12) which depends on a g iv en set U = ( U 1 , U 2 , . . . ) . In subsection 2.1 for arbitrary set U we shall obtain an explicit expression for the probability to achieve a given configurati on in T steps in case the ini tial configuratio n is the empty state (th e state with out particles), see fo rmulae (27), (34) and (35). For an arbit rary chosen external potenti al U it is i mpossibl e to obtain an asymptot ic limit in formulae (34) and (35) in the lar ge time limit. In subsection (2.2) we specify rate r ( n ) n = 1 , 2 , . . . by form ula (38 ), no w t he rate depends on two parameters: a constant denoted by r and an exponent β . In th is case we present t he asym ptotic formulae for the density of particles, σ ( n ) , see (42) and (44). W e shall show that for a gi ven and large enough T all characteristics of the asymptoti c configuration of parti cles: its size, the number of in volved particles, the center m ass e t.c. undergo a jum p at β = 1 (when T → ∞ ). W e shall show that t he normalization functi on Z λ (0) =0 ( T ) = Z ( r , β , T ) also has a j ump. In our problem this function plays a role quite si milar to the role of the partiti on function in statistical physics; in this sense in our m odel we m ay t reat t his ju mp at β = 1 as a first order phase transiti on. The expression for probability in dif ferent re gions of the parameter β is giv en by (55) and (56). Let us note that if we fix β = 1 , then, our model turns out to be a discrete time version of the model called asymmetric si mple e xclusi on process (ASEP), now the constant r plays a role of asymmetry parameter . In two parts of sub section 2.3 we shall lin k respectiv ely trans ition weight s (in it em I) and the normalization function (see item II) with tau fun ctions of two different BKP , in the item II the key rol e is the relation (64) which relates rates r ( n ) to the BKP higher times. Let us notice that a wid e us age of th e free fermio n approach t o random partit ions and certain random processes was presented by Andrei Okounko v in a series of papers, in particular see [2]. Our approach is di f ferent and based o n (14). It is also different from the approach in vented by H.Spohn and K-H.Gwa in [3] for the study of (the cont inues tim e) ASEP where s pin syst em was used for the coding of particle configura t ions. In this approach the state spin up codes the filled stated, spin down the empt y state. V ia Jordan-W igner t ranform this spin sys tem may be related to fermioni c on e. Howe ver in that case a quadric fermi onic Hamiltoi nian is used to describe the sto chastic dynamics of particles which was identified with Hamilto nian dynamics of quantum spi n (or , of nonli near fermion ic) s ystem where t he (real) wav e function yields t he probability distribution for the st ochastic process. Follo wing [4] we use free fermio ns and the 3 answer for the probabi lity is gi ven b y t he ratio of two factors (35) quite similar t o wh at we ha ve in therm odynamics where t he probabil ity of a s tate is th e ratio of the weight of the state and a normalization function (the partition function). The main part of this work was reported by one of t he authors (J.W .L) on the workshop ”Random and int egrable models in m athematics and physics” in Brussel, Septemb er 11-15, 2007. 2 F ock vectors and configurations of hard cor e particle s Neutral fermions . I n what foll ows we shall n eed neutral fermions, { φ n } n ∈ Z , as introduced by E. Date, M. Jimbo, M. Kashiwara and T . Miwa in [5], defined by the follo wi ng property: [ φ n , φ m ] + = ( − 1) n δ n, − m , (4) where [ , ] + denotes the anticommutat or . In particular , ( φ 0 ) 2 = 1 2 . Next we define an action on vac uum st ates by φ n | 0 i = 0 , h 0 | φ − n = 0 , n < 0 , φ 0 | 0 i = 1 √ 2 | 0 i , h 0 | φ 0 = 1 √ 2 h 0 | , (5) Note that the d efinition of these ”Fock spaces” is dif ferent fr om the usual one which was intro- duced in [5]. W e foll ow [6], wh ere the action of φ 0 (which is not a number) i s diffe rent. See Appendix for some more details. The basis of the corresponding right and left Fock spaces are formed by vectors | λ i := φ λ 1 · · · φ λ N | 0 i (6) and by h λ | := ( − 1) | λ | h 0 | φ − λ N · · · φ − λ 1 , (7) where λ 1 > · · · > λ N > 0 (8) and | λ | = λ 1 + · · · + λ N . W e have one to one corr espondence between configurations of hard core particles on the lattice 1 , 2 , 3 , . . . and th e basis Fock vectors. These configuration s are als o called Maya diagrams. Namely , the M aya di agram of the vector | λ i is a set of vertices 1 , 2 , 3 , . . . , where each vertex numbered by λ i is drawn as t he black ball (a hard core p article), all oth er vertices are white balls. As we see, by (4)-(5) we hav e h λ | µ i = δ λ,µ (9) Consider the following o perator B = B 1 ( U ) + B − 1 ( U ) (10) 4 where U = ( U 1 , U 2 , . . . ) , is a semi-infinite set of numbers (which may be also considered as var iabl es), and where B 1 ( U ) = ∞ X i> 0 ( − 1) i +1 φ i φ 1 − i e − U i + U i − 1 = φ 1 φ 0 e − U 1 − φ 2 φ − 1 e − U 2 + U 1 + φ 3 φ − 2 e − U 3 + U 2 − · · · (11) and B − 1 ( U ) = ∞ X i ≥ 0 ( − 1) i +1 φ i φ − 1 − i e − U i + U i +1 = − φ 0 φ − 1 e U 1 + φ 1 φ − 2 e − U 1 + U 2 − φ 2 φ − 3 e − U 2 + U 3 − · · · (12) It is straightforward to check the relations B − 1 B 1 − B 1 B − 1 = 1 2 , B − 1 | 0 i = 0 = h 0 | B 1 , (13) which we shall need soon for the calculation of (21). At last let us note that for d iffe rent purpose the o perators B 1 and B − 1 were u sed in [7] (namely , to construct examples of multiv ariable hypergeometric functions which are also multisol iton B KP tau functi ons [8]). 2.1 Stochastic system: description via fermions The birth-death of particles at the o rigin and their diffusi ve motio n may be described as fol lows. A sequence of Fock vec to rs | λ ′ i → ( B 1 ( U ) + B − 1 ( U )) | λ ′ i → · · · → ( B 1 ( U ) + B − 1 ( U )) T | λ ′ i → · · · (14) describes an e volution of the i nitial (basis) Fock vector | λ ′ i - where the variable T = 0 , 1 , 2 , . . . plays a role of discrete time - t o linear com binations of different basis F ock vectors. Du e to the correspondence bet ween basis Fock vectors and configurations of h ard core particles, this e volution may be interpreted as the random process described in section 1, wh ere each time step is numbered by T . The details will be presented below . Let us notice that each basis Fock v ector | λ i := φ λ 1 · · · φ λ N | 0 i (15) where λ is a strict partiti on which is λ = ( λ 1 , λ 2 , . . . , λ N ) , t hat is in one-to-one correspondence with a configuration of parti cles, located in the nodes with numb ers λ 1 > · · · > λ N > 0 . This is the reason, why we hav e chos en (5) as d efinition of the Fock space and not the one of [5]. For the F ock space introduced in that article this is not the case. W e are interested in the discrete-tim e version of t his random p rocess which is giv en by (14), where each time step is numbered by T . As we see via (4) and (5), the first term in the right hand side of (11) describes the creation of a hard-core particle l ocated at the node number 1 (the origin) (provided that this node i s free). Since φ 0 | 0 i = 1 √ 2 | 0 i we assign the weight 1 √ 2 e − U 1 to this creation p rocess. Each other 5 term of B 1 describes a hop to the right. Sim ilarly , the first t erm of B − 1 in th e righ t hand side of (12) describes the elim ination of a particle located at the node 1 (provided th at this node is occupied). Since again φ 0 | 0 i = 1 √ 2 | 0 i we assign the weight 1 √ 2 e U 1 to eac h elimi nation proce ss . Other terms of B − 1 describes the hops to the left (backward motion i n the direction of t he origin). It is not difficult to see that W λ ′ → λ ( U ; T ) := h λ | ( B 1 ( U ) + B − 1 ( U )) T | λ ′ i (16) is a sum of weights of paths o ver all paths of duration T starting at the configuration λ ′ and ending at t he configuration λ , and, therefore, yields t he transition weigh t of the T -step random process which describes a transit ion from an initial configuration of the hard-core particles described by coordi nates λ ′ 1 , . . . , λ ′ N ′ to a target configuration λ 1 , . . . , λ N defined in the In- troduction. As the nu mber of particles does not have to be conserved along the process, N is not necessarily equal to N ′ . Let u s mark that for eac h path from a configuration λ ′ to a configuration λ of a duration T we have N − N ′ = n + − n − (17) T = j + + j − + n + + n − (18) | λ | − | λ ′ | = j + − j − + n + − n − (19) where j + is the n umber of hop s to t he left during the ti me interval T , j − is the n umber of hop s to the r ig ht, n + is the number of creations of a particle and n − is the number of eliminations of particles at the node 1 . Let us no tice t hat from (18) and (19) it fol lows t hat T and | λ | − | λ ′ | ha ve the same parity . Consider the case λ ′ = 0 and look for the weight of the process which transports the initi al empty state (there are no particles at all) to a giv en configuration λ in T steps W 0 → λ ( U ; T ) := h λ | ( B 1 ( U ) + B − 1 ( U )) T | 0 i (20) In order to e valuate the right hand side we need the following formul ae e z ( B − 1 + B 1 ) = e z 2 4 e z B 1 e z B − 1 , e z B − 1 | 0 i = | 0 i , (21) (the first equation follows from the Baker -Campbell-Hausd orf f formula and fr om (13)), and also the formula h λ | e z B 1 ( U ) | 0 i = 2 − N 2 z | λ | e − P N i =1 U λ i N Y i =1 1 λ i ! N Y i ··· >λ N > 0 h λ | ( B − 1 + B 1 ) T | 0 i (28) = T ! ∞ X N =0 2 − N 2 X λ 1 > ··· >λ N > 0 T −| λ | ev en 2 | λ |− T Γ( T −| λ | 2 + 1 ) N Y i =1 e − U λ i λ i ! N Y i> U n (and, t herefore, left ho pping rates are much larger than right hopping rates), then th e configurations where T = | λ | are dominant in the sum for normalization function. Let us note that up without the factor 2 − N 2 e − P N i =1 U λ i the number W 0 → λ ( T ) is equal to t he number of s hifted st andard tableau of shape λ , see [16], that is the number of ways t he Y oung diagram of the strict partition λ may be cr eated by adding box by box to the empty partit ion in a way that on each step we ha ve the diagram of a strict partition. Finally , one can get rid of the restriction λ 1 > · · · > λ N in the s ummation (29) rewriting it as a sum over all non-negativ e inte gers λ 1 , · · · , λ N : Z ( U ; T ) = T ! ∞ X N =0 2 − N 2 N ! X λ 1 , ··· ,λ N > 0 T −| λ | ev en 2 | λ |− T Γ( T −| λ | 2 + 1 ) N Y i =1 e − U λ i λ i ! N Y i 1 d escribes a locking potential while in β < 1 case the potent ial try to drive particles to the right from the orig in. The case β = 1 may be considered as a dis crete tim e version o f the so-called asym metric simple exclusion process (ASEP) on t he half-line, now , the parameter r being an asymmetry parameter . Let notice that formally the point n = n ∗ = r 1 1 − β is a point of an extremum of t he potential U n where the left and the right hopping rates are equal. It means we want to find the configuration λ = λ ( T ) where for giv en T the weight is as follows W 0 → λ ( r , β ; T ) = T ! 2 | λ |− T  T −| λ | 2  ! 2 − N 2 e − E λ ( r ,β ) (39) here T − | λ | i s e ven, and the”electrostatic ener gy“ o f the configuration is E λ ( r , β ) = − log r | λ | N Y i =1 1 ( λ i !) β N Y i> T ∗ . In t he lar ge T limit as we see from (44) R = R ( β , T ) =          q 8 T β if 0 < β < 1 q 8 T 1+ r − 2 if β = 1 2 (2 r 2 T ) 1 2 β if 1 < β < 2 (45) Thus, when 0 ≤ β < 1 the size of the asym ptotic configuration is proport ional to √ T , wh ile in th e vi cinity of β = 2 we hav e the forth root behavior . As we see t he discont inuity appears at β = 1 in the l ar ge T limit. The same behavior has t he numb er of particl es in the asy mptotic configuration which is proportional to the size of the configuration: N ( r, β , T ) = R Z 1 0  β π arccos u  du = β R ( β , T ) π (46) The weight of the asymptotic configuration (43) for large e no ugh T is | λ ( T ) | = β R 2 8 ≈      T + O  T β  if 0 < β < 1 T 1+ r − 2 if β = 1 β 2 (2 r 2 T ) 1 β if 1 < β < 2 (47) 10 R 0 1/2 Figure 2: The ev olu tion of the empty configurati on in T → ∞ lim it for β = 1 . As ymptotic density of particles. Notice that the center mass of the configuration also has a jump at β = 1 : | λ ( T ) | N ( r, β , T ) ≈          π q T 8 β if 0 < β < 1 π q T 8(1+ r − 2 ) if β = 1 π 4 (2 r 2 T ) 1 2 β if 1 < β < 2 (48) For lar ge enough T , th e number m ( T ) = j − ( T ) + n − ( T ) depends on region as follows m ( T ) = 1 2 ( T − | λ | ) = 1 2 T − β R 2 16 =      O  T β  if 0 < β < 1 T 2(1+ r 2 ) if β = 1 T 2 − β 4 (2 r 2 T ) 1 β if 1 < β < 2 (49) In (47) and (49) we keep terms which we shall use in e valuations below . For lar ge enough T , th e electrostatic ener gy (40) of λ ( T ) is E λ ( T ) ( r , β ) = −| λ ( T ) | ln r +  β R 2 Z 1 0 σ ( u ) ( u ln( uR ) − u ) d u  − 1 2 R 2 Z 1 0 Z 1 0 σ ( u ) σ ( u ′ )ln u − u ′ u + u ′ dudu ′ (50) = | λ ( T ) | ( β ln R − ln r ) − R 2 β 2 A where β 2 A = − β Z 1 0 σ ( u ) ( u ln u − u ) du + Z 1 0 du ′ Z u 0 σ ( u ) σ ( u ′ )ln u − u ′ u + u ′ du = β 2  1 16 + ln 2 8  (51) see Appendix A.4 which yields A = 1 16 + ln 2 8 . By (45) and (47) we obtain 11 E λ ( T ) ( r , β ) ≈      β 2 T ln T − T  β 2 ln β 2 + β 2 + ln r  if 0 < β < 1 1 2 T ln T 1+ r − 2 + T 1+ r − 2  1 2 ln 8 1+ r − 2 − 8 A − ln r  if β = 1 β 4 (2 r 2 T ) 1 β (ln T + (1 + 2 β )ln2 − 16 β A ) if 1 < β < 2 (52) Using Stirling’ s approxi mation to ev aluate m ( T )! , for T large, we obtain the weigh t of the process as follows W 0 → λ ( T ) ( r , β ; T ) = T ! 2 − 2 m ( T ) m ( T )! 2 − N ( T ) 2 e − E λ ( T ) ( r ,β ) (53) =          T ! e − β 2 T ln T + T ( β 2 ln β 2 + β 2 +ln r ) + O ( √ T ) if 0 < β < 1 T ! e − T 2 ln T + T { ln 2(1+ r 2 ) 2(1+ r 2 ) + ln 2(1+ r − 2 ) 2(1+ r − 2 ) + 2ln 2 − 1+ln r 2 2(1+ r − 2 ) + 1 − ln 4 2(1+ r 2 ) } + O ( √ T ) if β = 1 W 0 → 0 ( T ) e − β 4 ( 2 r 2 T ) 1 β (1 − β )+ O „ T 1 2 β « if 1 < β < 2 (54) Then, as we see the n ormalization function Z ( T ) which according to the saddle point method has the same leading term in the lar ge T limit as W 0 → λ ( T ) ( r , β ; T ) has a discontuinity at β = 1 which may be interpreted as a s ort of th e first k ind phase transition in our non-equil ibrium system. Now we can e valuate t he t ype of asymp totic o f t he probabili ty to achie ve a given configu- ration in T → ∞ steps . W e have Z ( T ) = W 0 → λ ( T ) ( T ) e O (ln R ) , where t he last factor originates from the Gaussian int egral around the saddle poin t configuration λ ( T ) . Then for T ≫ | λ | we hav e p 0 → λ ( r , β , T ) ≈ W 0 → λ ( r , β , T ) W 0 → λ ( T ) ( r , β , T ) ≈ T | λ | 2 e − | λ | 2 2 | λ | e − E λ ( r ,β ) e ω ( T ,r,β ) (55) where ω does not depend on λ . (For the enum erator of ( 55 ) we used (3 1) a nd Stirling’ s approx- imation for (30).) The answer depends on the region of β e ω ( T , r,β ) =            e β − 1 2 T ln T − T ( β 2 ln β 2 + β 2 +ln r ) ) + T 2 ln(2 e )+ O ( √ T ) if 0 < β < 1 e − T  ln 2(1+ r 2 ) 2 + b 2 ( 1+ r − 2 ) ff + O ( √ T ) if β = 1 e β 4 ( 2 r 2 T ) 1 β (1 − β )+ O „ T 1 2 β « if 1 < β < 2 (56) where b = 2ln 2 − 1 ≈ 0 . 4 . As we see, in ea ch case, in the large T li mit e ω is v anis hing. At last let us note the following. As we see in case of a decreasing potential (or , the sam e, a increasing righ tward hopping rate), 0 < β < 1 , t he weight of the a sy mptotic configuration is equal to T which m eans that th e asym ptotic configuration is created b y only creating ev ents at th e origi n and rightward hops, there w ere n o eli mination events and backward hops in the history of this configuration, j − = n − = 0 . For β < 0 solution does not exists. In β → +0 limit the number of particles (46) vanishes, while | λ ( T ) | is equal to T . Indeed, the external potenti al U n is dec reasing so rapidly that the lar gest weight has the o ne particle configuration where t he particle moves in the ballis tic way: it is located at the distance T to the origin . When β > 2 we h a ve a locking pot ential which 12 forces particles to form a sort of a condensate near the origi n wh ere all sites are occupied. The size of the condensed phase i s defined by β . This problem is treated by a method s imilar to the suggested in [21]. Along this way we can sho w that the solution is giv en in terms of elliptic integrals of the first and thi rd kind. The problem will be cons idered in detail in the n ext version of this paper , where we will show that the normalization function Z ( T ) has singularity as function of the parameter β at β = 2 . 2.3 Relation to the BKP tau function The link of the described stochastic system to integrable equations is two-f ol d. (I) A BKP tau function as generating function for transition weights W λ ′ → λ ( T ) . Let us consider the following vacuum expectation v alue τ ( s, ¯ s ; U, z ) := h 0 | e H ( s ) e z B e ¯ H (¯ s ) | 0 i (57) where B is given by (10) and H ( s ) = 1 2 X n =1 , 3 , 5 ,... X m ∈ Z s n ( − 1) m +1 φ m φ − n − m , ¯ H ( ¯ s ) = 1 2 X n =1 , 3 , 5 ,... X m ∈ Z ¯ s n ( − 1) m +1 φ m φ n − m (58) with z , U = ( U 1 , U 2 , . . . ) and set s s = ( s 1 , s 3 , s 5 . . . ) and ¯ s = ( ¯ s 1 , ¯ s 3 , ¯ s 5 , . . . ) parameters. Function τ ( s, ¯ s ; U, z ) depends on U = ( U 1 , U 2 , . . . ) as the op erator B depends on th ese param- eters. The hierarchy of of Kadomts e v-Petviash vili equations of t ype B (the BKP hierarchy) was introduced in [5]. As we ha ve already mentioned we use its m odification suggested in [6]. These is a semi-in finite set of c om patible nonlinear diff erential e quati ons which m ay be viewed as a set of commutative time flows. It is com mon to enum erate the BKP equations (flows) by odd numbers. Expression (57) where U may be chosen as an arbitrary s et of numbers, provides an example of the BKP tau functio n constructed in [6], where s = ( s 1 , s 3 , s 5 . . . ) is the set of the so- called higher tim es (the ti mes of commutative flo ws related to d iff erent equations of the BKP hierarchy). Actually the set ¯ s = ( ¯ s 1 , ¯ s 3 , ¯ s 5 , . . . ) is also related to a (second) BKP hierarchy of equations, which is com patible with the first one; thus, (57) is an example of the t au fun ction of a coupled BKP hierarchy . Using results of [9] and doing si milar calculations in the frame work of BKP hiera rchy con- structed in [6] (see Appendix) one can show that the op erator e H ( s ) applied t o the left vacuum generates all basis l eft F ock ve ctors, quite similar e ¯ H ( ¯ s ) applied to the right vacuum vector gen- erates all basi s right Fock ve ctors. Namely , we ha ve the following left and right coherent states (see also (6) and (7)) h 0 | e H ( s ) = ∞ X N =0 2 − N 2 X λ 1 > ··· >λ N > 0 Q λ  s 2  h λ | , (59) e ¯ H ( s ) | 0 i = ∞ X N =0 2 − N 2 X λ 1 > ··· >λ N > 0 Q λ  s 2  | λ i (60) 13 where Q λ  s 2  , λ = ( λ 1 , λ 2 , . . . ) , are known as projectiv e Schur functions, see [16] 4 , which, in an appropriate space, form a complete set o f weighted po lynomials in the variables s 1 , s 3 , . . . . Using these formu lae we obtain that the BKP t au function (57) i s the g enerating function for (16), namely τ ( s, ¯ s ; U, z ) = ∞ X T =0 ∞ X N ,N ′ =0 2 − N 2 − N ′ 2 X λ 1 > ··· >λ N > 0 λ ′ 1 > ··· >λ ′ N ′ > 0 z T T ! Q λ  s 2  Q λ ′  ¯ s 2  W λ ′ → λ ( U ; T ) (61) What o ne obt ains in case of a general BKP tau function will be e xplain ed in a more detailed forthcoming paper . (II) A tau function of dual BKP as the generating funct io n f or the normalization fu nc- tion Z ( T ) Let us introduce the generating function for the normalization function (28) as follows Z ( U ; z ) := ∞ X N =0 X λ 1 > ··· >λ N > 0 h λ | e z B | 0 i = ∞ X T =0 z T T ! Z ( U ; T ) (62) By (21) and (22) we hav e Z ( U ; z ) = e z 2 4 ∞ X N =0 2 − N 2 X λ 1 > ··· >λ N > 0 z | λ | e − P N i =1 U λ i N Y i =1 1 λ i ! N Y i 0 N Y i =1 e λ i ln z − U λ i λ i ! N Y i 2 describes a mod el where a condensate of p articles (the region of full package) fills a region near the origin. W e shall explain the appearance of a phase transition at β = 2 which is rather similar to the transition studied in [21]. The other model where the injection r ate is a free parameter will be considered where an analog of the phase transition wi ll be presented, then it may be interesting to discus s lin ks wit h [15]. It i s interesting to understand links with results of [20] and wit h t he approach of [18]. It may be also interesting to link our results with t he results of a recent paper [24] where in the context of the quasiclassi cal l imit of T oda lattice hierarchy of inte grable equations the V ersh ik-Ker ov limit shape for random parti tions was reproduced. 15 Acknowledge ments W e are thankful to Joh n Harnad for kind hosp itality and numerous fruitful discuss ions wh ich allowed to create this paper which may be vi e wed as a cont inuation of [4] and [11]. W e t hank Marco Bertoll a and other participants of the working semi nar on Integrable Systems, Random Matrices and Random Processes in Concordia u niv ersity headed by J. Harnad for interesting joyful discussions. W e thank Anton Zabrodin for a discussion related to the methods presented in [22]. A A ppendic es A.1 A remarks o n BKP hierar chies [6] and [5] and r elated vacuum ex- pectation values Let us note that dif ferent va cuum s tates were used in the constructi ons of BKP hierarchy in versions [6] and [5]. If we deno te the left and right vacuum s tates used in [5] respectively by ′ h 0 | a nd | 0 i ′ then h 0 | = 1 √ 2 ′ h 0 | + ′ h 0 | φ 0 , | 0 i = 1 √ 2 | 0 i ′ + φ 0 | 0 i ′ (67) Introduce also ′ h 1 | = √ 2 ′ h 0 | φ 0 , | 1 i ′ = √ 2 φ 0 | 0 i ′ , (68) then ′ h 0 || 0 i ′ = ′ h 1 || 1 i ′ = 1 and instead of (5) we ha ve φ n | 0 i ′ = φ n | 1 i ′ = 0 , ′ h 0 | φ − n = ′ h 1 | φ − n = 0 , n < 0 (69) see [5] for details. Correspondingly Fock spaces used [6] and [5] are different. From the representational point of vie w this definition is s omewhat more con venient, since each Fock module remains irreducible for the algebra B ∞ which is the underlyi ng al gebra for KP equations of type B (BKP), see [6]. The vac uu m states ′ h 0 | and | 0 i ′ are mo re familiar objects in ph ysics. In particular any vac- uum expectation va lu e of an odd num ber of fermions vanishes, while, for instance, h 0 | φ 0 | 0 i = 1 √ 2 . Let F be a prod uct of e ven number of fermions. Then it is easy to see that h 0 | F | 0 i = ′ h 0 | F | 0 i ′ (70) Let us note that all vacuum expectation values (v . e.v .) used in the present paper (say , tau functions (57) and (65)) are sums of v . e.v . of mon omials containing e ven number of fermions. In the context of applications to random process we can cons truct a model of random mo tion on the semi-infinit e lattice based on the Fock space used i n [5], howe ver we find that t he Fock space used in [6] is much more natural to our point of view . 16 A.2 A remark on f ormulae contain ing Q λ functions Here we shall show th at formulae found in [9] are of use in our case (see also [8]). From [9] it is known t hat ′ h 0 | e H ( s ) φ λ 1 φ λ 2 · · · φ λ N | 0 i ′ = ( 2 − N 2 Q ( λ 1 ,λ 2 ,...,λ N ) ( s 2 ) for N e ven , 0 for N odd , √ 2 ′ h 0 | φ 0 e H ( s ) φ λ 1 φ λ 2 · · · φ λ N | 0 i ′ = ( 2 − N 2 Q ( λ 1 ,λ 2 ,...,λ N ) ( s 2 ) for N odd , 0 for N e ven , (71) where Q λ  s 2  , λ = ( λ 1 , λ 2 , . . . ) are the projecti ve Schur functions, see [16]. T hus h 0 | e H ( s ) φ λ 1 φ λ 2 · · · φ λ N | 0 i =  1 √ 2 ′ h 0 | + ′ h 0 | φ 0  e H ( s ) φ λ 1 φ λ 2 · · · φ λ N  1 √ 2 | 0 i ′ + φ 0 | 0 i ′  = 1 2 ′ h 0 | e H ( s ) φ λ 1 φ λ 2 · · · φ λ N | 0 i ′ + ′ h 0 | φ 0 e H ( s ) φ λ 1 φ λ 2 · · · φ λ N φ 0 | 0 i ′ + + 1 √ 2 ′ h 0 | φ 0 e H ( s ) φ λ 1 φ λ 2 · · · φ λ N | 0 i ′ + 1 √ 2 ′ h 0 | e H ( s ) φ λ 1 φ λ 2 · · · φ λ N φ 0 | 0 i ′ = 1 2 ′ h 0 | e H ( s ) φ λ 1 φ λ 2 · · · φ λ N | 0 i ′ + 1 2 ′ h 1 | e H ( s ) φ λ 1 φ λ 2 · · · φ λ N | 1 i ′ + + 1 √ 2 ′ h 0 | φ 0 e H ( s ) φ λ 1 φ λ 2 · · · φ λ N | 0 i ′ + 1 √ 2 ′ h 1 | φ 0 e H ( s ) φ λ 1 φ λ 2 · · · φ λ N | 1 i ′ = 2 − N 2 Q ( λ 1 ,λ 2 ,...,λ N )  s 2  , (72) since the role of ′ h 0 | a nd ′ h 1 | (resp. | 0 i ′ and | 1 i ′ ) is interchangeable. A.3 Case 0 < β < 2 : V ershik-K ero v type as ymptotic W e want to solve th e following singu lar integral equation where r is a giv en constant: ln r λ β + P Z R 0 σ ( xR − 1 ) dx λ − x − P Z R 0 σ ( xR − 1 ) dx λ + x + 1 2 ln2  T − Z R 0 xσ ( xR − 1 ) dx  = 0 (73) where λ ∈ [0 , R ] and where σ is d efined on th e interval x ∈ [0 , R ] . Here and belo w P serv es to denote the principal va lu e of integrals. Remark A.1 . Let us notice that th e secon d term in (73) describes the repulsio n of char ges distrib uted with densi ty σ a long (0 , R ) , whi le the third term may be interp reted as th e attraction of these charg es to their imag e in the mirror . Then, it i s natu ral to cont inues σ to the interv al ( − R, 0) such that σ ( − x ) : = − σ ( x ) (this describes the replacing of particles (with coordina te, say , x ) b y ho les (situated in − x ). P Z R 0 σ ( xR − 1 ) dx λ − x − P Z R 0 σ ( xR − 1 ) dx λ + x = P Z R − R σ ( xR − 1 ) dx λ − x 17 Notice th at in this case in gen eral w e get a j ump of σ in x = 0 . For our futur e purpose (of in- ver ting Hilbert type transforms ) we prefer to deal with continues functions. For this purpose w e shall modify σ in the reg ion ( − R, 0) by adding a constan t equal to this jump, ∆ , getting cont inues modified σ (compe nsating this cha nge of σ b y adding logarithmic term to the right hand side), see belo w Now , σ ( x ) is a function defined on the whole interv al [ − R, R ] via relation σ ( − x ) = ∆ − σ ( x ) , x ∈ [ − R, R ] (74) (As we shall see l ater ∆ = β if 0 < β < 2 , and ∆ = 2 if β > 2 ). Then , we re-write (73) in a more con venient form as P Z R − R σ ( xR − 1 ) dx λ − x = ln( R + λ ) ∆ + ln λ β − ∆ − ln( C r ) , λ ∈ [ − R, R ] (75) where we denoted C 2 := 2  T − Z R 0 xσ ( xR − 1 ) dx  (76) C is an independent of λ constant to be defined later . Singular integral equation (75) may be solved by standard method, see [22]. In case σ ( xR − 1 ) is continues and bounded on [ − R, R ] the solution is gi ven by formulae σ ( xR − 1 ) = − π − 2 P Z R − R √ x 2 − R 2 √ λ 2 − R 2 ln ( λ + R ) ∆ λ δ C r dλ x − λ , x ∈ [ − R, R ] , (77) ( δ := β − ∆ ) see formula (42 .26) in [22]. Let us ev aluate the i ntegral in an e xpli cit w ay . W e consider √ λ 2 − R 2 in the i ntegrand of (77) as singl e-v alued function wi th the cut [ − R, R ] whose upper limit on [ − R, R ] is positive and lowe r limit on [ − R, R ] is negati ve. Also w e shall consider ln( λ + R ) ∆ in th e in tegrand as the single valued function d efined on the wh ole complex p lane with the cut [ −∞ , − R ] . The cut of ln λ δ will be viewed o n the ray ( −∞ , 0) , a little bit above the real ax e. Then we ha ve P Z R − R √ x 2 − R 2 √ λ 2 − R 2 ln ( λ + R ) ∆ λ δ C r dλ x − λ = − 1 2 I C √ x 2 − R 2 √ λ 2 − R 2 ln ( λ + R ) ∆ λ δ C r dλ x − λ (78) where the contour C i s going crossing the poi nts 0 and 2 R : C = − C + + C − . One can inflate th e contour through the point 2 R as there are no ’bad’ sing ularities there. W e have a cut ( −∞ , 0) caused by the logarithm. Inflating the contour t o the in finity we see that the o nly contribution will be caused by t he cut of the logarit hm (we come to the contour which st arts on minus i nfinity going a l ittle bit upper the real line, than turning at the origin and going b ack to minus i nfinity a little bit under the real line. (The contribution of the circle embracing infinity v anishes as the integrand’ s asymptoti c is dλ λ 2 .) Thi s yields the integral along the cut P Z R − R √ x 2 − R 2 √ λ 2 − R 2 ln ( λ + R ) ∆ λ δ C r dλ x − λ = (79) − 1 2 Z 0 − R √ x 2 − R 2 √ λ 2 − R 2 (2 π δ √ − 1) dλ x − λ − 1 2 Z 0 −∞ p y 2 − 2 Ry √ h 2 − 2 Rh (2 π ∆ √ − 1) dh y − h (80) 18 where h = λ + R , y = x + R and 2 π ∆ √ − 1 and 2 π ∆ √ − 1 are the jumps of the logarithm . The first integral is [23] Z 0 − R √ x 2 − R 2 √ λ 2 − R 2 (2 π δ √ − 1) dλ x − λ = arcsin − R 2 x As we see taking small x this integral is not a real number; this results in the condition δ = 0 , or , the same ∆ = β For the sec on d integral of (80) we ha ve (see [23]): Z + ∞ 0 1 √ h 2 + 2 Rh dh y + h = 1 p − y 2 + 2 Ry arccos  y R − 1  (81) Inserting the last formula into (80) and then into (77) we finally obtain σ ( xR − 1 ) = β π arccos x R , x ∈ [0 , R ] , 0 ≤ β ≤ 2 (82) W e add the last un-equality to provide (37), 0 ≤ σ ≤ 1 . At last let us mark that one may sho w that if we relate the asymptotic configuration (82) to a strict partition, then, the shape of the Y oung diagram of the dou ble o f this partiti on wi ll coincide with the so-called V ershik-Ke rov asymptotic shape [12], [13]. A.4 T wo useful integrals One can show that 1 π Z 1 0 arccos u ( u ln u − u ) du = − 1 8 − ln 2 8 (83) 1 π 2 Z 1 0 du Z u 0 arccos u arccos u ′ ln u − u ′ u + u ′ du ′ = − 1 16 (84) which yields A = 1 16 + ln 2 8 . A.5 The rel at ion between f ormula (65) and (63) Recal from (66) t hat φ ( z ) = P i ∈ Z z i φ i . Using (4) and (5), we can also calculate t he va cuum expectation v alue for two of these fields: h 0 | φ ( z 1 ) φ ( z 2 ) | 0 i = X m> 0 ( − z 2 /z 1 ) m + 1 2 = 1 2 z 1 − z 2 z 1 + z 2 , (85) 19 (where we assume that | z 1 | > | z 2 | ). Using W ick’ s Theorem we obtain that h 0 | φ ( z 1 ) φ ( z 2 ) · · · φ ( z 2 m ) | 0 i =  1 2  m Y 1 ≤ i 0 h 0 | e H ( t ) φ ( x 1 ) φ ( x 2 ) · · · φ ( x 2 m ) | 0 i m Y i =1 sig n ( x 2 i − 1 − x 2 i ) = X all x i > 0 2 m Y k =1 b ( t, x k ) Y 1 ≤ i 0 2 m Y k =1 b ( t, x k ) Y 1 ≤ i 0 h 0 | e H ( t ) φ ( x 1 ) φ ( x 2 ) · · · φ ( x 2 m ) φ ( x 2 m +1 ) φ 0 | 0 i m Y i =1 sig n ( x 2 i − 1 − x 2 i ) = X all x i > 0 2 m +1 Y k =1 b ( t, x k ) Y 1 ≤ i 0 2 m +1 Y k =1 b ( t, x k ) Y 1 ≤ i ℓ . So if π − 1 =  1 2 · · · ℓ − 1 ℓ ℓ + 1 · · · 2 m 2 m + 1 j 1 j 2 · · · j ℓ − 1 j ℓ j ℓ +1 · · · j 2 m ℓ  , then ν =  1 2 · · · ℓ − 1 ℓ ℓ + 1 ℓ + 2 · · · 2 m 2 m + 1 j 1 j 2 · · · j ℓ − 1 ℓ j ℓ j ℓ +1 · · · j 2 m − 1 j 2 m  21 and ρ =  1 2 · · · ℓ − 1 ℓ ℓ + 1 · · · 2 m − 1 2 m i 1 i 2 · · · i ℓ − 1 i ℓ i ℓ +1 · · · i 2 m − 1 i 2 m  , i k = ( j k if j k < ℓ, j k − 1 if j k > ℓ, For instance for ℓ = 3 and m = 2 o ne has π − 1 =  1 2 3 4 5 5 1 4 2 3  → ν =  1 2 3 4 5 5 1 3 4 2  → ρ =  1 2 3 4 4 1 3 2  Hence, if we develop (65) as a po wer series in z and k eep the term e z 2 4 . Then substitut e (90) and (92) and use (64), we obtain (63). References [1] M. Fisher , “W alks , walls, wetting and melting”, J . Stat. Phys. 34 (1984) 667-729 [2] A. Ok oun kov , “R andom Matrices and Random Permutations”, arXiv:math-990317 6 ; A. Okounkov , “I nfinit e wedg e and random p artitions” arXiv:mat h-9907127 ; A. Ok- ounkov , N. R esheti khin, “Correlation function of Schur process with application to l o- cal g eometry of a random 3-dim ensional Y oung diagram”, arXiv:math/0107056 ; A. Okounkov , N. Reshetikhin, “Random ske w plane partitions and the Pearce y process”, arXiv:math/0503 508 [3] K-H. Gwa and H. S poh n, “Bethe solution for the dynamical-scaling exponent of the noisy Bur gers equatio n”, Phys. Rev . A 46 , 844-854 (1992) [4] J. H arnad and A.Y u. Orlov , “Fermionic construction of tau functions and random pro- cesses”, Physica D 235 (2007) pp. 168-206 [5] E. Date, M. Jimbo, M. Kashiwara a nd T . Miwa, “T ransformation groups for soliton equa- tions, IV A ne w hierarchy of solito n equations of KP-type”, Physica 4D (1982) 343-365 [6] V . Kac and J. van de Leur , “The Geom etry of Spinors and the M ulticomponent BKP and DKP Hierarchies”, CRM Proceedings and Lecture Notes 14 (1998) 159-202 [7] A.Y u. Orlov , “Hypergeometric functions related to Schur Q-po lynomials and BKP equation”, Theoretical and Mathematical Physics, 137 (2): 1573-1588 (2003); arXiv:math-ph/030 2011 [8] J.J.C. Nimmo and A.Y u. Orlo v , “ A relationship between rational and multi-soliton so- lutions of the BKP hierarchy”, Glasgow Mathemati cal Journal, 4 7 A (2005) 149-168; arXiv:nlin/04 05009 [9] Y . Y ou , “Polynomial solutions of the BKP hierarchy and projectiv e representations of symmetric groups”, in Infini te-dimensional Lie algebr as and gr oups , pp . 449-464, Adv . Ser . M ath. Phys., 7 . W orld Science Publis hing, T eaneck, Ne w Jersey 22 [10] J.J.C. Nim mo, “Hall-Litt lew ood symm etric fun ctions and the BKP equation”. J. Phys ics A, 23 , 751-760 (1990) [11] H.Braden, J. Ha rnad, J.W . van de Leur and A.Y u. Orlov , “Fermionic construction of grand partition function for supersymmetric matrix models and BKP hierarchies” [12] A.M.V ershik and S.V .K erov , “ Asym ptotics of the Plancherel measure of the symmetric group and the limiting form of Y oung tableaux”, So viet Mathemati cs Dok lady 18 (19 77) 527-531 [13] A.M. V ershik and S.V . K erov , “ Asymp totics of the largest and the typi cal dimensio ns of the irreducible representations of a symmetric group”, Funktsi onal’nyi Anali z i Ego Prilozheniya, V ol. 19, No. 1, (1985) pp. 25-36 [14] T . Eguchi, K. Hori, S.-K. Y ang, “T opological σ -Models and Lar ge- N M atrix Int egral”, arXiv:hep-th/9503017 [15] G.M. Sch ¨ utz, “Critical phenomena and un iv ersal dynamics in one-dimensional driven diffusi ve sys tems with two species of particles”, cond-mat/0308450 [16] I.G. M acdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1995 [17] M.L. Mehta, Random Matrices , 2nd edition (Academic, San Diego, 1991). [18] P .J . Forrester , “Log Gases and Random Matrices”, book in preparation http://www .ms.unim elb .edu.au/˜ m atpjf.matpjf.ht ml [19] I.M. Lou tsenko and V . Spiridonov , “Solito n Solutio ns of Integrable Hi erarchies and Coulomb Plasmas”, J . Stat . Phys , 99 , 751, (2000) [20] C. T racy and H. W idom, “ A Limit Theorem for Shifted Schur Measure” math.PR/0210 255 . [21] M.Douglas and V .Kazakov , “Lar ge N phase transitions in continuu m QC D 2 ”, arXiv: hep-th/9305047 [22] F .D.Gakhov , “Boundary Problems”, ed. 2, Mosco w , Nauka 1977 [23] A.P .Prudni kov , Y u.A.Bychkov and O.I.Mariche v “Inte grals and Se ries”, v ol I, Mosco w , Nauka 1981 [24] A.Marshakov , “Seiberg-W itten Theory and Extended T oda Hierarchy”, arXi v: hep- th/0712.280 2 23