Painleve Functions in Statistical Physics

P ainlev é F unctions in Statistical Ph ysics Craig A. T racy and Harold Widom Abstract W e review recent progres s in limit laws for the one-dimensional asymmetric simple exclusion pro cess (ASEP ) on the in teger lattice. The limit la ws are expressed in t er ms of a cer tain Painlev é II function. F urthermore, we take this o ppo rtunit y to give a brief survey of the app ear ance of P ainlevé functions in statistical ph ysics. Running Title: P ainlev é in Statistical Ph ysics MSC Num b ers: 34M55, 60K35, 82B23 Craig A. T racy Harold Widom Department of Mathematics Department of Mathematics Univ er sit y of California Univ er sit y of California Davis, CA 95 616, USA Santa Cruz, CA 95 064, USA email: tracy @math. ucdavis.edu email: widom@uc sc.ed u “It w as a pleasan t surprise to me that suc h s p ecial f unctions actually app eared in concrete problems of theoretica l ph ysics. . . ” Mikio S ato [4]. 1 In tro duction The app earance o f P ainlev é functions in the 2D Ising mo del is well-kno wn [37, 64]. Equally well - kno wn is that this pro vided one imp etus for M. Sato, T. Miw a and M. Jim b o [48] to develop their theory of holonomic quantum fields whic h connects the theory of isomondrom y preserving deformations of linear different ial equations with the n -p oint correlation functions of the 2D Is ing mo del. 1 The general consensus in the field of “exactly solv able mo dels” is that correlation functions are expressible in terms of P ainlev é f unctions only in mo dels that are fr e e fermion mo dels . More precisely , one exp ects that for the app earance of functions of the P ainlev é type, it is necessary for the underlying mo del or pro cess to b e a determinan tal pro ces s in the sense of Soshnik o v [52]. In addition to the 2D Is ing mo del, s ome notable examples where Painlev é f unctions arise in correlation functions include the one-dimensional imp enetrable Bose gas [21, 28, 33 , 34], the I s ing c hain in a tran v erse field [41], the distribution functions of random matrix theory [1, 5, 16, 22, 28, 56, 57 , 58], Hammersley’s gro wth pro cess [7, 8], corner and p olynucl ear grow th mo dels [9, 24, 29, 42 , 43] and the totally asymmetric simple exclusion pro cess (T ASEP) [12 , 29, 44]. Univ ersalit y theorems in random matrix theory ha v e extended the app earance of P ainlev é functions to a wide class of matrix ensem bles [13, 17, 18, 19, 51]. 2 In recent dev elopmen ts [3, 45, 46, 47 ] P ainlevé I I app ears in the long time asymptotics of ex plicit form ulas for the exact heigh t distribution for the KPZ equation [32] with narro w wed ge initial condition. As just noted, one do es not exp ect Pa inlevé functions to arise in correlatio n functions in models that are exactly solv able in the sense of Baxter [11] but are not free fermion models, e.g. 6-v ertex mo del, XXZ quan tum spin chain , Baxter’s 8-v ertex mo del. Ha ving said that, the universality c onj e ctur e arising in the theory of phase transitions suggests, for instance, that the scaling limit of a large class of ferromagnetic 2D Ising mo dels is the same as that of the Onsager 2D Ising model; and hence, P ainlev é functions are conjectured to app ear (in the massiv e scaling limit) in mo dels outside of the class of exactly solv able mo dels. This last statemen t is substan tiated by the dev elopmen ts in [3, 45, 46, 47]. In this pap er w e review recen t progress [59, 60, 61, 62 , 63] on the curren t fluctuations in the asymmetric simple exclusion pro cess (ASEP) on the in teger lattice Z [35, 36]. ASEP is in the class 1 A complete account o f the SMJ theory can b e found in the recent monograph by Pa lmer [39 ]. 2 It is also w orth noting th at due to t he close connection of random matrix theory to multiv ariate statistical analysis, these same distribution functions inv olving P ainlevé functions are no w routinely used in data analysis [30, 31, 40]. 1 of Bethe Ansatz solv able mo dels [23, 25] but only for certain v alues of the parameters is ASEP a determinan tal pro cess [29, 44, 49]. That ASEP is Bethe Ansatz s olv able comes as no surprise onc e one realizes that the generator of ASEP is a s imilarit y (not unitary!) transformation of the XXZ- quan tum spin Hamiltonian [2, 50, 65]. Our main results relate the limiting curren t fluctuations in ASEP f or certain initial conditions to the TW distributions F 1 and F 2 of random matrix theory [58, 59]. Both F 1 and F 2 are expressible in terms of the same Hastings-McLeod solution of P ainlevé I I [20, 26], see §4.2. 2 Master Equation and Bethe Ansatz Solution Since its in tro duction in 1970 b y F. Spitzer [53], the asymmetric s imple exclusion process (ASEP) has attracted considerable atten tion b oth in the mathematics and ph ysics literature due to the fact it is one of the simplest lattice models describing transport far from equilibrium. Recall [35, 36] that the A SEP on the in teger lattice Z is a conti nuou s time Marko v pro cess η t where η t ( x ) = 1 if x ∈ Z is o ccupied at time t , and η t ( x ) = 0 if x is v acan t at time t . P articles mo v e on Z according to tw o rules: (1) A particle at x w aits an exp onen tial time with parameter one, and then c ho oses y with probabilit y p ( x, y ) ; (2) If y is v acan t at that time it mo v es to y , while if y is o ccupied it remains at x . The adjectiv e “simple” refers to the f act that the allo w ed j umps are only one step to the righ t, p ( x, x + 1) = p , or one step to the left, p ( x, x − 1) = q = 1 − p . The totally asymmetric s imple exclusion process (T ASEP) allo ws jumps only to the righ t ( p = 1 ) or only to the left ( p = 0 ). 3 In the mapping from the XXZ quan tum spin c hain, the anisotrop y parameter ∆ of the spin c hain is related to the hopping probabilit ies p and q by ∆ = 1 2 √ pq ≥ 1 , the ferromagnetic regime of the XXZ spin c hain. W e b egin with a sys tem of N particl es and later tak e the limit N → ∞ . A configuration is sp ecified by giving the lo cation of the N particles. W e denote b y Y = { y 1 , . . . , y N } with y 1 < · · · < y N the initial configuration of the pro cess and write X = { x 1 , . . . , x N } ∈ Z N . When x 1 < · · · < x N then X represen ts a p ossible configuration of the system at a later time t . W e denote by P Y ( X ; t ) the probabilit y that the system is in configuration X at time t , giv en that it w as initially in configuration Y . Giv en X = { x 1 , . . . , x N } ∈ Z N w e set X + i = { x 1 , . . . , x i − 1 , x i + 1 , x i +1 , . . . , x N } , X − i = { x 1 , . . . , x i − 1 , x i − 1 , x i +1 , . . . , x N } . 3 It is T AS EP that is a determinan tal pro cess. 2 The master equation for a f unction u on Z N × R + is d dt u ( X ; t ) = N X i =1  p u ( X − i ; t ) + q u ( X + i ; t ) − u ( X ; t )  , (1) and the b oundary conditions are, for i = 1 , . . . , N − 1 , u ( x 1 , . . . , x i , x i + 1 , . . . , x N ; t ) = p u ( x 1 , . . . , x i , x i , . . . , x N ; t ) + q u ( x 1 , . . . , x i + 1 , x i + 1 , . . . , x N ; t ) . (2) The initial condition is u ( X ; 0) = δ Y ( X ) when x 1 < · · · < x N . (3) The basic fact is that if u ( X ; t ) satisfies the master equation, the boundary conditions, and the initial condition, then P Y ( X ; t ) = u ( X ; t ) when x 1 < · · · < x N . This is, of course, one of Bethe’s basic ideas (see, e.g., [10 ]): incorp orate the in teraction (in this case the exclusion prop ert y) into the b oundary conditions (2) of a free particle system (1). Recall that an in v ersion in a p ermuta tion σ is an ordered pair { σ ( i ) , σ ( j ) } in whic h i < j and σ ( i ) > σ ( j ) . W e define [65 ] S αβ = − p + q ξ α ξ β − ξ α p + q ξ α ξ β − ξ β (4) and then A σ = Y { S αβ : { α, β } is an inv ersion in σ } . W e also set ε ( ξ ) = p ξ − 1 + q ξ − 1 . In the next theorem w e shall assume p 6 = 0 , so the A σ are analytic at zero in all the v ariab les. Here and later all differen tials dξ incorporate the factor (2 πi ) − 1 . Theorem 2.1 . W e ha v e P Y ( X ; t ) = X σ ∈S N Z C r · · · Z C r A σ Y i ξ x i − y σ ( i ) − 1 σ ( i ) e P i ε ( ξ i ) t dξ 1 · · · dξ N , (5) where C r is a circle cen tered at zero with radius r so small that all the p oles of the integr and lie outside C r . The pro of that P Y ( X ; t ) satisfies (1) is immediate and the fact it satisfies the b oundary condi- tions (2) is exactly the same argumen t as in the XXZ problem [65]. The difficult y lies in sho wing (5) satisfies the initial condition (3). Observ e that the term in (5) corresp onding to the iden tit y 3 p erm utation do es satisfy the initial condition. Th us the pr o of will b e complete once one demon- strates that the remaining n ! − 1 other terms sum to zero at t = 0 . This is indeed the case (some are individually zero and others cancel in pairs) and the res ult dep ends crucially up on the choice of the con tours C r [59]. F or the sp ecial case of T ASEP , p = 1 , it follo ws from (4) and (5) that the righ t-hand side of (5) can b e expressed as a N × N determinan t as first obtained in [49]. W e note that unlik e the usual applications of Bethe Ansatz, it is not the sp ectral theory of the op erator that is of in terest but rather the transition probabilit y P Y ( X ; t ) . Th us there are no Bethe equations in our approac h; and hence, no issues concerning the completeness of the Bethe eigenfunctions. Indeed, there is not ev en an Ansatz in this approac h! W e remark that this result extends with only minor mo difications to the solution Ψ( x 1 , . . . , x N ; t ) of the time-dependen t Sc hrö dinger equation with XXZ Hamiltonian where the x i ’s denot e the lo cation of the N “up spins” in a sea of “do wn spins” on Z . 3 Marginal Distributions and the Large N Limit W e henceforth ass ume q > p so there is a net drift of particles to the lef t. Here we consider tw o differen t initial conditions. The first, called step ini tial c ondition , starts with particles lo cated at Z + = { 1 , 2 , . . . } . The second initial condition is the step Bernoul li i n itial c ondition : eac h site in Z + , independent ly of the others, is initially o ccupied with probabilit y ρ , 0 < ρ ≤ 1 ; all other sites are initially uno ccupied. In each of these cases it make s sense to sp eak of the p osition of the m th particle from the left at time t , x m ( t ) , and its distribution function P ( x m ( t ) ≤ x ) . It is elemen tary to relate P ( x m ( t ) ≤ x ) to the distribution of the total curren t T at p os ition x at time t , T ( x, t ) := n um b er of particles ≤ x at time t ; namely , P ( T ( x, t ) ≤ m ) = 1 − P ( x m +1 ( t ) ≤ x ) . F or this reason w e first concen trate on P Y ( x m ( t ) ≤ x ) and only at the end translate the results in to statemen ts concerning T . (The subscript Y denotes the initial configuration.) No w for finite Y P Y ( x m ( t ) = x ) = X x 1 < ··· k . Theorem 3.1 [59, 63]. Assume q > p , then P ρ ( x m ( t ) ≤ x ) = X k ≥ 1 q k ( k − 1) / 2 τ k ( k +1) / 2 k ! c m,k Z C R · · · Z C R Y 1 ≤ i 6 = j ≤ m ξ j − ξ i f ( i, j ) × Y i ρ ξ i − 1 + ρ (1 − τ ) m Y i =1 ξ x i e tε ( ξ i ) 1 − ξ i dξ i (8) The con tour C R , a circle of radius R ≫ 1 cent ered at the origin, is c hosen s o that all (finite) p oles of the in tegrand lie inside the con tour. W e remark that f or T ASEP , p = 0 , the ab o ve sum reduces to one term; and this term can b e sho wn to b e equal to a m × m determinan t. The final simplification results if w e use the iden tit y [60] det  1 f ( i, j )  1 ≤ i,j ≤ k = ( − 1) k ( pq ) k ( k − 1) / 2 Y i 6 = j ( ξ j − ξ i ) f ( i, j ) Y i 1 (1 − ξ i )( q ξ i − p ) in (8) and recognize the summand, a k -dimensional integr al, as the co efficien t of λ k in the F redholm expansion of det( I − λK ρ ) where K ρ acts on functions on C R b y f ( ξ ) − → Z C R K ρ ( ξ , ξ ′ ) f ( ξ ′ ) dξ ′ 6 where K ρ ( ξ , ξ ′ ) = q ξ x e tε ( ξ ) p + q ξ ξ ′ − ξ ρ ( ξ − τ ) ξ − 1 + ρ (1 − τ ) , τ = p q . (9) Note that when ρ = 1 , the case of step initial condition, the last factor in K ρ ( ξ , ξ ′ ) equals one. Since the co efficien t of λ k in the expansion of det( I − λK ρ ) is equal to ( − 1) k k ! Z det( I − λK ρ ) dλ λ k +1 , this fact together with the τ -binomial theorem giv es the final result for P ρ ( x m ( t ) ≤ x ) . Theorem 3.2 [59, 63]. Let P ρ denote the probabilit y measure for ASEP with step Bernoulli initial condition with densit y ρ and x m ( t ) denote the p osition of the m th particle from the left at time t , then P ρ ( x m ( t ) ≤ x ) = Z C det( I − λK ρ ) Q m − 1 j =0 (1 − λτ j ) dλ λ (10) where the con tour C is a circle cen tered at the origin enclosing all the singularities at λ = τ − j , 0 ≤ j ≤ m − 1 and K ρ is the in tegral op erator whose kernel is given by (9). 4 Limit Theorems 4.1 KPZ Scaling The scaling limit that is of most in terest is the KPZ sc aling limi t [32 , 54]. In the terminology here this scaling limit is m → ∞ , t → ∞ with σ = m t ≤ 1 fixed . As w e s hall see, the limiting distribution will dep end up on the relativ e sizes of σ and ρ 2 . F or the momen t we concen trate on the cases 0 < σ < ρ 2 and σ = ρ 2 with 0 < ρ ≤ 1 . As in any centr al limit theorem, to obtain a non trivial limit the x in P ρ ( x m ( t ) ≤ x ) m ust also b e scaled (this to o is part of KPZ scaling). In an ticipation of the theorem w e set x := c 1 t + c 2 t 1 / 3 s where the 1 3 is the famous KPZ univ ersalit y exp onen t [32, 38] and c 1 := − 1 + 2 √ σ , c 2 := σ − 1 / 6 (1 − √ σ ) 2 / 3 . The t w o distribution functions that arise in the KPZ scaling limit are defined in the next s ection. 7 4.2 Distributions F 1 and F 2 The distributions F 1 and F 2 can be defined b y either their F redholm determinan t represen tations or their represen tations in terms of a P ainlev é I I function. Here w e tak e the latter route. Let q denote the solution to the P ainlev é I I equation q ′′ = x q + 2 q 3 satisfying q ( x ) ∼ Ai ( x ) , x → ∞ , where Ai ( x ) is the Airy function. That such a solution exists and is unique was pro v ed b y Hastings and McCleo d [26]. 5 Then w e ha ve F 2 ( s ) = exp  − Z ∞ s ( x − s ) q 2 ( x ) dx  , (11) F 1 ( s ) = exp  − 1 2 Z ∞ s q ( x ) dx  ( F 2 ( s )) 1 / 2 . (12) The asymptotics of these distributions as x → ∞ is straigh tforw ard given the large x asymptotics of the Airy function; ho w ever, the complete asymptotic expansion as x → −∞ has only recen tly b een completed [6]. F or high-accuracy n umerical ev aluation of F 1 and F 2 , it turns out that it is b etter to start with their F redholm determinan t represen tations [15]. 4.3 Limit La ws The asymptotic analysis [61, 63] of the F redholm determinan t in the form ula f or P ρ ( x m ( t ) ≤ x ) in (10) required the develop men t of new metho ds since the operator K ρ is not of the usual “in tegrable in tegral op erator” form normally app earing in random matrix theory [14, 27, 57]. The main p oin t is that the k ernel K ρ has the same F redholm determinan t as sum of t wo k ernels; one has large norm but fixed sp ectrum and its resolve nt can b e computed exactly , and the other is b etter b eha ved [61]. W e no w state the results of this asymptotic analysis. Theorem 4.1 [61, 63]. When 0 ≤ p < q , γ := q − p , lim t →∞ P ρ  x m ( t/γ ) − c 1 t c 2 t 1 / 3 ≤ s  = F 2 ( s ) when 0 < σ < ρ 2 , (13) lim t →∞ P ρ  x m ( t/γ ) − c 1 t c 2 t 1 / 3 ≤ s  = F 1 ( s ) 2 when σ = ρ 2 , ρ < 1 . (14) 5 A modern account of P ainlev é transcendents can be found in the monograph by F ok as, et a l. [20]. 8 T able 1: The mean ( µ β ), v ariance ( σ 2 β ), skew ness ( S β ) and kurtosis ( K β ) of F β , β = 1 , 2 . The n um b ers are courtesy of F. Bornemann and M. Prähofer. β µ β σ 2 β S β K β 1 -1.206 533 574 582 1.607 781 034 581 0.293 464 524 08 0.165 242 9384 2 -1.771 086 807 411 0.813 194 792 8329 0.224 084 203 610 0.093 448 0876 This theorem implies a limit la w for the curren t fluctuations. Define v = x/t, a 1 = (1 + v ) 2 / 4 , a 2 = 2 − 4 / 3 (1 − v 2 ) 2 / 3 . Theorem 4.2 . When 0 ≤ p < q , γ := q − p , lim t →∞ P ρ  T ( v t , t/γ ) − a 1 t a 2 t 1 / 3 ≤ s  = 1 − F 2 ( − s ) when − 1 < v < 2 ρ − 1 , (15) lim t →∞ P ρ  T ( v t , t/γ ) − a 1 t a 2 t 1 / 3 ≤ s  = 1 − F 1 ( − s ) 2 when v = 2 ρ − 1 , ρ < 1 . (16) F or step initial condition with 0 < σ < 1 the limit laws are (13) and (15) [61, 62]. When σ > ρ 2 (or v > 2 ρ − 1 ) the fluctuations are of order t 1 / 2 and the limiting distribution is Gaussian, see [63] for details. F or T ASEP , p = 0 , with step initial condition the limit la w (15) w as first pro ved b y Johansson [29]. F or T ASEP with step Bernoulli initial condition the limit la ws (15) and (16) w ere conjectured b y Prähofer and Sp ohn [44] and pro ve d recen tly by Ben Arous and Corwin [12]. The fact that these limit la ws remain essenti ally iden tical (the only c hange is the factor γ in the time slot) is a very strong statemen t of KPZ Universalit y . F rom the inte grable systems p ersp ectiv e, these results are, to the b est of the authors’ knowle dge, the first limit la ws of Bethe ansatz solv able models (outside the class of determinan tal mo dels) where the correlation functions are expressible in terms of P ainlev é functions. 5 Conclusions T o da y Pain levé functions o ccur in man y areas of theoretical statistical ph ysics. In the case of KPZ fluctuations there are no w exp erimen ts [38, 55] on sto c hastically gro wing in terfaces where 9 quan tities s uch as the skewness and the kurtosis of F β (see T able 1), as well as the distribution functions themselv es, are compared with exp erimen t. In [55] K. T ak euc hi and M. 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