aa r X i v : . [ m a t h . P R ] J a n THE ( q, µ, ν ) -BOSON PROCESS AND ( q, µ, ν ) -TASEP IVAN CORWIN
Abstract.
We prove a intertwining relation (or Markov duality) between the ( q, µ, ν )-Boson processand ( q, µ, ν )-TASEP, two discrete time Markov chains introduced by Povolotsky in [16]. Using thisand a variant of the coordinate Bethe ansatz we compute nested contour integral formulas forexpectations of a family of observables of the ( q, µ, ν )-TASEP when started from step initial data.We then utilize these to prove a Fredholm determinant formula for distribution of the location ofany given particle. Introduction
We study the ( q, µ, ν )-Boson process and the ( q, µ, ν )-TASEP, two discrete time interacting par-ticle systems introduced by Povolotsky [16]. The primary contribution of this paper is the discoveryof an intertwining relationship between the Markov transition matrices for these processes. Thisgeneralizes the Markov duality previously discovered in [10, 5] for the q -Boson process and q -TASEP(special cases of the processes considered herein corresponding to setting ν = 0). Utilizing the co-ordinate Bethe ansatz solvability discovered in [16] and following the strategy laid out in [10, 5]for the ν = 0 case, we prove nested contour integral formulas for expectations of a large class ofobservables of the ( q, µ, ν )-TASEP, when started from step initial data. These formulas completelycharacterize the fixed time distribution of the process, and by following the approach of [4] (see also[8]) we extract two Fredholm determinant formulas for the one-point distribution of ( q, µ, ν )-TASEPwith step initial data.There are a variety of interesting interacting particle systems which arise through limits or spe-cializations of parameters of the processes considered herein. The Fredholm determinant formulawhich we prove here is likely to be amenable to asymptotic analysis (such as that performed in[4, 6, 7, 13] for the special cases of q -TASEP, the O’Connell-Yor semi-discrete directed polymer andthe Kardar-Parisi-Zhang equation).This introductory section contains most of the main results of the paper, in addition to thenotation and background necessary to state and prove them (in particular a brief introduction to q -hypergeometric series and Heine’s 1847 q -generalization of Gauss’ summation formula). Section 2provides a general parameter extension of the processes considered in the introduction, and showsthat the intertwining relationship and some of the Bethe ansatz solvability extends to this generalsetting. It also includes proofs of those results stated without proofs in the introduction. Section3 contains discussion of some related results in the literature, possible extensions to the presentresults and new directions of research.1.1. Notation.
We write Z > = { , , . . . } and Z ≥ = { , , . . . } . For N ∈ Z > define Y N := ( Z ≥ ) { , ,...,N } and Y Nk := (cid:8) ~y ∈ Y N : X y i = k (cid:9) . For a vector ~y ∈ Y N and a vector ~s = ( s , . . . , s N ) with integers s i ∈ { , . . . , y i } for all i , let ~y s i i,i − = (cid:0) y , . . . , y i − + s i , y i − s i , . . . , y N (cid:1) . q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 2 Define the Weyl chamber W Nk := (cid:8) ~n = ( n , . . . , n k ) : N ≥ n ≥ · · · ≥ n k ≥ (cid:9) . To every ~y ∈ Y Nk we may associate the ordered list of locations ~n ( ~y ) ∈ W Nk such that for each i ∈ { , . . . , N } the number of elements of ~n equal to i is exactly y i . Likewise to ~n ∈ W Nk weassociate the uniques ~y = ~y ( ~n ) ∈ Y Nk such that ~n ( ~y ) = ~n . For example for N ≥
2, if ~y was such that y = 3, y = 1, y = 1 and all other y i = 0, then ~y maps to ~n ( ~y ) = (2 , , , , I ⊆ { , . . . , k } let ~n − I denote the the vector ~n with n i replaced by n i − i ∈ I .For N ∈ Z > define X N := (cid:8) ~x = ( x , x , . . . , X N ) ⊂ Z N : + ∞ = x > x > · · · > x N (cid:9) . For a vector ~x ∈ X N and a vector ~j = ( j , . . . , j N ) with integers j i ∈ { , . . . , x i − − x i − } for all i ,let ~x j i i := (cid:0) x , . . . , x i + j i , . . . , x N (cid:1) . Hypergeometric series.
Fix | q | <
1. Define the q -Pochhammer symbol( a ; q ) n := n Y i =0 (1 − aq i ) and ( a ; q ) ∞ := ∞ Y i =0 (1 − aq i ) . We record three identities (which follow directly from the definition) satisfied by the q -Pochhammersymbol. These can be found as (1.2.30), (1.2.31) and (1.2.32) in [14].( A ) ( a ; q ) n = ( a ; q ) ∞ ( aq n ; q ) ∞ , ( B ) ( a − q − n ; q ) n = ( a ; q ) n ( − a − ) n q − n ( n − , (1.2.1)( C ) ( a ; q ) n − k = ( a ; q ) n ( a − q − n ; q ) k ( − qa − ) k q k ( k − − nk . Define the basic q -hypergeometric series φ as φ ( a, b ; c ; q, z ) := ∞ X n =0 ( a ; q ) n ( b ; q ) n ( q ; q ) n ( c ; q ) n z n . (1.2.2)Since | q | <
1, this is convergent for | z | <
1. Heine’s 1847 q -generalization of Gauss’ summationformula [14, Section 1.5] states that φ ( a, b ; c ; q, c/ab ) = ( c/a ; q ) ∞ ( c/b ; q ) ∞ ( c ; q ) ∞ ( c/ab ; q ) ∞ , (1.2.3)as long as | c/ab | <
1. A special degeneration of this summation formula states that for any n ≥ φ ( q − n , b ; c ; q, q ) = ( c/b ; q ) n ( c ; q ) n b n . (1.2.4)1.3. A ( q, µ, ν ) -deformed Binomial distribution. We define the three parameter deformation ofthe Binomial distribution introduced in [16] (see therein for references relating various applicationsfor limits of this distribution). For | q | <
1, 0 ≤ ν ≤ µ <
1, and integers 0 ≤ j ≤ m define thefunction ϕ q,µ,ν ( j | m ) := µ j ( ν/µ ; q ) j ( µ ; q ) m − j ( ν ; q ) m ( q ; q ) m ( q ; q ) j ( q ; q ) m − j . (1.3.1) HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 3
When m = + ∞ , extend this definition by setting ϕ q,µ,ν ( j | + ∞ ) := µ j ( ν/µ ; q ) j ( µ ; q ) ∞ ( ν ; q ) ∞ q ; q ) j . (1.3.2)The following lemma shows that for each m ∈ Z ≥ ∪ { + ∞} this defines a probability distributionon j ∈ { , . . . , m } . Lemma 1.1.
Fix any choice of parameters | q | < and ≤ ν ≤ µ < . Then for m ∈ Z ≥ ∪{ + ∞} , the function ϕ q,µ,ν ( j | m ) defines a probability distribution over the set of j ∈ { , . . . , m } .Equivalently, m X j =0 ϕ q,µ,ν ( j | m ) = 1 . Proof.
By using (1.2.1(C)) the desired identity can be written as m X j =0 ( q − m ; q ) j ( ν/µ ; q ) j ( q ; q ) j ( µ − q − m ; q ) j q j = ( ν ; q ) m ( µ ; q ) m . The left-hand side is readily identified as φ ( q − m , ν/µ ; µ − q − m ; q, q ) (which terminates for j > m due to the q − m argument). Applying (1.2.4) with b = µ/ν and c = µ − q − m , and subsequentlyapplying (1.2.1(B)) yields the desired right-hand side above, and hence proves the identity. (cid:3) The following proposition generalizes Lemma 1.1 (which corresponds to y = 0) and is essentiallyparamount to the intertwining relationship we prove later as Theorem 1.3. The proposition is provedin Section 2.5. The ν = 0 version of this result was proved earlier as [5, Lemma 3.7]. The proof wepresent herein follows that general approach, though that presence of ν = 0 necessitates our use ofHeine’s 1847 q -generalization of Gauss’ summation formula, given above as (1.2.3). Proposition 1.2.
Fix any choice of parameters | q | < and ≤ ν ≤ µ < . Then. for all m, y ∈ Z ≥ , m X j =0 ϕ q,µ,ν ( j | m ) q jy = y X s =0 ϕ q,µ,ν ( s | y ) q sm . Similarly, for all y ∈ Z ≥ , + ∞ X j =0 ϕ q,µ,ν ( j | + ∞ ) q jy = ϕ q,µ,ν (0 | y ) . Both Lemma 1.1 and Proposition 1.2 can be analytically continued in the parameter q within asuitable domain, though we do not expand on this observation or its applications herein.1.4. The ( q, µ, ν ) -Boson process. This discrete time interacting particle system was introducedby Povolotsky [16] and shown therein to be the most general zero range chipping model [12] withfactorized steady state which is also solvable via coordinate Bethe ansatz. See Sections 3.1, 3.2 and3.5 for further discussion on this; Section 2.1 for a general parameter version of this process forwhich some of our results still hold; and Section 3.3 for a discussion related to our choice of usingthe term
Boson in naming this process.Fix | q | <
1, 0 ≤ ν ≤ µ < N ≥
1. The N -site ( q, µ, ν )-Boson process is a discretetime Markov chain ~y ( t ) = { y i ( t ) } Ni =0 ∈ Y N . The values of y i ( t ) record the number of particles abovesite i at time t . At time t + 1 the state ~y ( t ) is updated to another state ~y ( t + 1) according to thefollowing procedure. For each site i ∈ { , . . . , N } , s i ∈ (cid:8) , , . . . , y i ( t ) (cid:9) particles are transferredto site i − t + 1) with probability ϕ q,µ,ν ( s i | y i ( t )) (see the left-hand side of Figure 1). HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 4 { iy i ( t ) s i x n ( t ) x n − ( t ) x n +1 ( t ) j n N Figure 1.
Left: A single time step of the ( q, µ, ν )-Boson process is illustrated. Foreach site i ∈ { , . . . , N } , a random number of the y i ( t ) particles above the site aremoved them to the left to site i −
1. The jumps are independent at each site anddistributed according to ϕ q,µ,ν (cid:0) j | y i ( t ) (cid:1) . Right: A single time step of the ( q, µ, ν )-TASEP is illustrated. Each particle x i ( t ) with i ∈ { , . . . , N } jumps to any of thesites x n ( t ) through x n +1 ( t ) −
1. The jumps are independent at each site and theirlength j is distributed according to ϕ q,µ,ν (cid:0) j | x n − ( t ) − x n ( t ) − (cid:1) This occurs in parallel for each site, so y i ( t + 1) is equal to y i ( t ) plus those particles which weretransferred from site i + 1 to site i and minus those particles which were transferred from site i tosite i − y i ( t + 1) = y i ( t ) + s i +1 − s i ). No particles are transferred into site N or out of site 0.For functions f : Y N → R define the operators (cid:2) A q,µ,ν (cid:3) i , i ∈ { , . . . , N } , via their action (cid:2) A q,µ,ν (cid:3) i f ( ~y ) = y i X s i =0 ϕ q,µ,ν ( s i | y i ) f (cid:0) ~y s i i,i − (cid:1) . The operator (cid:2) A q,µ,ν (cid:3) i encodes the movement of particles from site i to i −
1. The single timestep transition matrix P Boson for the N -site ( q, µ, ν )-Boson process (which is time homogeneous) iswritten via the product of these operators (cid:0) P Boson f (cid:1) ( ~y ) = (cid:2) A q,µ,ν (cid:3) · · · (cid:2) A q,µ,ν (cid:3) N f ( ~y ) . It is clear that the N -site ( q, µ, ν )-Boson process conserves the number of particles. Hence the N -site, k -particle ( q, µ, ν )-Boson process has the state space Y Nk . Rather than identifying the numberof particles per site, we may identify the state ~y via recording the ordered location ~n ( ~y ) ∈ W Nk ofeach of the k particles.1.5. The ( q, µ, ν ) -TASEP. This discrete time interacting particle system was introduced by Po-volotsky [16] by virtue of its relationship to the ( q, µ, ν )-Boson process via a simple ZRP-ASEP andparticle-hole transformation (see Remark 1.6 herein).Fix | q | <
1, 0 ≤ ν ≤ µ < N ≥
1. The N -particle ( q, µ, ν )-TASEP (totallyasymmetric simple exclusion process) is a discrete time Markov chain ~x ( t ) = { x n ( t ) } Nn =0 ∈ X N . Thepurpose of setting x ≡ + ∞ is to simplify notation via having a “virtual particle” at + ∞ . Thevalue of x n ( t ) records the location of particle n at time t . At time t + 1 the state ~x ( t ) is updated toanother state ~x ( t + 1) according to the following procedure. For each n ∈ { , . . . , N } , x n ( t ) updatesin parallel and independently to x n ( t + 1) = x n ( t ) + j n where j n ∈ (cid:8) , . . . , x n − ( t ) − x n ( t ) − (cid:9) isdrawn according to the probability distribution ϕ q,µ,ν ( j n | x n − ( t ) − x n ( t ) −
1) (see the right-handside of Figure 1).
HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 5
For functions f : X N → R define the operators (cid:2) B q,µ,ν (cid:3) n , n ∈ { , . . . , N } , via their action (cid:2) B q,µ,ν (cid:3) n f ( ~x ) = x n − − x n − X j n =0 ϕ q,µ,ν ( j n | x n − − x n − f (cid:0) ~y j n n (cid:1) . This operator encodes the one-step update of the location of particle x n ( t ). The single time steptransition matrix P TASEP for N -particle ( q, µ, ν )-TASEP (which is time homogeneous) is writtenvia the product of these operators (cid:0) P TASEP f (cid:1) ( ~x ) = (cid:2) B q,µ,ν (cid:3) · · · (cid:2) B q,µ,ν (cid:3) N f ( ~x ) . The update rule for ( q, µ, ν )-TASEP is one-sided in the sense that particle n only depends onparticle n − q, µ, ν )-TASEP in which configurations are given by particle locations x ( t ) > x ( t ) > · · · .Any event concerning particle x N ( t ) depends only upon the evolution of the first N particles andhence can be studied in reference to the N -particle process. We will be concerned herein with stepinitial data where x n (0) = − n for n ≥
1. See Sections 3.1 and 3.2 for mention of some additionaltypes of initial data.1.6.
Intertwining the ( q, µ, ν ) -Boson process and ( q, µ, ν ) -TASEP. We come now to the pri-mary contribution of this paper. Theorem 1.3 and its corollaries demonstrate an intertwining relationship between the ( q, µ, ν )-Boson process and ( q, µ, ν )-TASEP, generalizing the dualities dis-covered by [10, 5] between the continuous Poison and discrete geometric q -TASEP and q -Bosonprocesses. The following theorem is a special case ( a i ≡ µ t ≡ µ ) of Theorem 2.1. Since weprove that general statement later, we forego a proof of this result here. We do, however, remarkthat this essentially boils down to Proposition 1.2. Theorem 1.3.
Fix | q | < , ≤ ν ≤ µ < and an integer N ≥ . Define H : X N × Y N → R as H ( x ; y ) := N Y i =0 q y i ( x i + i ) , (1.6.1) with the convention that if y > then the above product is 0. Then H intertwines the N -site ( q, µ, ν ) -Boson process and N -particle ( q, µ, ν ) -TASEP in the sense that P TASEP H = H (cid:0) P Boson (cid:1) ⊤ , where (cid:0) P Boson (cid:1) ⊤ represents the transpose of P Boson . We record two corollaries of the above intertwining. In order to state the first corollary, we fixthe following notation. We say h : Z ≥ × Y N → R ≥ solves the true evolution equation with initialdata h ( ~y ) if it satisfies:(1) for all ~y ∈ Y N and t ≥ h ( t + 1; ~y ) = P Boson h ( t ; ~y );(2) for all ~y ∈ Y N , h (0; ~y ) = h ( ~y ).The upcoming corollary follows quite readily from Theorem 1.3 – see the proof of Corollary 2.2, ageneral parameter version of this result. It shows that a certain family of expectations of observablesof the ( q, µ, ν )-TASEP satisfy a closed, deterministic evolution equation. Corollary 1.4.
For any fixed ~x ∈ X N , h ( t ; ~y ) := E ~x (cid:2) H ( ~x ( t ) , ~y ) (cid:3) = E ~x h N Y i =0 q y i ( x i + i ) i HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 6 is the unique solution to the true evolution equation with initial data h ( ~y ) := Q Ni =0 q y i ( x i + i ) . The second corollary (which we state but do not utilize) is the Markov duality of the ( q, µ, ν )-Boson process and ( q, µ, ν )-TASEP. Recall that in general, two Markov chains x ( t ) and y ( t ) withstate spaces X and Y are said to be dual with respect to a duality functional H : X × Y → R if forall x ∈ X and y ∈ Y , and all t ≥ E x h H (cid:0) x ( t ) , y (cid:1)i = E y h H (cid:0) x, y ( t ) (cid:1)i , where E x is the expectation of the Markov chain x ( t ) stated from x (0) = x , and E y is the expectationof the Markov chain y ( t ) stated from y (0) = y . Corollary 1.5.
The N -particle ( q, µ, ν ) -TASEP ~x ( t ) and N -site ( q, µ, ν ) -Boson process ~y ( t ) aredual with respect to the duality functional H ( x ; y ) . Remark 1.6.
There is a simpler duality between these processes which should be distinguishedfrom that of Corollary 1.5. The gaps g i ( t ) := x i − ( t ) − x i ( t ) of the ( q, µ, ν )-TASEP evolve accordingto the update rule for the ( q, µ, ν )-Boson process in which particles now jump from site i to i + 1(instead of i to i − q, µ, ν )-TASEP corresponds with g (0) = + ∞ and g i (0) ≡ i = 1. This mapping between the two processes was discussed by Povolotsky [16]and called a ZRP-ASEP and particle-hole transformation.1.7. Free evolution equations with two-body boundary conditions.
We confront the ques-tion of how to solve the true evolution equation in Corollary 1.4 and hence compute formulas forthe expectations of ( q, µ, ν )-TASEP observables. A priori it is not clear how to proceed. The firstreduction is to recognize that the transition matrix P Boson is a direct sum of transition matrices forthe N -site, k -particle ( q, µ, ν )-Boson process which has state space Y Nk . In principal this reducesthe problem of computing h ( t ; ~y ) for any fixed N and k to a matter of finite matrix multiplication(or exponentiation). The challenge, however, is to figure out a way to perform this computationwith complexity which remains constant as N , k and t grow.Proposition 1.7 provides an important step towards this reduction in complexity. The N -site, k -particle ( q, µ, ν )-Boson process ~y ( t ) can be rewritten in terms of particle locations ~n ( t ) = ~n ( ~y ( t )).The idea (which dates back to Bethe’s solution [3] of the Heisenberg XXX quantum spin chain) is torewrite the k -particle true evolution equation for ~n ( t ) ∈ W Nk in terms of a k -particle free evolutionequation subject to k − two-body boundary conditions , but on a large state space ~n ∈ Z k . Inthe spirit of the reflection principal, any solution to the free evolution equation, which satisfies thetwo-body boundary conditions and satisfies the desired initial data when restricted to W Nk will thencoincide on W Nk with the unique solution to the true evolution equation.Most every k -particle system do not enjoy this reducibility as higher order boundary conditionsmust also be imposed. The reason Povolotsky introduced the ( q, µ, ν )-Boson process in [16] wasbecause it was the most general process within the class he was considering which enjoyed thisproperty. The following proposition is effectively contained in Section 3 of [16]. It is a special caseof Proposition 2.3, which contains details as to the proof. Proposition 1.7.
Fix | q | < , ≤ ν ≤ µ < , and integers N, k ≥ . If u : R ≥ × Z k → C solves: (1) ( k -particle free evolution equation) for all ~n ∈ Z k and t ≥ , u ( t + 1; ~n ) = k Y i =1 (cid:2) ∇ µ,ν (cid:3) i u ( t ; ~n ) , where [ ∇ µ,ν ] i u ( t ; ~n ) := µ − ν − ν u ( t ; ~n − i ) + − µ − ν u ( t ; ~n ) ; HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 7 /νγ γ Figure 2.
Possible contours of integration for Theorem 1.8 with k = 5(2) ( k − for all ~n ∈ Z k such that for some i ∈ { , . . . , k − } , n i = n i +1 , and all t ≥ , αu ( t ; n − i,i +1 ) + βu ( t ; ~n − i +1 ) + γu ( t ; ~n ) − u ( t ; ~n − i ) = 0 where the parameters α, β, γ are defined in terms of q and ν as α = ν (1 − q )1 − qν , β = q − ν − qν , γ = 1 − q − qν ;(3) (initial data) for all ~n ∈ W Nk , u (0; ~n ) = h (cid:0) ~y ( ~n ) (cid:1) ;then for all ~n ∈ W Nk , and all t ≥ , u ( t ; ~n ) = h (cid:0) t ; ~y ( ~n ) (cid:1) where h ( t ; ~y ) is the solution to the trueevolution equation with initial data h . Nested contour integral formula for step initial data.
Theorem 1.8 provides an exact,and concise nested contour integral formula for the expectations of the general class of observablesconsidered in Corollary 1.4, for the ( q, µ, ν )-TASEP started from step initial data. This achieves theaim of finding a solution to the true evolution equation which does not grow in complexity. At firstglance, this formula might seem to be pulled out of thin air. Formulas of this type, however, haveoccurred previously in the coordinate Bethe ansatz literature and the Macdonald process literature– see Sections 3.2 and 3.4 for further discussion as well as [18, 4, 10, 5, 9]. In any case, once givensuch a proposed formula, assisted by Proposition 1.7 it is quite simple to prove the theorem.
Theorem 1.8.
Fix | q | < , ≤ ν ≤ µ < , and integers N, k ≥ . Consider ( q, µ, ν ) -TASEPstarted from step initial data. Then for any ~n ∈ W Nk , E " k Y i =1 q x ni + n i = ( − k q k ( k − (2 π i ) k I γ · · · I γ k Y ≤ A A and all contours exclude and /ν (see Figure 2 for an illustration when k = 5 ). HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 8
Proof.
Call the right-hand side of (1.8.1) u ( t ; ~n ). Let us check that u ( t ; ~n ) satisfies the three condi-tions of Proposition 1.7 with initial data h ( ~y ) = ~y : y =0 (or in other words u (0; ~n ) = Q ki =1 n i > ).To check the k -particle free evolution equation observe that by bring (cid:2) ∇ µ,ν (cid:3) i inside the integration,it suffices to confirm that (cid:2) ∇ µ,ν (cid:3) i (cid:18) − νz i − z i (cid:19) n i (cid:18) − µz i − νz i (cid:19) t = (cid:18) − νz i − z i (cid:19) n i (cid:18) − µz i − νz i (cid:19) t +1 , as is readily done.To check the k − − ν ) (1 − qν )(1 − νz i )(1 − νz i +1 ) ( z i − qz i +1 ) . The factor of ( z i − qz i +1 ) cancels the pole separating the contours γ i and γ i +1 . We are now freeto deform the contours γ i and γ i +1 to lie along the same curve. Since n i = n i +1 (by hypothesis)the integrand is anti-symmetric in z i and z i +1 . This, however, implies that the integral is zero, andhence the second condition is confirmed.To check the initial data observe that when n k = 0, there is no pole at 1 for the z k integral.Thus, since γ k contains no poles, the integral (and likewise u (0; ~n )) is zero. The alternative n k > n i > ~n ∈ W Nk . In this case, we expand γ through γ k to infinity. There are no polesat 1 /ν since n i − ≥
0, and there is no pole at infinity due to quadratic decay. There are poles at z i = 0 which (through evaluating the residues) shows the integral is equal to one. Thus we haveshown u (0; ~n ) = Q ki =1 n i > as desired.By virtue of Proposition 1.7, this means that u ( t ; ~n ) = h (cid:0) t ; ~y ( ~n ) (cid:1) where h ( t ; ~y ) is the solutionto the true evolution equation with initial data h ( ~y ) = ~y : y =0 . Corollary 1.4 then implies that u ( t ; ~n ) = E ~x (cid:2) H ( ~x ( t ) , ~y ( ~n )) (cid:3) which is immediately matched to the left-hand side of (1.8.1). (cid:3) Fredholm determinant formula.
For ( q, µ, ν )-TASEP with step initial data, the formulasfrom Theorem 1.8 provide a complete characterization of the distribution of ~x ( t ). Indeed, eachrandom variable q x n ( t )+ n , 1 ≤ n ≤ N , is in (0 ,
1) and hence knowledge of all joint movement sufficesto characterize the joint distribution. Despite this fact, it is not obvious how to extract meaningfulasymptotic distribution information from these formulas. In the case of one-point distributions(i.e. the distribution of x n ( t ) for a single n ) this was achieved in [4]. We will apply the approachdeveloped in [4] (in particular, the general restatement of the calculation in [4] which can be foundin [10, Section 3]).Theorem 1.9 provides two Fredholm determinant formulas for the e q -Laplace transform of theobservable q x n ( t )+ n , and consequently for the one-point distribution of x n ( t ) (see Remark 1.10). Thistype of Fredholm determinant formula (in particular that of (1.9.1) is quite amenable to asymptoticanalysis – see Section 3.5 for further discussion. Theorem 1.9.
Fix q ∈ (0 , and ≤ ν ≤ µ < . Consider ( q, µ, ν ) -TASEP ~x ( t ) started from stepinitial data. Then for all ζ ∈ C \ R + , E " (cid:0) ζq x n ( t )+ n ; q (cid:1) ∞ = det (cid:0) I + K ζ (cid:1) (1.9.1) where det (cid:0) I + K ζ (cid:1) is the Fredholm determinant of K ζ : L ( C ) → L ( C ) for C a positively orientedcircle containing 1 with small enough radius so as to not contain 0, /q and /ν . The operator K ζ HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 9 is defined in terms of its integral kernel K ζ ( w, w ′ ) = 12 π i Z i ∞ +1 / − i ∞ +1 / π sin( − πs ) ( − ζ ) s g ( w ) g ( q s w ) 1 q s w − w ′ ds with g ( w ) = (cid:18) ( νw ; q ) ∞ ( w ; q ) ∞ (cid:19) n (cid:18) ( µw ; q ) ∞ ( νw ; q ) ∞ (cid:19) t νw ; q ) ∞ . The following second formula also holds: E " (cid:0) ζq x n ( t )+ n ; q (cid:1) ∞ = det (cid:0) I + ζ ˜ K (cid:1) ( ζ ; q ) ∞ (1.9.2) where det (cid:0) I + ζ ˜ K (cid:1) is the Fredholm determinant of ζ times the operator ˜ K ζ : L ( C , ) → L ( C , ) for C , a positively oriented circle containing 0 and 1 (but not /ν ). The operator ˜ K is defined interms of its integral kernel ˜ K ( w, w ′ ) = g ( w ) /g ( qw ) qw ′ − w where the function g is as above. As this type of deduction of Fredholm determinant formulas from q -moment formulas has ap-peared before in [4, 10, 5], we provide the steps of the proof, without going into too much detail asto how they are justified. We also do not recall the definition of Fredholm determiants, but ratherrefer the reader to [4, Section 3.2.2]. Proof.
The first Fredholm determinant formula is referred to by [10] as
Mellin-Barnes type, and thesecond as
Cauchy type. In order to prove the Mellin-Barnes type formula we utilize the formulafor E (cid:2) q k ( x n ( t )+ n ) (cid:3) from specializing all n i ≡ n in Theorem 1.8. For the purpose of this proof define µ k := E (cid:2) q k ( x n ( t )+ n ) (cid:3) . The reason we permit this abuse of notation (this µ k of course has a differentmeaning than the parameter µ , or µ t from Section 3) is that we can now identify µ k with the formulapresent in [10, Definition 3.1], subject to defining f ( w ) := g ( w ) /g ( qw ), with g from the statement ofTheorem 1.9. We may then apply [10, Propositions 3.3 and 3.6] with the contour C A = C chosento be a very small circle around 1, and D R,d , D
R,d ; k specified by setting R = 1 / d arbitrary,as it does not matter for this choice of R ). The outcome of this is that X k ≥ µ k ζ k k q ! = det (cid:0) I + K ζ (cid:1) . In the course of this application, it is necessary to check that a few technical conditions on thecontours, as well as ζ and g are satisfied. These are easily confirmed for ζ with | ζ | small enough,and C a small enough circle around 1. The only condition depending on the function g is that | g ( w ) /g ( q s w ) | remain uniformly bounded as w ∈ C , k ∈ Z > and s ∈ D R,d ; k varies. This is readilyconfirmed for g from the statement of Theorem 1.9.Now, observe that for ζ with | ζ | small enough, we also have that X k ≥ µ k ζ k k q ! = E " (cid:0) ζq x n ( t )+ n ; q (cid:1) ∞ . This is justified (as in [4, Theorem 3.2.11]) by the deterministic fact that q x n ( t )+ n ∈ (0 ,
1) and anapplication of the q -Binomial theorem. This establishes (1.9.1) for | ζ | sufficiently small. However, HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 10 both sides are analytic over C \ R + and thus the claimed Mellin-Barnes type result of (1.9.1) followsvia analytic continuation.The Cauchy type formula (1.9.2) follows from [10, Proposition 3.10] along with a small amountof algebra. The proof essentially follows that of [4, Theorem 3.2.16]. (cid:3) Remark 1.10.
Just like the usual Laplace transform of a positive random variable, the e q -Laplacetransform in Theorem 1.9 can be readily inverted (see [4, Proposition 3.1.1] or [10, Proposition 7.1])to give the distribution of x n ( t ). Remark 1.11.
Setting g i ( t ) := x i − ( t ) − x i ( t ), Remark 1.6 shows that ~g ( t ) evolves as the ( q, µ, ν )-Boson process with particles moving from i to i + 1. The ( q, µ, ν )-TASEP step initial data corre-sponds with having g (0) = + ∞ and g i (0) = 0 for i >
1. Let C s ( t ) = P ∞ i = s +1 g i ( t ) be the number ofparticles of ~g ( t ) strictly to the right of site s at time t . Then clearly { x n ( t ) + n ≥ s } = { C s ( t ) ≥ n } and hence Theorem 1.9, in light of the inversion formula mentioned in Remark 1.10, provides anexact formula for the distribution of C s ( t ) as well.1.10. Acknowledgements.
The author extends thanks for A. Povolotsky for an early draft of thepaper [16] as well as useful discussions, and also appreciates discussions with A. Borodin and B.Vet˝o related to this work. The author was partially supported by the NSF through DMS-1208998as well as by Microsoft Research and MIT through the Schramm Memorial Fellowship, and by theClay Mathematics Institute through the Clay Research Fellowship.2.
General parameter intertwining
We introduce a more general version of the ( q, µ, ν )-Boson process and ( q, µ, ν )-TASEP whichincludes site/particle dependent jump parameters a i , i ∈ { , . . . , N } and time dependent jumpparameters µ t , t ∈ Z > . The processes considered in Section 1 correspond to setting all a i ≡ µ t ≡ µ . We state and prove a general parameter analog of the intertwining relationship givenearlier as Theorem 1.3 (below Theorem 2.1) and an analog of Proposition 1.7 (below Proposition2.3) providing the reduction of the associate true evolution equation to a free evolution equationwith two-body boundary conditions. For general µ t but a i ≡ a i parameters. For the case ν = 0 analogs of Theorem 1.8 and 1.9 do hold [5] for general a i parameters – see Section 3.4 for a brief discussion of the relation between the inclusion of theseparameters and Macdonald processes [4].2.1. Site/particle dependent and time dependent jump parameters.
Fix | q | < ν ∈ [0 , N ≥
1. Further, fix site/particle dependent jump parameters a i > i ∈ { , . . . , N } , and time dependent jump parameters µ t ∈ [ ν,
1) for all t ∈ Z > . We define ageneral parameter version of the ( q, µ, ν )-Boson process and ( q, µ, ν )-TASEP from Sections 1.4 and1.5 with respect to these site/particle dependent and time dependent jump parameters as follows.For the general parameter version of the ( q, µ, ν )-Boson process, s i ∈ { , , . . . , y i ( t ) } particlesare transferred from site i to site i − t + 1 with probability ϕ q,a i µ t +1 ,ν ( s i | y i ( t )). Forthe general parameter version of the ( q, µ, ν )-TASEP, the particle x n ( t ) updates its location to x n ( t + 1) = x n ( t ) + j n where j n ∈ (cid:8) , . . . , x n − ( t ) − x n ( t ) − (cid:9) is drawn according to the probabilitydistribution ϕ q,a n µ t +1 ,ν ( j n | x n − ( t ) − x n ( t ) − t ≥ P Boson t and P TASEP t denote the respective transition matrices from time t − t for the general parameter ( q, µ, ν )-Boson process and ( q, µ, ν )-TASEP. Generalizing what HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 11 is written in Sections 1.4 and 1.5, respectively, we may write explicitly (cid:0) P Boson t f (cid:1) ( ~y ) = (cid:2) A q,a µ t ,ν (cid:3) · · · (cid:2) A q,a N µ t ,ν (cid:3) N f ( ~y ) , (2.1.1) (cid:0) P TASEP f (cid:1) ( ~x ) = (cid:2) B q,a µ t ,ν (cid:3) · · · (cid:2) B q,a N µ t ,ν (cid:3) N f ( ~x ) , (2.1.2)where in the first line f : Y N → R and in the second line f : X N → R .2.2. Intertwining.
The following theorem is a general parameter analog of Theorem 1.3. Theproof essentially amounts to Proposition 1.2.
Theorem 2.1.
Fix | q | < , ν ∈ [0 , , and an integer N ≥ . Further, fix site/particle dependentjump parameters a i > for all i ∈ { , . . . , N } , and time dependent jump parameters µ t ∈ [ ν, forall t ∈ Z > . Recall H : X N × Y N → R from (1.6.1). H intertwines the general parameter versionof the N -particle ( q, µ, ν ) -TASEP ~x ( t ) and the the N -site ( q, µ, ν ) -Boson process ~y ( t ) in the sensethat for all t ≥ P TASEP t H = H (cid:0) P Boson t (cid:1) ⊤ . Proof.
Recalling the definition of (cid:2) B q,µ,ν (cid:3) i from Section 1.5 and employing (2.1.1) we readily findthat P TASEP t H ( ~x, ~y ) = N Y i =1 x i − − x i − X j i =0 ϕ q,a i µ t ,ν ( j i | x i − − x i − q j i y i ! N Y i =0 q y i ( x i + i ) . Applying Proposition 1.2 to each term of the product i = 1 through N above we find that P TASEP t H ( ~x, ~y ) = ϕ q,a µ t ,ν (0 | y ) N Y i =2 y i X s i ϕ q,a i µ t ,ν ( s i | y i ) q s i ( x i − − x i − ! N Y i =0 q y i ( x i + i ) = H (cid:0) P Boson t (cid:1) ⊤ , where we have employed (2.1.1) and the definition of [ A q,µ,ν (cid:3) i from Section 1.4. (cid:3) We say that h : Z ≥ × Y N → R ≥ solves the general parameter version of the true evolutionequation with initial data h ( ~y ) if:(1) for all ~y ∈ Y N and t ≥ h ( t + 1; ~y ) = P Boson t +1 h ( t ; ~y );(2) for all ~y ∈ Y N , h (0; ~y ) = h ( ~y ). Corollary 2.2.
For any fixed ~x ∈ X N , h ( t ; ~y ) := E ~x (cid:2) H ( x ( t ) , y ) (cid:3) = E ~x h N Y i =0 q y i ( x i + i ) i is the unique solution to the general parameter version of the true evolution equation with initialdata h ( ~y ) := Q Ni =0 q y i ( x i + i ) .Proof. It is clear from Theorem 2.1 that h ( t ; ~y ) solves the general parameter version of the trueevolution equation with initial data as given. To show the uniqueness of solutions to the trueevolution equation observe first that the matrix P Boson t +1 splits into a direct sum of matrices actingonly on the subspace Y Nk , over k ≥
0. Each of these matrices is finite and triangular in the sensethat ~y ∈ Y Nk , h ( t + 1; ~y ) depends only upon the values of h ( t ; ~y ′ ) for those ~y ′ such that for all i ∈ { , . . . , N } , y ′ i + · · · + y ′ N ≤ y i + · · · + y N . This clearly implies uniqueness (and existence) ofsolutions. (cid:3) HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 12
Theorem 2.1 also implies a form of Markov duality, though this requires employing a time reversalof one of the chains. Since this duality is not utilized, we forego stating it.2.3.
Equivalence of true and free evolution equations.
The following theorem is a generalparameter analog of Proposition 1.7. The proof is essentially given in [16, Section 3] (though thatis stated for all a i ≡
1) and relies upon (2.3.1) a three parameter generalization of the Binomialexpansion.
Proposition 2.3.
Fix | q | < , ν ∈ [0 , , and integers N, k ≥ . Further, fix site/particle dependentjump parameters a i > for all i ∈ { , . . . , N } , and time dependent jump parameters µ t ∈ [ ν, forall t ∈ Z > . If u : R ≥ × Z k → C solves: (1) ( k -particle free evolution equation) for all ~n ∈ Z k and t ≥ u ( t + 1; ~n ) = k Y i =1 (cid:2) ∇ a ni µ t +1 ,ν (cid:3) i u ( t ; ~n ) where [ ∇ µ,ν ] i u ( t ; ~n ) := µ − ν − ν u ( t ; ~n − i ) + − µ − ν u ( t ; ~n ) ; (2) ( k − for all ~n ∈ Z k such that for some i ∈ { , . . . , k − } , n i = n i +1 and all t ≥ , αu ( t ; n − i,i +1 ) + βu ( t ; ~n − i +1 ) + γu ( t ; ~n ) − u ( t ; ~n − i ) = 0 where the parameters α, β, γ are defined in terms of q and ν as α = ν (1 − q )1 − qν , β = q − ν − qν , γ = 1 − q − qν ;(3) (initial data) for all ~n ∈ W Nk , u (0; ~n ) = h (cid:0) ~y ( ~n ) (cid:1) ;then for all ~n ∈ W Nk , and all t ≥ , u ( t ; ~n ) = h (cid:0) t ; ~y ( ~n ) (cid:1) where h ( t ; ~y ) is the solution to the generalparameter version of the true evolution equation with initial data h .Proof. This result is essentially contained in [16, Section 3]. In order to prove it, it suffices to show˜ h ( t ; ~y ) := u (cid:0) t ; ~n ( ~y ) (cid:1) solves the general parameter version of the true evolution equation with initialdata h , when restricted to ~y ∈ Y Nk . When k = 1 this is quite evident, though when k > u ( t + 1; ~n ) for ~n ∈ W Nk in terms of u ( t ; ~n ′ ) with some ~n ′ which may not be in W Nk . Specifically, this happens when there are clusters of equal coordinates in ~n , in which case theboundary condition satisfied by u enables us to reexpress u ( t + 1; ~n ) in terms of u ( t ; ~n ′ ) with all ~n ′ ∈ W Nk . This is facilitated by the following result. Lemma 2.4.
If a function f : Z m → R satisfies the boundary conditions αu ( t ; n − i,i +1 ) + βu ( t ; ~n − i +1 ) + γu ( t ; ~n ) − u ( t ; ~n − i ) = 0 , for all ~n ∈ Z k such that for some i ∈ { , . . . , m − } , n i = n i +1 , then it also satisfies m Y i =1 (cid:2) ∇ µ,ν (cid:3) i f ( n, . . . , n ) = m X j =0 ϕ q,µ,ν ( j | m ) f ( n, . . . , n | {z } m − j , n − , . . . , n − | {z } j ) . Proof.
This is shown in Section 3.1 of [16] via the following generalization of the Binomial expansion.Consider an associative algebra generated by
A, B obeying the quadratic homogeneous relation BA = αAA + βAB + γBB. HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 13
Then for p = µ − ν − ν , (cid:0) pA + (1 − p ) B (cid:1) m = m X j =0 ϕ q,µ,ν ( j | m ) A j B m − j . (2.3.1) (cid:3) Lemma 2.4 may be applied to each cluster of equal elements in ~n , starting with the clusterincluding n k and ending with the cluster including n . The repeated application of the lemmashows that u ( t + 1; ~n ) = k Y i =1 (cid:2) ∇ a ni µ t +1 ,ν (cid:3) i u ( t ; ~n ) = (cid:2) A q,a µ t +1 ,ν (cid:3) · · · (cid:2) A q,a N µ t +1 ,ν (cid:3) N ˜ h ( t ; ~y ) = P Boson t ˜ h ( t ; ~y ) . Since the initial data matches, this shows (by the uniqueness of solutions to the true evolutionequation) that ˜ h ( t ; ~y ) = h ( t ; ~y ) (cid:3) Nested contour integral and Fredholm determinant formulas.
Theorem 1.8 has astraightforward analog for the general parameter version of the N -particle ( q, µ, ν )-TASEP if all a i ≡ µ t ∈ [ ν,
1) by vary). The only difference in the nested contour integral formulais the replacement (cid:18) − µz j − νz j (cid:19) t t Y s =1 − µ s z j − νz j . Using this minor modification, Theorem 1.9 can likewise be modified by changing the definition ofthe function g ( w ) by the replacement (cid:18) ( µw ; q ) ∞ ( νw ; q ) ∞ (cid:19) t t Y s =1 ( µ s w ; q ) ∞ ( νw ; q ) ∞ . It is not clear how to construct a similar sort of nested contour integral solution when the a i arenot all equal. A similar difficulty arises in the context of the ASEP with bond dependent jumpparameters where [10] shows that duality and a reduction of the associated true evolution equationto free evolution equation with boundary conditions holds, yet there is no clear nested contourintegral formulas for moments.However, if the parameter ν = 0, then [5, Theorem 2.1(2)] did find general a i and µ t parameternested contour integral solutions given by the following replacement of terms in the integrand ofright-hand side of (1.8.1): k Y j =1 (cid:18) − νz j − z j (cid:19) n j (cid:18) − µz j − νz j (cid:19) t dz j z j (1 − νz j ) k Y j =1 n j Y i =1 a i a i − z j t Y s =1 (1 − µ s z j ) dz j z j . See Section 3.4 for figure discussion.2.5.
Proof of the ( q, µ, ν ) -deformed Binomial distribution identity. The proof of Proposition1.2 which we now present is a modification of that of [5, Lemma 3.7]. That case corresponds withsetting ν = 0. This leads to a simplification since the desired equality (2.5.8) within the below proofbecomes ℓ X r =0 ( − r q r ( r − ( q ; q ) y ( q ; q ) r ( q ; q ) y − r = ( ℓ = 00 otherwise . HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 14
That identify is [1, Corollary 10.2.2(c)]. The present proof requires further manipulations andultimately appeals to the more general identity in (1.2.3) of Heine’s 1847 q -generalization of Gauss’summation formula for the q -hypergeometric series φ . Proof of Proposition 1.2.
We prove the first identity of the proposition, as the second one followssimilarly (or through taking m → + ∞ ). Define S m,y = m X j =0 ϕ q,µ,ν ( j | m ) q jy . (2.5.1)To prove the lemma we must show that S m,y = S y,m for all y, m . In order to show this we will firstprove that for each m ≥ T m = { T mi,j : i, j ≥ } such that T m (cid:0) S m, , S m, , . . . (cid:1) ⊤ = (1 , , . . . ) ⊤ . (2.5.2)Since T m is lower triangular, this relation uniquely characterizes the elements of S m, · . Therefore,to show that S m,y = S y,m it suffices to prove that T m (cid:0) S ,m , S ,m , . . . (cid:1) ⊤ = (1 , , . . . ) ⊤ . (2.5.3)We start by finding T m so that (2.5.2) holds. Applying Lemma 1.1 with µ, ν replaced by q y µ, q y ν we find that ( ν ; q ) y ( µ ; q ) y ( νq m ; q ) y m X j =0 ϕ q,µ,ν ( j | m ) q yj ( µq m − j ; q ) y = 1 . (2.5.4)In the above expression we have used that ϕ q,q y µ,q y ν ( j | m ) = ϕ q,µ,ν ( j | m ) q yj ( ν ; q ) y ( µq m − j ; q ) y ( µ ; q ) y ( νq m ; q ) y . We may use the expansion ( a ; q ) y = y X r =0 ( − a ) r q r ( r − ( q ; q ) y ( q ; q ) r ( q ; q ) y − r (2.5.5)to rewrite (2.5.4) as the identity( ν ; q ) y ( µ ; q ) y ( νq m ; q ) y y X r =0 ( − µ ) r q mr q r ( r − ( q ; q ) y ( q ; q ) r ( q ; q ) y − r S m,y − r = 1 , where S m,y − r is defined in (2.5.1). This identity can be rewritten in the form of (2.5.2) with T m read off from the above expression. In order to prove (2.5.3) we must prove that( ν ; q ) y ( µ ; q ) y ( νq m ; q ) y y X r =0 ( − µ ) r q mr q r ( r − ( q ; q ) y ( q ; q ) r ( q ; q ) y − r S y − r,m = 1 (2.5.6)holds ( S m,y − r has been replaced by S y − r,m ).The rest of this proof is devoted to showing (2.5.6). After some minor manipulations to thisdesired identity, we will reduce it to an application of Heine’s q -Gauss summation formula. We mayuse the expansion (2.5.5) rewrite (2.5.6) as( ν ; q ) y ( µ ; q ) y y X r =0 ( − µ ) r q mr q r ( r − ( q ; q ) y ( q ; q ) r ( q ; q ) y − r S y − r,m = y X r =0 ( − νq m ) r q r ( r − ( q ; q ) y ( q ; q ) r ( q ; q ) y − r . (2.5.7) HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 15
We prove (2.5.7) by matching coefficients of powers of q m on both sides of the desired identity.Expanding both sides of (2.5.7) we can gather the coefficients of ( q m ) ℓ for ℓ = 0 , . . . , y . In orderthat all of the coefficients match, we must show that for all ℓ = 0 , . . . , y we have( ν ; q ) y ( µ ; q ) y ℓ X r =0 ( − µ ) r q r ( r − ( q ; q ) y ( q ; q ) r ( q ; q ) y − r ϕ q,µ,nu ( ℓ − r | y − r ) = ( − ν ) ℓ q ℓ ( ℓ − ( q ; q ) y ( q ; q ) ℓ ( q ; q ) y − ℓ . From the definition of ϕ this is equivalent to showing that for all ℓ = 0 , . . . , y we have ℓ X r =0 ( − r q r ( r − ( q ; q ) y ( q ; q ) r ( q ; q ) y − r ( ν/µ ; q ) ℓ − r ( ν ; q ) y − r = ( − ν/µ ) ℓ q ℓ ( ℓ − ( µ ; q ) y ( ν ; q ) y ( µ ; q ) y − ℓ . (2.5.8)Using the identity (1.2.1(C)) we can rewrite the left-hand side of (2.5.8) asLHS(2.5.8) = ( ν/µ ; q ) ℓ ( ν ; q ) y ℓ X r =0 ( q − ℓ ; q ) r ( ν − q − y ; q ) r ( q ; q ) r ( µν − q − ℓ ; q ) r ( µq y ) r (2.5.9)= ( ν/µ ; q ) ℓ ( ν ; q ) y φ ( q − ℓ , ν − q − y ; µν − q − ℓ ; q ; µq y ) . (2.5.10)The second equality follows from (1.2.2) and the fact that having q − ℓ as an argument cuts theinfinite summation in φ off for all r > ℓ . Using this φ expression for the left-hand side of (2.5.8)we can rewrite (2.5.8) as φ ( q − ℓ , ν − q − y ; µν − q − ℓ ; q ; µq y ) = ( − ν/µ ) ℓ q ℓ ( ℓ − ( µ ; q ) y ( ν ; q ) y ( µ ; q ) y − ℓ . We must show this identity for all ℓ = 0 , . . . , y . Applinyg (1.2.3) with a = q − ℓ , b = ν − q − y , and c = µν − q − ℓ reduces this desired identity to( ν/µ ; q ) ℓ ( ν ; q ) y ( µν − q ; q ) ∞ ( µq y − ℓ ; q ) ∞ ( µν − q − ℓ ; q ) ∞ ( µq y ; q ) ∞ = ( − ν/µ ) ℓ q ℓ ( ℓ − ( µ ; q ) y ( ν ; q ) y ( µ ; q ) y − ℓ which is easily confirmed by using the identities (1.2.1(A)) and (1.2.1(C)). This proves the identity(2.5.6) and hence completes the proof of Proposition 1.2. (cid:3) Discussion of results, extensions, open problems and relation to literature
Without going into too much detail, we provide some discussion below with a focus towardspossible extensions and new directions of research related to this work. We do not attempt a fullsurvey the literature related to the present work or to these extension and new directions.3.1.
Stationary version of the ( q, µ, ν ) -Boson process and ( q, µ, ν ) -TASEP. Evans-Majumdar-Zia [12] characterized the jump distributions for spatially homogeneous discrete time zero rangeprocesses (called mass transport models in [12] or zero range chipping models in [16]) on periodic do-mains which have factorized steady states (or in other words, invariant measures which are productmeasures). This class of processes involve moving s i out of y i ( t ) particles from site i to i − t +1 (independently and in parallel for all i ) according to a jump distribution ϕ ( s i | y i ( t )). Povolotsky[16] sought to characterize those jump distributions ϕ which additionally led to processes solvablevia Bethe ansatz (see Sections 1.7 and 3.2) and found that ϕ q,µ,ν constitutes that set.This paper primarily focuses on the N -site ( q, µ, ν )-Boson process in which there are no invariantmeasures (eventually all particles move to site 0). For the moment consider the model on Z withstate space ( Z ≥ ) Z , so that a state ~y = { y i } i ∈ Z . Taking an infinite volume analog of the factorized HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 16 steady states from [12] we arrive at a class of product measures on ~y indexed by a parameter ρ ∈ [0 ,
1) in which for each i ∈ Z , P ( y i = n ) = ρ n ( ν ; q ) n ( q ; q ) n ( ρ ; q ) ∞ ( ρν ; q ) ∞ . (3.1.1)We speculate that these constitute invariant measures for the ( q, µ, ν )-Boson process on Z (we do notconfirm that here). It would be interesting to classify the full set of translation invariant stationarymeasures for this process.The continuous time counterparts of the mass transport models in [12] are the totally asymmetriczero range processes (TAZRPs) in which a single particle moves from site i to i − g ( y i ),independently and in parallel over all i ∈ Z . The rate function g plays an analogous role to thejump probability distribution, though in continuous time only one particle can jump (as opposed toclusters which can move in the discrete time models). Under very mild growth conditions on g , itis known that TAZRPs have invariant measures which are product measures [4, Proposition 3.3.11](with one point distribution related to the rate g ). This should be compared to the discrete timeprocesses in which very particular conditions on the jump distribution must be satisfied, as shownin [12].Bal´azs-Komj´athy-Sepp¨al¨ainen [2] used second class particle and coupling methods to prove that awide class of TAZRPs demonstrate cube root fluctuations (i.e. t / ) in their particle current through characteristics . This cube root behavior is an indication of membership in the KPZ universality(see the review [11]). It would be interesting to develop the methods used in [2] to this discrete timesetting and prove cube root fluctuations in this manner. Note, it may only be possible to implementthis approach in the case of product form invariant measures.3.2. Plancherel theory and coordinate Bethe ansatz.
Utilizing the coordinate Bethe ansatz,Povolotsky [16] constructed eigenfunctions for k -particle restriction of the ( q, µ, ν )-Boson processtransition matrix on a periodic domain and on Z . Let us focus on the case of Z . In this case, theeigenfunctions are indexed by k complex numbers (sometimes called quasi-momenta). In order tosolve the true evolution equation for a specific space of initial data, it is necessary to determinewhich subset of the eigenfunctions constitute a complete basis for the desired space and how theseeigenfunctions should be normalized in such a decomposition. This problem goes under the generalname of completeness of the coordinate Bethe ansatz and is achieved by proving a Plancherel theory .For the case of the ( q, µ, ν )-Boson process with ν = 0 this has been achieved in [9]. It would beinteresting to develop the analogous theory for general ν = 0. Note that [16, Conjecture 2] providesa conjecture (which agrees with the ν = 0 case proved in [9]) for a portion of the desired results.One output of an analogous Plancherel theory to that of [9] would be a systematic and direct routeto solve the ( q, µ, ν )-Boson process true evolution equation for more general initial data.3.3. Algebraic Bethe ansatz.
The continuous time q -Boson process (see Section 3.5) was intro-duced by Sasamoto-Wadati [17] in the language of the algebraic Bethe ansatz. The generator for theprocess arises from a certain representation of the q -Boson Hamiltonian, which is built (in a standardway) from quantum L and R matrices involving the q -Boson algebra. In principal the coordinateeigenfunctions produced in [16] should be accessible from the algebraic Bethe ansatz (though thismapping of the algebraic to coordinate eigenfunctions have not been performed). The coordinateeigenfunctions of the ( q, µ, ν )-Boson process are different than those for the q -Boson process (theydepend non-trivially on ν ). This suggests that if one could fit the discrete time ( q, µ, ν )-Bosonprocess into the algebraic Bethe ansatz it would require use of a modified L matrix. When ν = 0the coordinate eigenfunctions for the ( q, µ, q -Boson process.However, even with ν = 0, it has not yet been determined how the ( q, µ, HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 17 from the algebraic Bethe ansatz transfer matrix. For a further discussion on this, see [9, Section1.2.4].3.4.
Measures on interlacing partitions and symmetric functions.
The nested contour in-tegral moment formulas of Theorem 1.8 are reminiscent of formulas which arise in the theory ofMacdonald processes [4]. When ν = 0, [5, Section 6] makes a very clear link between these formulasand Macdonald processes (as well as between the q -Boson process and commutation relations involv-ing Macdonald first difference operators). This ν = 0 link is facilitated by the fact that there existnested contour integral formulas for moments of the general a i parameter ( q ; µ ; 0)-Boson processconsidered in Section 2. As observed in Section 2.4, it is not clear how to produce such formulaswhen ν = 0 (or whether such formulas exist). Hence, it remains unclear whether there exists ananalogous theory to that of Macdonald processes which relates exactly to the ( q, µ, ν )-Boson process.Also in the ν = 0 context, [5] introduced another discrete time variant of q -TASEP (differentthan the ( q, µ, β specialization of Macdonald processes). Thisprocess, called Bernoulli q -TASEP, was studied in [5] via the same sort duality approach utilizedherein. It is not clear whether there is a ν = 0 generalization of this Bernoulli q -TASEP which issimilarly solvable.3.5. Limits of the ( q, µ, ν ) -Boson process and ( q, µ, ν ) -TASEP. We have already alluded tothe ν = 0 degeneration of the ( q, µ, ν )-Boson process and ( q, µ, ν )-TASEP. This process coincideswith the discrete time geometric q -TASEP (and the q -Boson process related through the ASEP-ZRPand particle-hole transform) which was studied in [5]. The results of this paper generalize some ofthose in [5] to this ν = 0 setting. If we further set µ = (1 − q ) ǫ and scale time like ǫ − , then as ǫ → q -TASEP converges to the continuous time Poisson q -TASEP of [4, 10].A further limit (cf. [10, Section 6]) involving q → q, µ, ν )-Boson process and( q, µ, ν )-TASEP including: a TASEP with generalized update, a continuous time fragmentationmodel, a multiparticle hopping asymmetric diffusion process which interpolated between TASEPand the drop-push model, a discrete time zero range process involving at most one particle jumpingper-site, and the Asymmetric Avalanche Process – see [16] for the relevant degenerations, descrip-tions and references for these processes. It would be interesting to likewise degenerate Theorems1.8 and 1.9 to these models and consequently study their long-time and large-scale behavior. Forcontinuous time Poisson q -TASEP, the O’Connell-Yor semi-discrete directed polymer model and theKardar-Parisi-Zhang equation this has been done in [4, 6, 7, 13].Let us briefly illustrate one of the other degenerations mentioned above. The multiparticle hop-ping asymmetric diffusion process arises by setting µ = q ∈ (0 , ν = q − ǫ − ǫ and rescaling time bya factor of ǫ − (i.e. let t = ǫ − τ where τ will represent a continuous time parameter in the ǫ → ϕ q,µ,ν ( j | m ) = ǫ j ] q − + O ( ǫ ) where [ j ] q − = − q − j − q − .(Note: [16] contains a small mistake in the rate, which would correspond in the present notationto replacing j by m .) Hence, the continuous time limit as ǫ → q, µ, ν )-Boson processunder this scaling is as follows: for each site i and each j ∈ { , . . . , y i ( τ ) } there is an exponentialalarm clock which rings at rate j ] q − . When this occurs, j particles are moved from site i to site i − q, µ, ν )-TASEP has particle x n ( τ ) jumping to site x n ( τ ) + j for j ∈ { , . . . , x n − ( τ ) − x n ( τ ) − } according to an exponential alarm with the rate j ] q − . HE ( q, µ, ν )-BOSON PROCESS AND ( q, µ, ν )-TASEP 18
Theorems 1.8 and 1.9 both have clear degenerations for this later model. Recall that we arepresently considering the scaling µ = q , ν = q − ǫ − ǫ and t = ǫ − τ . Focusing on Theorem 1.9, the onlyterm which requires a bit of care as ǫ → g ( w )) (cid:18) ( µw ; q ) ∞ ( νq ; q ) ∞ (cid:19) t → e τ P ∞ i =0 qiw − qi +1 w . Thus Theorem 1.9 holds for the multiparticle hopping asymmetric diffusion process with g ( w ) = (cid:18) − w (cid:19) n e τ P ∞ i =0 qiw − qi +1 w qw ; q ) ∞ . We do not pursue further asymptotics of this process here.
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