Tracy-Widom limit of q-Hahn TASEP

We consider the q-Hahn TASEP which is a three-parameter family of discrete time interacting particle systems. The particles jump to the right independently according to a certain q-Binomial distribution with parallel updates. It is a generalization of the discrete time q-TASEP which is the q-deformed totally asymmetric simple exclusion process (TASEP) on Z for q in [0,1). For step initial condition, we prove that the current fluctuation of q-Hahn TASEP at time t is of order $t^{1/3}$ and asymptotically distributed as the GUE Tracy-Widom distribution. We verify the KPZ scaling theory conjecture for the q-Hahn TASEP.


Introduction
In the totally asymmetric simple exclusion process (TASEP) on the one-dimensional integer lattice Z, particles with vacant right neighbour jump to the right by 1 according to independent Poisson processes with unit rate.However it is a simple non-reversible stochastic interacting particle system, the exclusion constraint produces an interesting behaviour.There has been a lot of studies around this model and its discrete time versions.Due to the determinantal structures of correlation functions, the limiting process for particle positions or for the current fluctuations were found to be given by the Airy processes [8,9,16,19].
For a parameter q ∈ [0, 1), the q-deformation of TASEP is a particle system on Z where the jumps are independent of each other and happen with rate 1 − q gap where the gap is the number of consecutive vacant sites next to the particle on its right.This particle system is referred to as q-TASEP and it reduces to TASEP for q = 0.The q-TASEP belongs to the Kardar-Parisi-Zhang (KPZ) universality class.As it was expected by the universality conjecture, the q-TASEP shows the characteristic asymptotic fluctuation statistics of the KPZ class.Indeed, it was first shown by Ferrari and Vető in [14] that the large time current fluctuations are governed by the (GUE) Tracy-Widom distribution.This confirms the KPZ scaling theory conjecture, see also [21].
The q-TASEP was first introduced by Borodin and Corwin in [4].They investigated the q-Whittaker 2d growth model which is an interacting particle system in two space dimensions on the space of Gelfand-Tsetlin patterns.The q-TASEP is a Markovian subsystem of this two-dimensional process.Due to the connection to the q-Whittaker process, the study of Macdonald processes enabled to derive explicit formulas for expectations of relevant observables of the q-TASEP for a certain class of initial conditions.In particular for step initial condition, a Fredholm determinant formula was given by Borodin, Corwin and Ferrari in [5] for the q-Laplace transform of the particle position.The asymptotic analysis performed in [14] is based on this formula.A technical limitation of [14] was recently removed by Barraquand in [2] and the analysis was extended to the case of finitely many extra slow particles.
The totally asymmetric zero range process with a special choice of rate function, the q-Boson particle system was first introduced by Sasamoto and Wadati in [20].Its duality to q-TASEP was proved in [7] and, as a consequence, joint moment formulas for multiple particle positions were obtained for q-TASEP which in principle characterize their distribution, however they are not of Fredholm determinant form.More recently, the q-Boson particle system was analyzed in [6] and in [17] by using the Bethe ansatz: [6] provides a spectral theory for the particle system, i.e. the eigenfunctions and their properties are described; [17] focuses directly on transition probabilities of the process with finitely many particles.In these two papers, the distribution of the left-most particle's position in a finite system after a fixed time for general initial condition was characterized via moment formulas in [6] and explicitly via a multiple contour integral in [17], but neither expression seems particularly amenable to asymptotic analysis.
In [3], two natural discrete time versions of q-TASEP were introduced and Fredholm determinant expressions were proved for the q-Laplace transform of the particle positions.A further extension of q-TASEP appears in [12], the q-PushASEP which is yet another integrable particle system, a q-generalization of the PushASEP which was studied in [8].An explicit contour integral formula was derived for the joint moments of particle positions in [12] and a Fredholm determinant formula for the q-Laplace transform of particle positions was conjectured.
In the recent paper [18], Povolotsky introduced a three-parameter family of discrete time particle systems on Z which was referred to as q-Hahn TASEP or (q, µ, ν)-TASEP in subsequent works.Using the duality of the q-Hahn Boson process and the q-Hahn TASEP, Corwin derived a Fredholm determinant formula for the q-Laplace transform of the particle position in q-Hahn TASEP with step initial condition in [10].This formula is used as a starting point of the asymptotic analysis carried out in the present paper, see Theorem 3.1 below.A certain part of the proof of this formula was recently simplified by Barraquand in [1].
The asymptotic analysis performed in this paper shows similarities with the one in [5] and the one in [14].In all of these cases, one of the main difficulties lies in the choice of the contours for the Fredholm determinant: they have to be a steep descent paths for the asymptotic analysis but also the extra singularities of the integrand have to be controlled.The contours that we choose in the present case are circular and they are shown on Figure 3. Since the present analysis covers a model with three parameters, the proof of the steep descent property along the contours here is more general and parallelly also more involved as in earlier works.It was necessary however to impose the technical conditions (2.14)-(2.15) on the parameters of the model.The present proof of the steep descent property only works under the technical condition (2.14) which however excludes the application of the present results to the discrete time geometric q-TASEP, see [3].On the other hand, condition (2.15) ensures that no residues coming from the sine in the denominator of the integrand have to be encountered.For the removal of this condition, another version of Theorem 3.1 has to be deduced where the contour for the Fredholm determinant is pulled over certain singularities.This extension requires further ideas about the blow up near the singularities and it is subject to future work.
The paper is organized as follows.We introduce the q-Hahn TASEP model and describe the main result on the fluctuation of the particle position in Section 2. Section 3 contains the pre-asymptotic Fredholm determinant formula for q-Hahn TASEP which was proved in [10].We also show how the main result of the paper follows from the convergence of the corresponding Fredholm determinants.The rest of the paper is devoted to the asymptotic analysis: Section 4 contains the main steps of the analysis as propositions; the complex contours which are suitable for asymptotics are given and proved to be steep descent in Section 5; finally the propositions are proved in Section 6.

Model and main result
We start with the definition of the q-Hahn TASEP with step initial condition and further notations.Let q ∈ (0, 1).The q-Pochhammer symbol is given by for any a ∈ C and n integer.The definition naturally extends to the infinite q-Pochhammer symbol (a; q) ∞ which is meant as an infinite product.For a fixed q ∈ (0, 1) and 0 < ν < µ < 1 and integers 0 ≤ j ≤ m, define the weights of the q-Binomial distribution as 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 .
When m = ∞, extend this definition by setting ϕ q,µ,ν (j|∞) = µ j (ν/µ; q) j (µ; q) ∞ (ν; q) ∞ 1 (q; q) j . (2. 3) The q-Hahn TASEP is a discrete time interacting particle system on Z with parallel updates that consists of the evolution of particles X(τ ) = (X N (τ ) : N ∈ Z or N ∈ N) for τ ≥ 0. The particles are numbered from right to left.For the Nth particle at time τ , given that the number of vacant sites to the right of it is m = X N −1 (τ ) − X N (τ ) − 1, the particle at X N (τ ) jumps to the right by j with probability ϕ q,µ,ν (j|m) independently of the others.Jumps of different particles happen at the same time with parallel updates.Note that the dynamics preserves the order of particles.
Step initial condition means that the particles are initially at all negative integer positions, i.e. there are only particles with labels N = 1, 2, . . .and they are initially at be the q-gamma function.Then the q-digamma function is defined by  The macroscopic shape of the positions of q-Hahn TASEP particles for q = 0.2, µ = 0.4, ν = 0.3 which is given by the parametric curve (f /κ, 1/κ).
Definition 2.2.Let q ∈ (0, 1) be fixed and choose a parameter θ > 0. To these values, we associate the parameters f ≡ f (q, µ, ν, θ) = κ Ψ q (θ + log q µ) − Ψ q (θ + log q ν) + Ψ q (θ + log q ν) − Ψ q (θ) log q , (2.7) It turns out that explicit formulas for the quantities above are only available in terms of the parameter θ which appears naturally in the asymptotic analysis of the problem.It could however be possible to parameterize the problem by κ since it corresponds to the macroscopic position where we focus on as explained below.The parameters f and κ describe the global behaviour of the particle system.The law of large numbers holds for the position of the Nth particle after time κN as N → ∞.
In order to visualize the macroscopic behaviour (2.10), consider the evolution of the points (X N (τ ) + N, N) in the coordinate system.For τ = 0, these points all lie on the positive half of the vertical axis.For τ large and after rescaling the picture by τ , the points are macroscopically around (f /κ, 1/κ) which is a curve that can be parameterized by θ and it is shown on Figure 1.By computing limits using (2.6)-(2.7),one can see that the curve (f /κ, 1/κ) touches the axes at ((Ψ q (log q µ) − Ψ q (log q ν))/ log q, 0) for θ → 0 and at (0, (µ − ν)/(1 − ν)) for θ → ∞.It means that the right-most q-Hahn TASEP particle has speed (Ψ q (log q µ) − Ψ q (log q ν))/ log q and that the left-most particle which has already started moving after time τ is around the position −(µ − ν)τ /(1 − ν).
In this paper, we study the fluctuations of particle X N around the deterministic macroscopic position given by (2.10).One expects by KPZ universality that these fluctuations are of order O(N 1/3 ) and have Tracy-Widom statistics (see the review [13]).Further, at a given time τ = κN, particles are correlated over a scale O(N 2/3 ) and their limit process is the Airy 2 process.It is also expected by [11] that the same limit process arises for the position X N at times of order N 2/3 away from κN as it was shown for TASEP in [15].Therefore it is natural to consider for any c ∈ R the scaling with κ, f , χ and φ given in Definition 2.2.It means that on the top of the macroscopic behaviour given by (2.10) and governed by the parameter θ through κ and f , we allow for a smaller N 2/3 time scale on which the parameter c in (2.11) is understood as the time parameter of the expected Airy 2 process scaling limit.Hence the rescaled tagged particle position given by is expected to converge to the Airy 2 process as a process in c.Our main result is the convergence of one-point distribution of ξ N to the Tracy-Widom distribution function [22].
For our proof to work, we have to assume that for the parameters of the q-Hahn TASEP the technical conditions hold.
where F GUE is the GUE Tracy-Widom distribution function.
Remark 2.4.The technical condition (2.14) is needed for the proof of Proposition 5.2 and 5.3 to establish the steep descent property along the contours C θ and D θ .The origin of this condition is more explained in Remark 5.4.The condition (2.15) is already used in the first step of the proof of Proposition 4.2 in the contour deformation in order to make sure that no poles coming from the sine in the kernel K x given by (4.3) have to be encountered.The upper bound on θ is plotted as a function of q on Figure 2. Note that the condition (2.15) implies in particular that q < 1/3 and θ ∈ (0, 1).We expect that condition (2.15) could be eliminated or weakened via establishing another version of Theorem 3.1 that allows for reaching larger values of θ.This requires new ideas about the blow up of the kernel near the singularities and a control on the function f 0 at the points of poles coming from the sine in the denominator as in [5] and in [14].

Finite time formula and proof of the main result
The first part of Theorem 1.10 in [10] gives the following Fredholm determinant expression for the q-Laplace transform of the particle position in q-Hahn TASEP with step initial condition.
where C 1 is a positively oriented circle containing 1 with small enough radius so as to not contain 0, 1/q and 1/ν.The operator K ζ is defined in terms of its integral kernel where Proof of Theorem 2.3.With the scaling (3.4) of ζ on the left-hand side of (3.1), one has This implies that (5.3) of [14] also holds in this case, so the rest of the proof follows exactly the same steps as in Section 5 of [14] about the convergence of the q-Laplace transform, hence we only give a brief outline below.In particular, it follows from (3.6) that the left-hand side of (3.1) converges to P(ξ N < x) the distribution function of ξ N at x as N → ∞ when ζ is rescaled via (3.4).As it is proved in Section 5 of [14], Theorem 4.1 on the convergence of the right-hand side of (3.1) to the GUE Tracy-Widom distribution function is enough for the weak convergence of ξ N and for the proof of Theorem 2.3.

Asymptotic analysis
The theorem below is the most important input for the Tracy-Widom limit of the rescaled position in q-Hahn TASEP.
Theorem 4.1.Let x ∈ R be fixed and choose ζ according to (3.4).Suppose that for the parameters of the q-Hahn TASEP, the conditions (2.14)-(2.15)hold.Then as N → ∞.
In order to perform the asymptotic analysis, we substitute (2.11) and (3.4) for the values of τ and ζ into (3.2) and perform the change of variables 2) The kernel which we get is and β x as in (3.5).The contours for the Fredholm determinant and for the integral defining the kernel transform under the change of variables (4.2) as follows.The contour for w and w ′ was originally C 1 , a small circle around 1, hence the contour for W and W ′ can be chosen to be C 0 which is a small circle around 0 that does not contain the singularities at −1 and at − log q ν.If this circle is small enough, the contour for Z becomes a small perturbation of 1/2 + iR which can be shifted to θ + iR without crossing any singularity of the integrand since (2.15) means in particular that θ ∈ (0, 1).Hence the choice for the Z contour in (4.3) is appropriate and we can write the equality of the Fredholm determinants Theorem 4.1 follows from the series of propositions below.In the propositions without repeating everywhere, we assume that for the parameters of the q-Hahn TASEP, the conditions (2.14)-(2.15)hold.To state the first proposition, we introduce the V-shaped contour where θ > 0 is the tip of the V, ϕ ∈ (0, π) is its angle and δ ∈ R + ∪ {∞}.We also introduce the kernel where W, W ′ ∈ V δ θ,π−ϕ .The dependence of the kernel on ϕ is not indicated in the notation.Note that K x,δ only differs from K x by the integration contours.Proposition 4.2.For any fixed δ > 0 and ε > 0 small enough, there are ϕ ∈ (0, π/2) and N 0 such that for all N > N 0 By defining the rescaled kernel the change of variables shows that (4.12) Next we show that on the contour V δN 1/3 0,π−ϕ , the kernel K N x,δ can be replaced by the one obtained by using the Taylor series.Proposition 4.3.For any fixed ε > 0 small enough, there is a small δ > 0 and an N 0 such that for any N > N 0 , where as N → ∞.
Proposition 4.5.We can rewrite the Fredholm determinant where and F GUE is the GUE Tracy-Widom distribution function.

Steep descent contours
This section is devoted to establish contours along which the function with principal contribution in the exponent is steep descent.We first define these contours which are also shown on Figure 3 along with the contourplot of the function Re(f 0 ).
and let C θ be the image of C θ under the map w → log q w.Proposition 5.2.Suppose that for the parameters of the q-Hahn TASEP, (2.14) holds.
Then the contour C θ is steep descent for the function − Re(f 0 ) in the sense that the function attains its maximum at q θ corresponding to s = 0, it increases for s ∈ (−π, 0) and it decreases for s ∈ (0, π).
Proposition 5.3.Suppose that for the parameters of the q-Hahn TASEP, (2.14) holds.Then the contour D θ is steep descent for the function Re(f 0 ) in the sense that the function attains its maximum at q θ corresponding to t = 0, it increases for t ∈ (−π, 0) and it decreases for t ∈ (0, π).
Remark 5.4.The condition q ≤ ν is needed for the use of the first part of Lemma 5.5 to obtain (5.13) which is an ingredient to the steep descent property in Proposition 5.2.This condition is also used in the proof of Proposition 5.3 for the application of (5.23) with α = νq k and β = q k+1 .The condition µ ≤ 1/2 is imposed in Proposition 5.3, because the inequality (5.23) could be proved for β ≤ 1/2.The 1/2 seems numerically to be close to optimal.The condition µ ≤ 1/2 could be weakened in Proposition 5.2, because (1 − q θ )µ/(1 − µ) ≤ 1 is enough to obtain (5.12) from the first part of Lemma 5.5.The latter is a weaker condition, but µ ≤ 1/2 has to be assumed for our proof of Proposition 5.3 to work.
The function given by is useful for the proof of the propositions about steep descent contours.It has the following properties.
and the inequality above is sharp for s ∈ (0, π) if b < c.
Remark 5.6.The first part of Lemma 5.5 is designed to compare the different terms in the derivative of f 0 along a circular contour given in (5.11).The inequality (5.3) is sharp in the sense that the two sides as functions of s are tangential at s = 0.The factor 1/2 in the second part of the lemma seems numerically to be close to optimal. Proof.
1.The inequality is the consequence of the fact that ∂ ∂b is non-positive for −1 < b < 1 and 0 ≤ s ≤ π and it is strictly negative for −1 < b < 1 and 0 < s < π.

Calculations show that
which is non-negative for the given parameter values and strictly positive for for t ∈ (0, π).
The series representations of the q-digamma function and that of its derivative are used in the proof below.They are expressed as (5.8) Proof of Proposition 5.2.We investigate the function − Re(f 0 ) along C θ , i.e. w(s) = 1 − re is for s ∈ [0, π] where we use the notation r = 1 − q θ as we will do throughout this proof.The case s ∈ (−π, 0] is similar.One can check by calculation that Re d ds log w(s) = g(r, s) (5.9) and that for any 0 where the k = 0 term for α = 1 is understood as the limit lim b→−∞ g(b, s) = 0. Using (5.9)-(5.10)for (4.4), we have (5.11) Note that the k = 0 term of the last summand is 0 by the observation above.
The first part of Lemma 5.
Hence in order to complete the argument, one has to compare the remainder of the terms g(−rµq k /(1 − µq k ), s) and g(−rνq k /(1 − νq k ), s) in (5.11) with g(r, s).To this end, we use the first part of Lemma 5.5 for b = −rµq k /(1 − µq k ) and c = r which gives us that and for b = −rνq k /(1 − νq k ) and c = r, we have (5.15) What remains to show is that because of the following.Let us multiply (5.14) by κ times the factor between parentheses in the first sum of (5.16) and multiply (5.15) by the factor between parentheses in the second sum of (5.16) and sum these up for k.Then add κ times (5.12) and (5.13) to the sum.This altogether is to be compared to (5.11).Note that the coefficients of g(−rµq k /(1−µq k ), s), g(−rνq k /(1−νq k ), s) and g(−rq k /(1−q k ), s) coincide.(Remember that the last term for k = 0 in (5.11) is 0.) On the other hand, the coefficients of g(r, s) are exactly the two sides of (5.16), therefore if (5.16) holds true, then the derivative (5.11) is non-positive for s ∈ [0, π] and negative for s ∈ (0, π).By multiplication in (5.16), for α ∈ (0, 1], we get terms of the form where we used that r = 1 − q θ and the series expansions (5.7)-(5.8).The right-hand side of (5.16) equals κ times the difference of (5.17) for α = µ and for α = ν and the difference of (5.17) for α = ν and for α = q.Note that the sum in (5.17) for α = q can be replaced by the one for α = 1, because the extra term is 0 as it can be seen from the second expression in (5.17).Hence the right-hand side of (5.16) can be written as 1 log q κ Ψ q (θ + log q µ) − Ψ q (θ + log q ν) + Ψ q (θ + log q ν) − Ψ q (θ) which is exactly f by (2.6)-(2.7)as required.

Proofs of propositions
This section contains the proofs of the proposition which lead to Theorem 4.1.For later use, note that differentiation of (4.4)-(4.6)gives the following Taylor series expansions 4 ), (6.1) Proof of Proposition 4.2.The proof consists of the following three steps.We first deform the integration contour for the kernel K x to the steep descent contour.Then we show that Proposition 4.2 holds with ϕ = π/2 instead of ϕ ∈ (0, π/2).In the last step, we deform the short contours so that we get the statement for ϕ ∈ (0, π/2).
Step 1: Contour deformation.The first observation is that as long as conditions (2.14)-(2.15)hold, then i.e. the contour C 0 can be blowed up to C θ .This simply follows from the Cauchy theorem since the singularities coming from f 0 at − log q µq k , − log q νq k and log q q k for k = 0, 1, 2, . . .are by condition (2.14) all smaller than log q 2 and the point of C θ with the smallest real part is log q (2 − q θ ).One the other hand, the condition (2.15) ensures that no pole coming from the sine in the denominator is crossed along the deformation, since the real part of the points of C θ is between log q (2 − q θ ) and θ and the difference of the two is assumed to be less than 1 by (2.15).
Step 2: Localization to short contours.In this step, we prove the statement of the proposition with ϕ = π/2 instead of ϕ ∈ (0, π/2).Recall from Definition 5.1 that s → w(s) parametrizes the contour C θ and hence s → log q w(s) parametrizes the contour C θ .It allows for writing the Fredholm determinant as . ( The kernel in (6.5) diverges logarithmically in the neighbourhood of s i = 0, but it is bounded otherwise.More precisely, there is a constant C such that |K x (log q w(s), log q w(s On the other hand, we use the fact that the contour C θ is steep descent for the function W → − Re(f 0 (q W )) as an immediate consequence of Proposition 5.2.Since this function gives the main contribution in the exponent in the W variable, the kernel converges to 0 exponentially as N → ∞ for all W ∈ C θ except for a δ-neighbourhood of θ.Hence by dominated convergence, the integral along the contour C θ in the Fredholm series of K x in (6.5) can be neglected apart from a δ-neighbourhood of θ by making an error of order O(exp(−cδ 3 N)) for some c > 0. Keeping the endpoints of the remaining contour, it can be replaced by V δ θ,π−ϕ for some ϕ ∈ (0, π/2) by Cauchy's theorem.Note that in the last step, the orientation of the contour changes.
With a similar argument, we can localize the Z-contour as well.By linearity, one can take out the Z integrations from the determinant in (6.5) to obtain a sum where the kth term is a 2k-fold integration.It is still integrable, since the behaviour in the Z variables is e −π Im(Z) due to the sine in the denominator.The function Re(f 0 (q Z )) is periodic along the contour θ + iR with period 2π/| log q| in the imaginary direction.The contour {θ + it : t ∈ [π/ log q, −π/ log q]} is however steep descent for Re(f 0 (q Z )) by Proposition 5.3.Therefore, the steep descent property and the periodicity implies that by making an exponentially small error in N, we can restrict the Z integral to the set ∪ k∈Z I k where Now we argue that in the limit, only the integral over I 0 survives.Let us consider the change of variables Now the term N −1/3 / sin(π(W − Z)) for k = 0 converges to 1/(π(w − z)), whereas for k = 0, we have which is a summable but it is of smaller order than the term for k = 0.It means that the integral over ∪ k∈Z I k can be replaced by the one over I 0 in the limit.This proves the proposition for ϕ = π/2.
Step 3: Deformation of short contours.What we show in this step is that the contour for the Z integral in the kernel K x,δ that can be taken to be a segment from θ −iδ to θ + iδ can be replaced by V δ θ,ϕ by possibly choosing a smaller δ.First with the ϕ obtained in the localization of the W contour and by using Cauchy's theorem, we replace the integration path for Z by the union of where we mean four segments in the complex plane with the given endpoints.The function Z → f 0 (q Z ) behaves around θ as f (q θ ) + (Z − θ) 3 in the leading order by the Taylor expansion (6.1).On the other hand by Proposition 5.3, we know that the value of Re(f 0 (q Z )) as Z ∈ θ + iR is smaller than f 0 (q θ ).Hence by taking δ sufficiently small, we can achieve that Re(f 0 (q Z )) as Z ∈ S 1 ∪ S 4 is strictly smaller than f 0 (q θ ).Similarly to the second step of this proof, we can further reduce the integration path to S 2 ∪ S 3 = V δ θ,ϕ by making an exponentially small error in N.This completes the proof.
In order to get that the Fredholm determinants are also close, we need a uniform fast decaying bound on K N x,δ (w, w ′ ).The main term in the exponent is − N Re f 0 (q W ) = −χ Re for any ǫ > 0 where O ǫ (w 3 ) means that the error term is at most ǫw 3 .By taking δ small enough, ǫ can be arbitrarily small, hence the error term is negligible compared to the cubic behaviour of −χ Re w 3 /3.The error terms coming from f 1 and f 2 in the exponent are similarly dominated.Hence the difference of the Fredholm determinants goes to 0 as N → ∞ by dominated convergence, which proves the proposition.
Proof of Proposition 4.4.Since the integrand in (4.14) has cubic exponential decay in w and z along the given contours V ∞ 0,π−ϕ and V ∞ 0,ϕ respectively, the convergence of the Fredholm determinants follows by dominated convergence similarly to the earlier proofs.
Hence we can write as required.