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The KPZ fixed point for discrete time TASEPs

We consider two versions of discrete time totally asymmetric simple exclusion processes (TASEPs) with geometric and Bernoulli random hopping probabilities. For the process mixed with these and continuous time dynamics, we obtain a single Fredholm determinant representation for the joint distribution function of particle positions with arbitrary initial data. This formula is a generalization of the recent result by Mateski, Quastel and Remenik and allows us to take the KPZ scaling limit. For both the discrete time geometric and Bernoulli TASEPs, we show that the distribution function converges to the one describing the KPZ fixed point.


Introduction
The totally asymmetric simple exclusion process (TASEP) is a typical interacting stochastic particle system and can be interpreted as a stochastic growth model of an interface, which turns out to belong to the Kardar-Parisi-Zhang(KPZ) universality class introduced in [18]. In addition, the TASEP is one of the most basic model in the integrable probability [11]. Remarkable algebraic structures allow us to obtain exact explicit forms of distribution functions for some quantities.
On a macroscopic level, the particle density evolves deterministically according to the Burgers equation [30,31]. Therefore, a natural question is to focus on fluctuation properties around the determistic growth, which exhibit universal properties characterizing the KPZ class. There are many important results in the literature of the integrable probability. First, for the step initial condition, the one-point limiting distribution for the particle current in the TASEP has been obtained by Johansson [15] by converting the problem to the last passage percolation and then using the RSK correspondence. It turned out that the limiting distribution is the GUE Tracy-Widom distribution. In [21,29], this result can also be obtained by using an explicit form of the the transition probability in the TASEP [33]. Similar results have been found for symmetrized last passage problems by Baik-Rains [2]. The results include the one-point limiting distribution of the particle current for the alternating initial condition in the language of the TASEP or equivalently, the height distribution for the flat initial condition in the language of the growth process called the polynuclear growth (PNG) model [24]. In this case, the limiting distribution is turned out to be the GOE Tracy-Widom distribution.
These results on the one-point fluctuations have been generalized to the case of the multipoint fluctuations. For the case corresponding to the step initial condition, a Fredholm dterminant formula for the limiting multi-point distributions has been first obtained in the PNG model with space-time continuous setting [25] by using the technique related to the RSK correspondence. The same result has been obtained for the discretized PNG model [16]. The limiting process characterized by the multi-point distribution is called the Airy 2 process. On the other hand, for the other conditions, the first important result has been given in [32]. Sasamoto has developed the technique for obtaing the mult-point function in terms of the transition probability in TASEP [33] not only for the step initial condition but also for the alternating one and has obtained a Fredholm determinant formula for the limiting functions in the alternating case. The process characterized by the multi-point distribution is now called the Airy 1 process. This approach in [32] has been further studied and been applied to the TASEP and the PNG model with different settings [5,7,8,9].
We have been interested in the entire structure of the universal limiting process for more general initial data. Our understanding of this problem has been advanced by the recent result by Matetski, Quastel, and Remenik [20]. A Fredholm determinant formula for an arbitrary initial data has already been obtained in [8] based on the approach developed in [32]. The correlation kernel for the Fredholm determinant can be expressed in terms of the biorthogonal functions, say Φ k (x) and Ψ k (x). The problem is that one of them, say Φ k (x) does not have an explicit representation while Ψ k (x) does. Thus it had not been clear how to take the KPZ scaling limit of this kernel. [20] has overcome this situation. They represent the function in terms of a stopping time of the geometric random walk. This expression allows us to take the KPZ scaling limit since by Donsker's invariance principle, we easily find this stopping time converges to the one for the Brownian motion in the KPZ scaling limit. Based on this technique, the limiting multi-point distribution functions for the particle positions in the arbitrary initial condition has been obtained. The process with this multi-point distribution is called the KPZ fixed point. Recently various interesting progresses on this problem have been made for example in [22,23,27].
In this paper, we show that the technique in [20] can be applicable to different versions of the TASEP beside the usual continuous time one. In particular, we focus on two versions of the discrete time TASEPs: the geometric random hopping with parallel update and the Bernoulli random hopping with sequential update. Furthermore, in both case, we consider the case where the hopping probabilities are time-dependent. For the step initial condition these dynamics has appear as a special case of the higher spin vertex model and has been recently studied in [19]. To the best of our knowledge, however, the analyses for the arbitrary initial condition has not been studied yet. We show Schütz's type determinantal formulas for transition probabilities for both the geometric and Bernoulli TASEP with time dependent hopping probabilities. Combining these with Schütz's formula for the continuous time TASEP, we get the determinantal transition probability for the system mixed with the three types of dynamics. Using this, we obtain a Fredholm determinant formula for the multi-point distribution for the particle positions, in which we can take the KPZ scaling limit. This is a generalized formula to the one [20]: When we vanish the whole parameters of the mixed dy-namics except the part of the continuous time TASEP, the determinantal formula is reduced to the result in [20]. Finally taking the KPZ scaling limit for both discrete time geometric and Bernoulli TASEP, we see that the multi-point distribution functions converges to the one describing the KPZ fixed point.
The paper is organized as follows. In Sec. 2, we state the three versions of the TASEPs, continuous time and two types of discrete time version: geometric and Bernoulli random hopping. Their mixed version is also stated. We also give our main result: the Fredholm determinant formula for the mixed TASEP (Theorem 2.4) and the KPZ scaling limit in two cases of the geometric and Bernoulli TASEPs (Theorem 2.11, and Propositions 2.14 and 2.15). In Sec. 3, after giving the determinantal formulas for the transition probabilities for the above three types of TASEPs, we give the proof of Theorem 2.4 using the framework developed in [20]. In Sec. 4, we give the proofs of Theorem 2.11, and Propositions 2.14 and 2.15. The crucial step is the saddle point analysis for the kernels.

Models and results
In this section we define three versions of the TASEP and introduce our main results.

Models
In this paper we consider the TASEPs on Z. Each particle jumps only to the right independently and stochastically if the target site is empty. If the site is occupied by the other particle, it cannot move, which represents the exclusion interaction.
In the TASEPs we mainly focus on the position of each particle. Let X t (i) ∈ Z be a position of the ith particle at time t. We set t ∈ Z or t ∈ R according to the version. Since the dynamics of the TASEPs preserves the order of the particles, we can always assume, · · · < X t (2) < X t (1) < X t (0) < X t (−1) < X t (−2) < · · · . The particles at ±∞ are playing no role in the dynamics when adding ±∞ into the state space.

Continuous time TASEP
The continuous time TASEP on Z was introduced in [35] in the literature of mathematics. In this case t ∈ R ≥0 and each particle independently attempts to jump to the right neighboring site at rate γ ∈ R ≥0 provided this site is empty. It is a continuous time Markov process with the generator L defined as follows: Let η = (η x ) x∈Z ∈ {0, 1} Z be a particle configuration. η x = 1 means the site x is occupied by a particle while η = 0 means it is empty. The generator L acting on cylinder functions f : where η x = 1, if the site is occupied by a particle, 0, if the site x is empty, and η x,x+1 η y otherwise.

Discrete time Bernoulli TASEP with sequential update
We define the discrete time Bernoulli TASEP with sequential update on Z. This version was studied previously in [6] as a marginal of dynamics on Gelfand-Tsetlin patterns which preserve the class of Schur processes and more recently in [4,19] in the studies of the integrable probability. Let us assume the particle configurations at time t ∈ Z ≥0 as X t (j) = a j , j ∈ Z. The particle positions at time t + 1 are determined by the following update rule: We update the position of the ith particle X t+1 (i) in increasing order. Suppose that we already updated the i − 1th particle and its position is b i−1 i.e. X t+1 (i − 1) = b i−1 . Then the update rule is given as follows: for a = a i + 1, 0 otherwise.
This dynamics mean that starting from right to left, for the time step t → t + 1, the ith particle jumps to the right neighboring site with probability p t+1 ∈ (0, 1) provided this site is empty. Since the update is sequential from right to left, during a time step, a block of consecutive particles can jump. For later use, we define β t , t = 0, 1, 2, · · · by

Discrete time geometric TASEP with parallel update
We define the discrete time geometric TASEP with parallel update on Z. This was studied previously in [36] as a marginal of dynamics on Gelfand-Tsetlin patterns which preserve the class of Schur processes. More recently it has been also investigated in [4,19]. Let us assume that for t ∈ Z ≥0 and j ∈ Z, X t (j) = a j . The update rule of the positions at time t + 1 are given as follows: For each 1 ≤ i ≤ N , where the update is independent for each i and t.
Note that in this dynamics, the jth particle can jump with multiple cites according to the truncated geometric distribution defined in (2.2) with parameter α t .
2.1.4 TASEP α,β,γ : TASEP mixed with the continuous time TASEP and the discrete time TASEPs In this paper, we consider the TASEP combined with the above three versions. First we take three time parameters t 1 , t 2 ∈ Z ≥0 and t 3 ∈ R ≥0 . Then particles evolve according to the discrete time geometric TASEP with parameter α := {α 1 , α 2 , · · · , α t 1 }(Sec. 2.1.3) from time 0 to t 1 , the discrete time Bernoulli TASEP with parameter β = {β t 1 +1 , β t 1 +2 , · · · , β t 1 +t 2 } (Sec. 2.1.2) from time t 1 to t 1 + t 2 , and the continuous time TASEP with parameter γ (Sec. 2.1.1) from t 1 + t 2 to t 1 + t 2 + t 3 . In this paper we denote this mixed TASEP as TASEP α,β,γ This type of the mixed TASEP with t 3 = 0 has been introduced in [19]. We decided the order of the three dynamics as above. In fact the distribution of the particles' positions remain unchanged if we freely exchange order of these dynamics since the semigroups of all the three dynamics are shown to be exchangeable thanks to the Yang-Baxter relations [10,12].

Results
In this subsection, we give our main results.

Joint distribution of the particle positions
Here we give a single Fredholm determinant formula for joint distribution of the particle position in TASEP α,β,γ defined in Sec. 2.1.4. For the descriptions of the results below including the followin one, we state some definitions.
Definition 2.2. Let RW m , m = 0, 1, 2 · · · be the position of a random walker with Geom[ 1 2 ] jumps strictly to the left starting at some fixed site c, i.e., where χ i , i = 1, 2, · · · are the i.i.d. random variable with P(χ i = k) = 1/2 k+1 , k = 0, 1, 2, · · · . We also define the stopping time where τ is the hitting time of the strict epigraph of the curve (X 0 (k + 1)) k=0,...,n−1 by the random walk RW k . When the number of particles is N, X 0 (m) is constant and defined only m ≤ N .
Remark 2.5. In the case of continuous time TASEP, i.e. the special case α i = β j = 0 with 1 ≤ i ≤ t 1 , t 1 + 1 ≤ t 1 + t 2 , this formula has been obtained in Theorem 2.6 in [20]. Theorem 2.4 above is the generalization of the result in [20] which involves the cases of the discrete time TASEPs as well as the continuous time one.

The Kardar-Parisi-Zhang (KPZ) scaling limit
Here we state our result on the scaling limit of the joint distribution function in Theorem 2.4. We focus on the following two cases: • The discrete time Bernoulli TASEP (Sec. 2.1.2) In the TASEP α,β,γ introduced in Sec. 2.1.4, the case is realized by the specialization • The discrete time geometric TASEP (Sec. 2.1.3) As above, it is realized by To see the universal behavior of the fluctuations, we focus on the height function defined as follows.
Definition 2.6. For z ∈ Z, the TASEP height function related to X t is given by denote the label of the rightmost particle which sits to the left of, or at, u at time t and we fix h 0 (0) = 0.
Note that it can be represented as We can extend the height function to a continuous function of x ∈ R by linearly interpolating between the integer points.
It is well known that the TASEP belongs to the Kardar-Parisi-Zhang (KPZ) universality class. Thus we expect that the proper scaling of the height function is On average the height of the TASEP grows as t 1 with speed A, which is a constant. On the other hand the fluctuation of the height around the average is of order t 1/3 contrary to the t 1/2 of the usual scaling in the central limit theorem. The scaling exponent of the x-direction is 2/3, the twice of the one in h-direction 1/3, which suggest that the path of the height function becomes the Brownian motion like. The exponents (1/3, 2/3) are known to be universal and characterizing the KPZ universality class while the constants A, B, C are not universal and depend on the models. As shown in Sec. 4, we have Based on the property of the height function, we define the scaled height, which is equivalent to (2.17) and (2.18) but a slightly different form appearing as the "1:2:3 scaling" in [28].
Definition 2.7. For t ∈ R ≥0 and x ∈ R, we define the scaling height function as the following.
• The discrete time Bernoulli TASEP where t and x are scaled as where t and x are scaled as Our goal is to compute the ε → 0 limit of the joint distribution function, Here P h ε 0 (·) represents the probability measure in which the initial height profile is h ε (0, x). We will show that the limit converges to the joint distribution function characterizing the KPZ fixed point introduced in [20].
Here we introduce the KPZ fixed point. First we define UC and LC as follows.
Definition 2.8. (UC and LC [20]). We define UC as the space of upper semicontinuous functions h : Now we are ready to state the KPZ fixed point. For more detailed information, see [20].
Definition 2.9 (The KPZ fixed point [20]). The KPZ fixed point is the unique Markov process on UC, ( h(t, ·)) t>0 with transition probabilities given by Here in LHS, x 1 < x 2 < · · · < x m ∈ R and a 1 , . . . , a m ∈ R, h 0 ∈ UC and P h 0 means the measure on the process with initial data h 0 . In RHS, the kernel is given by where (A) * represents the adjoint of an integral operator A, and B(x) is a Brownian motion with diffusion coefficient 2 and τ ′ is the hitting time of the hypograph of the function h.
Remark 2.10. (2.25) and (2.26) can be written in terms of the differential operators S t,x = In addition using the integral representation for the Airy function where is the positively oriented contour going the straight lines from e − iπ 3 ∞ to e iπ 3 ∞ through 0, we find that S t,x (v, u) (2.26) can be expressed as Now we assume that the limit exists. Note that by(2.13) and (2.29), (2.31), this assumption is rewritten as and the left hand side is interpreted as a linear interpolation to make it a continuous function of x ∈ R and we chose the frame of reference by i.e. the particle labeled 1 is initially the rightmost in Z <0 . Under this assumption, we have the following result for the limiting joint distribution function (2.23).
Remark 2.12. The One-sided fixed point formula for the continuous time TASEP has been given in Proposition 3.6 in [20]. Our theorem 2.11 indicates that Bernoulli TASEP and geometric TASEP settle into the same class "KPZ fixed point. The KPZ fixed point is believed to be the universal process for the KPZ class with arbitrary fixed initial data. Our result supports this universality.
Remark 2.13. In fact we can remove the assumption h 0 (x) = −∞ for x > 0 in the above theorem by using the similar argument in Theorem 3.8. in [20].
To prove Theorem 2.11, we use the following relationship between the particle positions X j (t) and the height function h t (z) (2.13). Let s 1 , . . . , s k , m 1 , . . . , m k ∈ R and z 1 , . . . , z k , n 1 , . . . , n k ∈ Z. We have which follows from the definitions of h t (x) (2.13). By this relation, we see where a 1 , . . . , a m ∈ R and t, n j , x j are scaled as (2.36) Thus we find that our goal, LHS of (2.34), can be obtained by taking the ε → 0 limit of the expression (2.5) t,x (v, u) are given by (2.26) and (2.27) respectively and with Γ 0 being a simple counterclockwise loop around 0 not enclosing 1, 1/p and (1 − p)/p.
Next, we state that the point wise convergence for the discrete time geometric TASEP is obtained as the following.

Transition probabilities
Let be the Weyl chamber, whose elements express the particle positions of the TASEPs.
The main object of this subsection is the transition probability of the TASEP: For x, y ∈ Ω N , we define which means the probability that at time t the particles are at positions x N < · · · < x 2 < x 1 provided that initially they are at positions y N < · · · < y 2 < y 1 . For all the three types of the TASEPs introduced in Sec.
where Γ 0,1 is any simple loop oriented anticlockwise which includes w = 0 and w = 1.
Next we introduce the result on the discrete time Bernoulli TASEP as follows.
Proof. This determinantal formula has been obtained for the time homogeneous case β 1 = β 2 = · · · = p 1 − p in [7] and [29]. If we confirm that the following two equations hold, one easily find that the result can be extended to the time inhomogeneous case: It is easy to see that the above two equations hold.
We also give the result on the discrete time geometric TASEP introduced in Sec.2.1.3.
Proof. We will check that the determinantal representation (3.8) satisfies the Kolmogorov forward equation.
where µ can take the empty set φ,μ := {1, . . . , N − 1} \ µ, |μ| means the number of elements inμ, and we define k i and x (µ) := (x µ N , . . . , x µ 1 ) by RHS in (3.9) consists of 2 N −1 terms and each element j in the subset µ represents the label of the particle which is on x j+1 + 1 at time t. Taking the hopping probability (2.2) in the geometric TASEP into account, we see that when j ∈ µ, we should assign the weight α ♯jump t+1 without the factor 1 − α t+1 for the jump of the j + 1th particle. Thus for the jumps of the N − |µ| = |μ| + 1 particles, we put the factor (1 − α t+1 ) |μ|+1 . In Appendix, we explain (3.9) in the case of N = 3.
We see that (3.9) is equivalent to the following two conditions for k = 1, . . . , N − 1. Now we show that (3.11) and (3.12) imply (3.9). First, we show below the equivalence by using the equation (3.12) with k = 1 and the version of N − 1 particles in (3.13), In LHS of (3.13), we divide the sum of a 1 as follows.
For n > 0, F (α) n is determined by the recurrence relation together with the property that F (α) 0 (x, t) = 0 for x large enough. In order for (3.27) to be satisfied for all n, we need to take the poles both at 0 and 1.

Biorthogonal ensembles for the joint distribution functions
In the following we consider the joint distribution function of the particle positions in the TASEP α,β,γ introduced in Sec. 2.1.4. We will give a formula in terms of a Fredholm determinant whose kernel can be written in an explicit form.

(3.30)
Here the right hand side is a Fredholm determinant with the kernel where Q n (x i , x j ) represents n-times convolution of Q(x, y) = 1/2 x−y · 1 x>y . The functions Ψ n k (x) and Φ n k (x), k = 0, . . . , n − 1 are defined as follows: For Ψ n k (x) with k ≤ n − 1, we define where Γ 0 is any positively oriented simple loop including the pole at w = 0 and f α,β,γ (w, t) is defined by (3.29). The functions Φ n k (x), k = 0, . . . , n − 1, are defined implicitly by (1) The biorthogonality relation x∈Z Ψ n k (x)Φ n l (x) = 1 k=l ; (2) 2 −x Φ n k (x) is a polynomial of degree at most n − 1 in x for each k.
Remark 3.6. The fact that the joint distribution of particle positions can be expressed by the Fredholm determinant is proved by [7] for the discrete time Bernoulli TASEP and by [8] for the continuous time TASEP. The above result includes a generalization of the initial conditions for particle position in the results of distribution of particle position of [7] and [8], and is the result when The continuous time TASEP, the discrete time Bernoulli TASEP and the discrete time geometric TASEP are mixed.
Proof. This proof can be proved in the same way as Theorem 4.3 in [28] by using the propositions and lemmas written in Chapter 4 of [28]. Therefore, only the outline of the proof is described below. (See [28] for more details.) From the proof of Proposition 3.2 in [7] and the proof of Theorem 2.1 in [8], we can found that if the following three equations are satisfied, the proof can be done regardless of the form of f α,β,γ (w, t): Therefore, it is sufficient for us to check the above three equations, but it is not hard to confirm that the above three equations hold. This completes the proof.
In the following, we will write Φ n k (x) that was not explicitly written in previous research [7] in an explicit form.
First, we prepare the tools to use. Q m can easily be taken from definition Q; As operators on ℓ 2 (Z), Q and Q m are invertible; Note that Ψ n 0 = R α,β,γ,t δ X 0 (n) with δ y (x) = 1 x=y . Then, the following lemma holds.
Theorem 3.8. Fix 0 ≤ k < n and consider particles at X 0 (1) > X 0 (2) > · · · > X 0 (n). Let h n k (l, z) be the unique solution to the initial-boundary value problem for the backwards heat equation Then the functions Φ n k from Theorem 3.5 are given by Here Q * (x, y) = Q(y, x) is the kernel of the adjoint of Q (and likewise for R * α,β,γ,t ).
Remark 3.9. It is not true that in general Q * h n k (l + 1, z) = h n k (l, z). In fact, Q * h n k (k, z) is divergent from the following.
Proof. This proof is almost the same as [20]. We show that the same can be proved in discrete time.
The existence and uniqueness of solutions of (3.39a)-(3.39c) is elementary consequence of the fact that ker(Q * ) −1 has dimension 1 and it is spanned by the function 2 z , which allows us to march forwards from the initial condition h n k (k, z) = 2 z−X 0 (n−k) uniquely solving the boundary value problem h n k (l, X 0 (n − k)) = 0 at each step. First, we prove that 2 −x h n k (0, x) is a polynomial of degree at most k. We use the mathematical induction. By (3.39b), ) is a polynomial of degree at most k − l for some 0 < l ≤ k. By (3.39a) and (3.35), Taking the sum over x ≥ X 0 (n − l + 1), one sees Therefore, using (3.39c), we have h n k (l − 1, x) = − x y=X 0 (n−l+1)+1 2 −y h n k (l, y).
By the induction hypothesis, h n k (l − 1, x) is a polynomial of degree at most k − l + 1 because h n k (l, y) is a polynomial of degree at most k − l *1 .
Similarly, taking the sum x < X 0 (n − l + 1), we get h n k (l − 1, x) = X 0 (n−l+1) y=x+1 h n k (l, y),which is again a polynomial of degree at most k − l + 1. From the above, it was shown that 2 −x h n k (0, x) is a polynomial of degree at most k. Now, we show that y∈Z h n k (0, y)R −1 α,β,γ,t (y, z), which is the rhs of (3.40), satisfies the condition *1 This can be understood from Faulhaber's formula : Bernoulli number.
(2) in Theorem 3.5. By (3.38), we have (3.42) Because 2 −z h n k (0, z) is a polynomial of degree at most k, it is enough to note that the sum is a polynomial of degree at most k in z as well. Next, we check the biorthogonality relation (1) of Theorem 3.5. Using (3.37), we get where in the first equality we have used the decay of R α,β,γ,t and the fact that 2 −x h n k (0, x) is a polynomial together with the fact that the z 1 sum is finite to apply Fubini.

Representation of the TASEP kernel in terms of a hitting probability
Combining Theorem 3.5 with (3.37) and (3.40), we have obtained the following expression of the kernel K t (3.31), Here Q, R α,β,γ,t and R −1 α,β,γ,t are given by (3.34), (3.36) and (3.38) respectively and G 0,n j is defined by G 0,n (z 1 , z 2 ) = n−1 k=0 Q n−k (z 1 , X 0 (n − k))h n k (0, z 2 ), (3.45) where h n k is the solution of (3.39a)-(3.39c). In this subsection, following the method in [20], we further rewrite the kernel in order to take the KPZ scaling limit. We use the fact that h n k can be written as hitting probabilities of random walk. Let RW * m with RW * 0 = c be the position of the random walk with Geom[ 1 2 ] jumps strictly to the right starting from c ∈ Z, i.e.
where χ j , j = 1, 2, · · · , m are i.i.d. random variables with P(χ j = k) = 1/2 k+1 , k ∈ Z ≥0 . Note that Q * defined below (3.40) represents the transition kernel of the random walk: for m = 0, 1, · · · , we have For 0 ≤ l ≤ k ≤ n − 1 we define the stopping times where we set min ∅ = ∞. Then, we have the following. which is the probability of the walk starting at z ∈ Z at time l − 1 ∈ Z and hitting X 0 (n − k) at time k ∈ Z.
Remark 3.11. Eq. (3.48) is written in [20] and the proof is left to the readers as Exercise 5.17 in [28]. Here we give an answer.
From the memoryless property of geometric distribution we get for ∀y > X 0 (n − k), and as a consequence we get for z 2 ≤ X 0 (n), G 0,n (z 1 , z 2 ) (3.3) can be expressed as = P RW * −1 =z 2 (τ 0,n < n, RW * n−1 = z 1 ), (3.55) where in the second equality we used (3.34) and (3.54). while in the third one we used (3.46). Note that RHS of the above equation represents the probability for the walk starting at z 2 ∈ Z at time −1 to end up at z 1 ∈ Z after n steps, having hit the curve (X 0 (n − m)) m=0,...,n−1 in between. The next step is to extend the region z 2 ≤ X 0 (n) in (3.55) to z 2 ∈ Z. We begin by observing that for each fixed y 1 and n ≥ 1, 2 −y 2 Q n (y 1 , y 2 ) extends in y 2 to a polynomial 2 −y 2Q (n) (y 1 , y 2 ) of degree n − 1 withQ Now, for y 1 − y 2 ≥ 1, we note thatQ (n) (y 1 , y 2 ) = Q n (y 1 , y 2 ). Remark 3.12. We note that Using the extension of Q m , we have the following lemma. For all z 1 , z 2 ∈ Z, we have where RW m and τ are defined by Definition. 2.2.
Proof. Although the proof is given in Lemma 2.4 of [20], we give its outline for self-containedness. For z 2 ≤ X 0 (n), (3.55) can be written as where in the second equality we used the fact Q n (x, y) (3.34) represents the n-step transition probability of RW m . Let From the relation z 2 < RW τ < z 1 for ∀τ < n and (3.57) we see that forḠ 0,n (z 1 , z 2 ) = G 0,n (z 1 , z 2 ) for z 2 ≤ X 0 (n). Furthermore we find that 2 −z 2 G 0,n (z 1 , z 2 ) is polynomial in z 2 with degree at most k, and similarly 2 −z 2Ḡ 0,n (z 1 , z 2 ) is polynomial in z 2 with degree at most k since 2 −y 2 h n k (0, y 2 ) is polynomial in y 2 with degree at most k. From the above, we find that the equalityḠ 0,n (z 1 , z 2 ) = G 0,n (z 1 , z 2 ) holds for the all z 2 ∈ Z.

Asymptotics
In this section we take the KPZ scaling limit for the discrete time Bernoulli and geometric TASEP and prove Proposition 2.14 and 2.15.
Calculating the derivatives of f (x) up to the third order, we have Thus we see that f (x) has the double saddle point at x = 0, Therefore, for small ε, f (x) is expanded as For small ε, we also have Now, we see the convergence of the integration path. First, we deform C ε to the contour ε where ε is the part of Airy contour within the ball of radius ε − 1 2 centred at ε − 1 2 , and C π 3 ε is the part of C ε to the right of . From where S t,x (y) is defined by (2.26).
Thus the remaining part is to show that the integral over C π 3 ε converges to 0. To see this note that the real part of the exponent of the integral over C ε in (4.1), parametrized as y = ε − 1 2 (1 − e iθ ), is given by Note that we used an expression transform Because the y ∈ C π 3 ε correspond to 0 < |θ| < π 3 *3 , using log(1+ x) < x *4 for x ∈ (−1, ∞)\{0}, we get Therefore, for sufficiently small ε, the exponent there is less than −ε − 3 2 κt for some κ > 0. Hence we see that the part C π 3 ǫ of the integral vanishes and this completes the proof of (2.37). We can also prove(2.38) in the similar way to (2.37) thus omit the proof.

Proof of Proposition 2.15
Proposition 2.15 can be shown in a similar manner to Proposition 2.14. Here we give only the proof of (2.43). (2.44) can be obtained in a parallel way to (2.43) whereas (2.45) follows from (2.44) and the Donsker's invariance principle as in the case of (2.39) in Proposition 2.14.

Proof of Theorem 2.11
By using Propositions 2.14 or 2.15, we can prove Theorem 2.11 as following. This proof is almost the same as [20]. First, we change variables in the kernel as in Proposition 2.14 (resp. Proposition 2.15), so that for z i = p(2−p) 4(1−p) ε − 3 2 t + 2ε −1 x i + ε − 1 2 (u i + a i ) − 2 (resp. z i = − α(2−α) 4(1−α) ε − 3 2 t + 2ε −1 x i + ε − 1 2 (u i + a i ) − 2) we need to compute the limit of ε − 1 2 (χ 2ε −1 x−2 K tχ2ε −1 x−2 )(z i , z j ). Note that the change of variables turnsχ 2ε −1 x−2 (z) intō χ −a (u). We have n i < n j for small ε if and only if x j < x i and in this case we have, under our scaling, ε − 1 2 Q n j −n i (z i , z j ) − → e (x i −x j )∂ 2 (u i , u j ), (4.19) as ε − → 0. For the second term in (2.6), by Proposition 2.14 we get −t,−x j (u i , u j ). (4.20) Therefore, we have a limiting kernel surrounded by projectionχ −a . It is nicer to have projection χ a , so we change variables u i → −u i and replace the Fredholm determinant of the kernel by that of its adjoint to get By using (S t,x ) * S t,−x = I and S [20] for more information on these equations), we get the following: A The Kolmogorov forward equation for the discrete time geometric TASEP with N = 3 Here, we explain (3.9) in more detail in the case of N = 3. In this case (3.9) can be decomposed into four terms, At time t, the first particle (from the right) is at x 2 + 1 which leads to the blocking of the second particle. (c) The case µ = {2}. At time t, the second particle is at x 3 + 1 which leads to the blocking of the third particle. (d) The case µ = {1, 2}. At time t, the first and second particles are at x 2 + 1 and x 3 + 1 respectively which leads to the blockings of both the second and the third paricles.