Multi-point distribution of periodic TASEP

Transcription

1 Multi-point distribution of periodic TASEP Jinho Baik and Zhipeng Liu October 9, 207 Abstract The height fluctuations of the models in the KPZ class are expected to converge to a universal process. The spatial process at equal time is known to converge to the Airy process or its variations. However, the temporal process, or more generally the two-dimensional space-time fluctuation field, is less well understood. We consider this question for the periodic TASEP totally asymmetric simple exclusion process). For a particular initial condition, we evaluate the multi-time and multi-location distribution explicitly in terms of a multiple integral involving a Fredholm determinant. We then evaluate the large time limit in the so-called relaxation time scale. Introduction The models in the KPZ universality class are expected to have the :2:3 scales for the height fluctuations, spatial correlations, and time correlations as time t. This means that the scaled two-dimensional fluctuation field h t γ, τ) := Hc γτt) 2/3, τt) c 2 τt) + c 3 τt) 2/3).) c 4 τt) /3 of the height function Hl, t), where l is the spatial variable and t is time, is believed to converge to a universal field which depends only on the initial condition. Here c, c 2, c 3, c 4 are model-dependent constants. Determining the limiting two-dimensional fluctuation field γ, τ) hγ, τ) := lim t h t γ, τ).2) is an outstanding question. By now there are several results for the one-point distribution. The one-point distribution of hγ, τ) for fixed γ, τ) is given by random matrix distributions Tracy-Widom distributions) or their generalizations. The convergence is proved for a quite long list of models including PNG, TASEP, ASEP, q-tasep, random tilings, last passage percolations, directed polymers, the KPZ equation, and so on. See, for example, 2, 24, 38,, 6, and the review article. These models were studied using various integrable methods under standard initial conditions. See also the recent papers 2, 34 for general initial conditions. The spatial one-dimensional process, γ hγ, τ) for fixed τ, is also well understood. This process is given by the Airy process and its variations. However, the convergence is proved rigorously only for a smaller number of models. It was proved for the determinantal models like PNG, TASEP, last passage percolation, 2 Department of Mathematics, University of Michigan, Ann Arbor, MI, baik@umich.edu Department of Mathematics, University of Kansas, Lawrence, KS zhipeng@ku.edu For some initial conditions, such as the stationary initial condition, one may need to translate the space in the characteristic direction. 2 See, for example, 3, 25, 23, 9, 8, 0, 3 for special initial conditions. See the recent paper 28 for general initial conditions for TASEP.

2 but not yet for other integrable models such as ASEP, q-tasep, finite-temperature directed polymers, and the KPZ equation. The two-dimensional fluctuation field, γ, τ) hγ, τ), on the other hand, is less well understood. The joint distribution is known only for the two-point distribution. In 205, Johansson 26 considered the zero temperature Brownian semi-discrete directed polymer and computed the limit of the two-point in time and location) distribution. 3 The limit is obtained in terms of rather complicated series involving the determinants of matrices whose entries contain the Airy kernel. Two other papers studied qualitative behaviors of the temporal correlations. Using a variational problem involving two independent Airy processes, Ferrari and Spohn 8 proved in 206 the power law of the covariance in the time direction in the large and small time limits, τ /τ 2 0 and τ /τ 2. Here, τ and τ 2 denote the scaled time parameters. De Nardis and Le Doussal 3 extended this work further and also augmented by other physics arguments to compute the similar limits of the two-time distribution when one of the arguments is large. It is yet to be seen if one can deduce these results from the formula of Johansson. The objective of this paper is to study the two-dimensional fluctuation field of spatially periodic KPZ models. Specifically, we evaluate the multi-point distribution of the periodic TASEP totally asymmetric simple exclusion process) and compute a large time limit in a certain critical regime. We denote by L the period and by N the number of particles per period. Set ρ = N/L, the average density of particles. The periodic TASEP of period L and density ρ) is defined by the occupation function η j t) satisfying the spatial periodicity: η j t) = η j+l t), j Z, t 0..3) Apart from this condition, the particles follow the usual TASEP rules. Consider the limit as t, L, N with fixed ρ = N/L. Since the spatial fluctuations of the usual infinite TASEP is Ot 2/3 ), all of the particles in the periodic TASEP are correlated when t 2/3 = OL). We say that the periodic TASEP is in the relaxation time scale if t = OL 3/2 )..4) If t L 3/2, we expect that the system size has negligible effect and, therefore, the system follows the KPZ dynamics. See, for example, 5. On the other hand, if t L 3/2, then the system is basically in a finite system, and hence we expect the equilibrium dynamics. See, for example, 4. Therefore, in the relaxation time scale, we predict that the KPZ dynamics and the equilibrium dynamics are both present. Even though the periodic TASEP is as natural as the infinite TASEP, the one-point distribution was obtained only recently. Last year, in a physics paper 33 and, independently, in mathematics papers 4, 27, the authors evaluated the one-point function of the height function in finite time and computed the large time limit in the relaxation time scale. The one-point function follows the the KPZ scaling Ot /3 ) but the limiting distribution is different from the infinite TASEP. 4 This result was obtained for the three initial conditions of periodic step, flat, and stationary. Some of the previous results can be found in physics papers 22, 4, 32, 9, 20, 30, 2, 29; most of them considered the spectral gap of the generator of the system. In this paper, we extend the analysis of the papers 4, 27 and compute the multi-point in time and location) distribution of the periodic TASEP with a special initial condition called the periodic step initial condition. Here we allow any number of points unlike the previous work of Johansson on the infinite TASEP. It appears that the periodicity of the model simplifies the algebraic computation compared with the infinite 3 There are non-rigorous physics papers for the two-time distribution of directed polymers 5, 6, 7. However, another physics paper 3 indicates that the formulas in these papers are not correct. 4 The formulas obtained in 4, 27 and 33 are similar, but different. It is yet to be checked that these formulas are the same. 2

3 TASEP. In a separate paper we will consider flat and stationary initial conditions. The main results are the following:. For arbitrary initial conditions, we evaluate finite-time joint distribution functions of the periodic TASEP at multiple points in the space-time coordinates in terms of a multiple integral involving a determinant of size N. See Theorem 3. and Corollary For the periodic step initial condition, we simplify the determinant to a Fredholm determinant. See Theorem 4.6 and Corollary We compute the large time limit of the multi-point in the space-time coordinates) distribution in the relaxation time scale for the periodic TASEP with the periodic step initial condition. See Theorem 2.. One way of studying the usual infinite TASEP is the following. First, one computes the transition probability using the coordinate Bethe ansatz method. This means that we solve the Kolmogorov forward equation explicitly after replacing it which contains complicated interactions between the particles) by the free evolution equation with certain boundary conditions, and then solve it explicitly. In 37, Schütz obtained the transition probability of the infinite TASEP. Second, one evaluates the marginal or joint distribution by taking a sum of the transition probabilities. It is important that the resulting expression should be suitable for the asymptotic analysis. This is achieved typically by obtaining a Fredholm determinant formula. In 35, Rákos and Schütz re-derived the famous finite-time Fredholm determinant formula of Johansson 24 for the one-point distribution in the case of the step initial condition using this procedure. Subsequently, Sasamoto 36 and Borodin, Ferrari, Prähofer, and Sasamoto 9 obtained a Fredholm determinant formula for the joint distribution of multiple points with equal time. This was further extended by Borodin and Ferrari 7 to the points in spatial directions of each other. However, it was not extended to the case when the points are temporal directions of each other. The third step is to analyze the finite-time formula asymptotically using the method of steepest-descent. See 24, 36, 9, 7 and also a more recent paper 28. In the KPZ :2:3 scaling limit, the above algebraic formulas give only the spatial process γ hγ, τ). We applied the above procedure to the one-point distribution of the periodic TASEP in 4. We obtained a formula for the transition probability, which is a periodic analogue of the formula of Schütz. Using that, we computed the finite-time one-point distribution for arbitrary initial condition. The distribution was given by an integral of a determinant of size N. We then simplified the determinant to a Fredholm determinant for the cases of the step and flat initial conditions. The resulting expression was suitable for the asymptotic analysis. A similar computation for the stationary initial condition was carried out in 27. In this paper, we extend the analysis of 4, 27 to multi-point distributions. For general initial conditions, we evaluate the joint distribution by taking a multiple sum of the transition probabilities obtained in 4. The computation can be reduced to an evaluation of a sum involving only two arbitrary points in the spacetime coordinates with different time coordinates.) The main technical result of this paper, presented in Proposition 3.4, is the evaluation of this sum in a compact form. The key point, compared with the infinite TASEP 9, 7, 28, is that the points do not need to be restricted to the spatial directions. 5 The final formula is suitable for the large-time asymptotic analysis in relaxation time scale. If we take the period L to infinity while keeping other parameters fixed, the periodic TASEP becomes the infinite TASEP. Moreover, it is easy to check see Section 8 below) that the joint distributions of the periodic TASEP and the infinite TASEP are equal even for fixed L if L is large enough compared with the times. 5 In the large time limit, we add a certain restriction when the re-scaled times are equal. See Theorem 2.. The outcome of the above computation is that we find the joint distribution in terms of a multiple integral involving a determinant of size N. For the periodic step initial condition, we simplify the determinant further to a Fredholm determinant. 3

4 Hence, the finite-time joint distribution formula obtained in this paper Theorem 3. and Corollary 3.3) in fact, gives a formula of the joint distribution of the infinite TASEP; see the equations 8.7) and 8.8). This formula contains an auxiliary parameter L which has no meaning in the infinite TASEP. From this observation, we find that if we take the large time limit of our formula in the sub-relaxation time scale, t L 3/2, then the limit, if it exists, is the joint distribution of the two-dimensional process hγ, τ) in.2). However, it is not clear at this moment if our formula is suitable for the asymptotic analysis in the subrelaxation time scale; the kernel of the operator in the Fredholm determinant does not seem to converge in the sub-relaxation time scale while it converges in the relaxation time scale. The question of computing the limit in the sub-relaxation time scale, and hence the multi-point distribution of the infinite TASEP, will be left as a later project. This paper is organized as follows. We state the limit theorem in Section 2. The finite time formula for general initial conditions is in Section 3. Its simplification for the periodic step initial condition is obtained in Section 4. In Section 5, we prove Proposition 3.4, the key algebraic computation. The asymptotic analysis of the formula obtained in Section 4 is carried out in Section 6, proving the result in Section 2. We discuss some properties of the limit of the joint distribution in Section 7. In Section 8 we show that the finite-time formulas obtained in Sections 3 and 4 are also valid for infinite TASEP for all large enough L. Acknowledgments The work of Jinho Baik was supported in part by NSF grants DMS-36782, DMS and DMS , and the Simons Fellows program. The work was done in part when Zhipeng Liu was at Courant Institute, New York University. 2 Limit theorem for multi-point distribution 2. Limit theorem Consider the periodic TASEP of period L with N particles per period. We set ρ = N/L, the average particle density. We assume that the particles move to the right. Let η j t) be the occupation function of periodic TASEP: η j t) = if the site j is occupied at time t, otherwise η j t) = 0, and it satisfies the periodicity η j t) = η j+l t). We consider the periodic step initial condition defined by η j 0) = and η j+l 0) = η j 0). We state the results in terms of the height function { for N + j 0, 0 for j L N, 2.) hp) where p = le + te 2 = l, t) Z R ) Here e =, 0) and e 2 = 0, ) are the unit coordinate vectors in the spatial and time directions, respectively. The height function is defined by l 2J 0 t) + 2η j t)), l, j= hle + te 2 ) = 2J 0 t), l = 0, 2.3) 0 2J 0 t) 2η j t)), l, j=l+ 4

5 Figure : The pictures represent the density profile at time t = 0, t = L and t = 0L when ρ = /2. The horizontal axis is scaled down by L. Figure 2: The pictures represent the limiting height function at times t = 0.5nL for n = 0,, 2,. The left picture is when ρ = /2 and the right picture is when ρ = 2/5. Both horizontal axis location) and vertical axis height) are scaled down by L. where J 0 t) counts the number of particles jumping through the bond from 0 to during the time interval 0, t. The periodicity implies that hl + nl)e + te 2 ) = hle + te 2 ) + nl 2N) 2.4) for integers n. See Figure for the evolution of the density profile and Figure 2 for the limiting height function. The formula of the density profile and the limiting height function can be deduced from 5. 6 Note that the step initial condition 2.) generates shocks. The characteristic directions of the shocks are given by the equations l + nl = 2ρ)t for integer n. We represent the space-time position in new coordinates. Let e c := 2ρ)e + e 2 2.5) be a vector parallel to the characteristic directions. If we represent p = le + te 2 in terms of e and e c, then Consider the region p = se + te c where s = l t 2ρ). 2.6) R := {le + te 2 Z R 0 : 0 l 2ρ)t L} = {se + te c Z R 0 : 0 s L}. 2.7) See Figure 3. Due to the periodicity, the height function in R determines the height function in the whole space-time plane. The following theorem is the main asymptotic result. We take the limit as follows. We take L, N in such a way that the average density ρ = N/L is fixed, or more generally ρ stays in a compact subset of 6 The limit of the height function L hxl, tl) as L in probability) is given by the formula hx, t) = min n hnx, t) where h nx, t) = 2t x n)2 + t 2 + 2n 2ρ)t) when t > /4ρ). The formula is different when t < /4ρ). 5

6 Figure 3: Illustration of the points p j, j =,, m, in the region R. the interval 0, ). We consider m distinct points p i = s i e + t i e c in the space-time plane such that their temporal coordinate t j and satisfy the relaxation time scale t j = OL 3/2 ). The relative distances of the coordinates are scaled as in the : 2 : 3 KPZ prediction: t i t j = OL 3/2 ) = Ot i ), s i = OL) = Ot 2/3 i ), and the height at each point is scaled by OL /2 ) = Ot /3 i ). Theorem 2. Limit of multi-point joint distribution for periodic TASEP). Fix two constants c and c 2 satisfying 0 < c < c 2 <. Let N = N L be a sequence of integers such that c L N c 2 L for all sufficiently large L. Consider the periodic TASEP of period L and average particle density ρ = ρ L = N/L. Assume the periodic step initial condition 2.). Let m be a positive integer. Fix m points p j = γ j, τ j ), j =,, m, in the region R := {γ, τ) R R >0 : 0 γ }. 2.8) Assume that τ < τ 2 < < τ m. 2.9) Let p j = s j e + t j e c be m points 7 in the region R shown in Figure 3, where e =, 0) and e c = 2ρ, ), with s j = γ j L, t j = τ j L 3/2 ρ ρ). 2.0) Then, for arbitrary fixed x,, x m R, m { lim P hp j ) 2ρ)s j 2ρ + 2ρ 2 } )t j x L 2ρ /2 ρ) /2 L /2 j = Fx,, x m ; p,, p m ) 2.) j= where the function F is defined in 2.5). The convergence is locally uniform in x j, τ j, and γ j. If τ i = τ i+ for some i, then 2.) still holds if we assume that x i < x i+. Remark 2.2. Suppose that we have arbitrary m distinct points p j = γ j, τ j ) in R. Then we may rearrange them so that 0 < τ τ m. If τ j are all different, we can apply the above theorem since the result holds 7 Since p j should have an integer value for its spatial coordinate, to be precise, we need to take the integer part of s j +t j 2ρ) for the spatial coordinate. This small distinction does not change the result since the limits are uniform in the parameters γ j, τ j. Therefore, we suppress the integer value notations throughout this paper. 6

7 for arbitrarily ordered γ j. If some of τ j are equal, then we may rearrange the points further so that x j are ordered with those τ j, and use the theorem if x j are distinct. The only case which are not covered by the above theorem is when some of τ j are equal and the corresponding x j are also equal. The case when m = was essentially obtained in our previous paper 4 and also 33.) In that paper, we considered the location of a tagged particle instead of the height function, but it straightforward to translate the result to the height function. Remark 2.3. We will check that Fx,, x m ; p,, p m ) is periodic with respect to each of the space coordinates γ j in Subsection 2.2. By this spatial periodicity, we can remove the restrictions 0 γ j in the above theorem. Remark 2.4. Since we expect the KPZ dynamics in the sub-relaxation scale t j L 3/2, we expect that the τ j 0 limit of the above result should give rise to a result for the usual infinite TASEP. Concretely, we expect that the limit lim τ 0 Fτ τ) /3 x,, τ m τ) /3 x m ; γ τ τ) 2/3, τ τ), γ m τ m τ) 2/3, τ m τ)) 2.2) exists and it is the limit of the multi-time, multi-location joint distribution of the height function Hs, t) of the usual TASEP with step initial condition, { m Hγj lim P τ 2/3 } j T 2/3, 2τ j T ) τ j T x T j= τ /3 j. 2.3) j T /3 See also Section Formula of the limit of the joint distribution 2.2. Definition of F Definition 2.5. Fix a positive integer m. Let p j = γ j, τ j ) for each j =,, m where γ j R and Define, for x,, x m R, F x,, x m ; p,, p m ) = 0 < τ < τ 2 < < τ m. 2.4) Cz)Dz) dz m dz 2.5) 2πiz m 2πiz where z = z,, z m ) and the contours are nested circles in the complex plane satisfying 0 < z m < < z <. Set x = x,, x m ), τ = τ,, τ m ), and γ = γ,, γ m ). The function Cz) = Cz; x, τ ) is defined by 2.2) and it depends on x and τ but not on γ. The function Dz) = Dz; x, τ, γ) depends on all x, τ, and γ, and it is given by the Fredholm determinant, Dz) = det K K 2 ) defined in 2.38). The functions in the above definition satisfy the following properties. The proofs of P), P3), and P4) are scattered in this section while P2) is proved later in Lemma 7.. P) For each i, Cz) is a meromorphic function of z i in the disk z i <. It has simple poles at z i = z i+ for i =,, m. P2) For each i, Dz) is analytic in the deleted disk 0 < z i <. P3) For each i, Dz) does not change if we replace γ i by γ i +. Therefore, F is periodic, with period, in the parameter γ i for each i. 7

8 P4) If τ i = τ i+, the function F is still well-defined for x i < x i+. Remark 2.6. When m =, the function F is same as the one-point distribution given in 4.0) of 4. For general m, we have 0 Fx,, x m ; p,, p m ) and F is a non-decreasing function of x k for each k m since it is a limit of a joint distribution. We discuss a few other properties in Section Definition of Cz) Let log z be the principal branch of the logarithm function with cut R 0. Let Li s z) be the polylogarithm function defined by z k Li s z) = k s for z < and s C. 2.6) k= It has an analytic continuation using the formula Li s z) = z x s Γs) 0 e x dx z for z C \, ) if Rs) > ) Set A z) = 2π Li 3/2 z), A 2 z) = 2π Li 5/2 z). 2.8) For 0 < z, z <, set Bz, z ) = zz 2 ηξ log ξ + η) dξ dη e ξ2 /2 z)e η2 /2 z ) 2πi 2πi = 4π k,k z k z ) k k + k ) kk 2.9) where the integral contours are the vertical lines Rξ = a and Rη = b with constants a and b satisfying log z < a < 0 < b < log z. The equality of the double integral and the series is easy to check see 9.27) 9.30) in 4 for a similar calculation.) Note that Bz, z ) = Bz, z). When z = z, we can also check that Bz) := Bz, z) = z Li/2 y) ) 2 dy. 2.20) 4π 0 y Definition 2.7. Define Cz) := where we set z m+ := 0. m l= z l z l z l+ m l= e x la z l )+τ l A 2z l ) e x la z l+ )+τ l A 2z l+ ) e2bz l) 2Bz l+,z l ) 2.2) Since A, A 2, B are analytic inside the unit circle, it is clear from the definition that Cz) satisfies the property P) in Subsubsection Definition of Dz) The function Dz) is given by a Fredholm determinant. Before we describe the operator and the space, we first introduce a few functions. For z <, define the function hζ, z) = 2π ζ Li /2 ze ζ 2 y 2 )/2 ) dy for Rζ) < ) 8

9 Figure 4: The pictures represent the roots of the equation e ζ2 /2 = z dots) and the contours Rζ 2 ) = 2 log z solid curves) for z = 0.05e i, 0.4e i, 0.8e i, from the left to the right. and hζ, z) = 2π ζ Li /2 ze ζ 2 y 2 )/2 ) dy for Rζ) > ) The integration contour lies in the half-plane Ry) < 0, and is given by the union of the interval, R±ζ) on the real axis and the line segment from R±ζ) to ±ζ. Since ze ζ2 y 2 )/2 < on the integration contour, Li /2 ze ζ2 y 2 )/2 ) is well defined. Thus, we find that the integrals are well defined using Li /2 ω) ω as ω 0. Note that the function hζ, z) does not extend continuously at Rζ) = 0. Observe the symmetry, hζ, z) = h ζ, z) for Rζ) < ) We also have hζ, z) = i i log ze ω2 /2 ) ω ζ dω 2πi for Rζ) < ) This identity can be obtained by the power series expansion and using the fact that 2π u e ω2 /2 dω = i e u2 +ω 2 )/2 dω i ω u 2πi for u with argu) 3π/4, 5π/4); see 4.8) of 4. From 2.24) and 2.25), we find that hζ, z) = Oζ ) as ζ in the region argζ) ± π > ɛ 2.26) 2 for any fixed z satisfying z <. Let x, τ, and γ be the parameters in Definition 2.5. We set f i ζ) := for i =,, m, where we set τ 0 = γ 0 = x 0 = 0. { e 3 τi τi )ζ3 + 2 γi γi )ζ2 +x i x i )ζ e 3 τi τi )ζ3 2 γi γi )ζ2 x i x i )ζ for Rζ) < 0 for Rζ) > ) Now we describe the space and the operators. For a non-zero complex number z, consider the roots ζ of the equation e ζ2 /2 = z. The roots are on the contour Rζ 2 ) = 2 log z. It is easy to check that if 0 < z <, the contour Rζ 2 ) = 2 log z consists of two disjoint components, one in Rζ) > 0 and the other in Rζ) < 0. See Figure 4. The asymptotes of the contours are the straight lines of slope ±. For 0 < z <, we define the discrete sets L z := {ζ C : e ζ2 /2 = z} {Rζ) < 0}, R z := {ζ C : e ζ2 /2 = z} {Rζ) > 0}. 2.28) 9

10 Figure 5: Example of S block dots) and S 2 white dots) when m = 3. The level sets are shown for visual convenience. For distinct complex numbers z,, z m satisfying 0 < z i <, define the sets { Rzm if m is even, S := L z R z2 L z3 L zm if m is odd, and { Lzm if m is even, S 2 := R z L z2 R z3 if m is odd. See Figure 5. Now we define two operators R zm 2.29) 2.30) by kernels as follows. If for some i, j {,, m}, then we set Similarly, if K : l 2 S 2 ) l 2 S ), K 2 : l 2 S ) l 2 S 2 ) 2.3) ζ L zi R zi ) S and ζ L zj R zj ) S ) K ζ, ζ ) = δ i j) + δ i j + ) i )) f iζ)e 2hζ,zi) hζ,zi )i ) hζ,z j ) j ) ζζ ζ Q j). 2.33) ) for some i, j {,, m}, then we set ζ L zi R zi ) S 2 and ζ L zj R zj ) S 2.34) K 2 ζ, ζ ) = δ i j) + δ i j ) i )) f iζ)e 2hζ,zi) hζ,zi+ )i ) hζ,z j+ ) j ) ζζ ζ Q 2 j). 2.35) ) Here we set z 0 = z m+ = 0 so that We also set Definition 2.8. Define Q j) = z j ) j z j for z = z,, z m ) where 0 < z i < and z i are distinct. e hζ,z0) = e hζ,zm+) =. 2.36) and Q 2 j) = z j+ )j z j. 2.37) Dz) := det K K 2 ) 2.38) 0

11 In this definition, we temporarily assumed that z i are distinct in order to ensure that the term ζ ζ in the denominators in 2.33) and 2.35) does not vanish. However, as we stated in P2) in Subsubsection 2.2., Dz) is still well-defined when z i are equal. See Lemma 7.. The definition of L z and R z implies that argζ) 3π/4 as ζ along ζ L z and argζ) π/4 as ζ along ζ R z. Hence, due to the cubic term ζ 3 in 2.27), f i ζ) 0 super exponentially as ζ on the set L z R z if τ < < τ m. Hence, using the property 2.26) of h, we see that the kernels decay super-exponentially fast as ζ, ζ on the spaces. Therefore, the Fredholm determinant is well defined if τ < < τ m. We now check the property P4). If τ i = τ i+, the exponent of f i has no cubic term ζ 3. The quadratic term contributes to O) since e ζ2 /2 = z i for ζ L zi R zi, and hence e cζ2 = O). On the other hand, the linear term in the exponent of f i has a negative real part if x i < x i+. Hence, if τ i = τ i+ and x i < x i+, then f i ζ) 0 exponentially as ζ along ζ S S 2 and hence the kernel decays exponentially fast as ζ, ζ on the spaces. Therefore, the Fredholm determinant is still well defined if τ i = τ i+ and x i < x i+. This proves P4). 2.3 Matrix kernel formula of K and K 2 Due to the delta functions, K ζ, ζ ) 0 only when for some integer l, and similarly K 2 ζ, ζ ) 0 only when ζ L z2l R z2l and ζ R z2l L z2l 2.39) ζ L z2l R z2l+ and ζ R z2l L z2l+ 2.40) for some integer l. Thus, if we represent the kernels as m m matrix kernels, then they have 2 2 block structures. For example, consider the case when m = 5. Let us use ξ i and η i to represent variables in L zi and R zi, respectively: ξ i L zi, η i R zi. 2.4) The matrix kernels are given by kξ, η ) kξ, ξ 2 ) kη 2, η ) kη 2, ξ 2 ) K = and kη, ξ ) K 2 = kξ 2, η 2 ) kξ 2, ξ 3 ) kη 3, η 2 ) kη 3, ξ 3 ) kξ 3, η 3 ) kξ 3, ξ 4 ) kη 4, η 3 ) kη 4, ξ 4 ) kξ 5, η 5 ) kξ 4, η 4 ) kξ 4, ξ 5 ) kη 5, η 4 ) kη 5, ξ 5 ) 2.42) 2.43) where the empty entries are zeros and the function k is given in the below. When m is odd, the structure is similar. On the other hand, when m is even, K consists only of 2 2 blocks and K 2 contains an additional non-zero block at the bottom right corner. We now define k. For i m, writing ξ = ξ i, η = η i, ξ = ξ i+, η = η i+, 2.44)

12 we define kξ, η) kξ, ξ ) fi ξ) kη, η) kη, ξ = ) f i+ η ) e hη,z i+ ) e hξ,z i ) e 2hξ,z i ) ξe hξ,z i+ ) e 2hη,z i+ ) η e hη,z i ) z i+ z i zi z i+ ξ η η η. ξ ξ η ξ 2.45) The term kξ m, η m ) is defined by the, ) entry of 2.45) with i = m where we set z m+ = 0. The term kη, ξ ) is defined by the 2, 2) entry of 2.45) with i = 0 where we set z 0 = Series formulas for Dz) We present two series formulas for the function Dz). The first one 2.50) is the series expansion of Fredholm determinant using the block structure of the matrix kernel. The second formula 2.5) is obtained after evaluating the finite determinants in 2.50) explicitly. To simplify formulas, we introduce the following notations. Definition 2.9 Notational conventions). For complex vectors W = w,, w n ) and W = w,, w n ), we set W) = i<jw j w i ) = det, W; W ) = w i w i ). 2.46) For a function h of single variable, we write w j i hw) = i n i n n hw i ). 2.47) i= We also use the notations for finite sets S and S. S; S ) = s s ), fs) = fs) 2.48) s S s S s S The next lemma follows from a general result whose proof is given in Subsection 4.3 below. Lemma 2.0 Series formulas for Dz)). We have Dz) = n!) 2 D nz) 2.49) n Z 0 ) m with n! = m l= n l! for n = n,, n m ), where D n z) can be expressed as the following two ways. i) We have, for n = n,, n m ), D n z) = ) n U l) L zl ) n l V l) R zl ) n l l=,,m det K ζ i, ζ j) n i,j= det K 2ζ i, ζ j ) n i,j= 2.50) 2

13 where U = U ),, U m) ), V = V ),, V m) ) with U l) = u l),, ul) n l ), V l) = v l),, vl) n l ), and { l) u k if i = n + + n l + k for some k n l with odd integer l, and ζ i = ζ i = Here, we set n 0 = 0. ii) We also have with d n,z U, V) := where m l=2 v l) k if i = n + + n l + k for some k n l with even integer l, { v l) k if i = n + + n l + k for some k n l with odd integer l, m u l) k if i = n + + n l + k for some k n l with even integer l. l= D n z) = U l) L zl ) n l V l) R zl ) n l l=,,m U l) ) 2 V l) ) 2 U l) ; V l) ) 2 ˆfl U l) )ˆf l V l) ) U l) ; V l ) ) V l) ; U l ) )e hvl),z l ) hv l ),z l ) U l) ; U l ) ) V l) ; V l ) )e hul),z l )+hu l ),z l ) 2.5) 2.52) d n,z U, V) 2.53) z ) nl l z ) nl l z l z l 2.54) ˆfl ζ) := ζ f lζ)e 2hζ,z l). 2.55) Recall 2.22), 2.23), and 2.27) for the definition of h and f j. The property P3) in Subsubsection 2.2. follows easily from 2.54). Note that γ i only appears in the factor ˆf l U l) )ˆf l V l) ) for l = i or i +. If we replace γ i by γ i +, then f i ζ) and f i+ ζ) are changed by z i f i ζ) and z i+ f i+ ζ) if Rζ) < 0, or z i f i ζ) and z i+ f i+ζ) if Rζ) > 0. But U l) has the same number of components as V l) for each l. Therefore ˆf l U l) )ˆf l V l) ) does not change. We can also check P3) from the original Fredholm determinant formula. The analyticity property P2) is proved in Lemma 7. later using the series formula. 3 Joint distribution function for general initial condition We obtain the limit theorem of the previous section from a finite-time formula of the joint distribution. In this section, we describe a formula of the finite-time joint distribution for an arbitrary initial condition. We simplify the formula further in the next section for the case of the periodic step initial condition. We state the results in terms of particle locations instead of the height function used in the previous section. It is easy to convert one to another; see 6.5). The particle locations are denoted by xit) where < x0t) < xt) < x2t) <. 3.) Due to the periodicity of the system, we have xit) = xi+nn t) nl for all integer n. 3

14 Figure 6: The pictures represent the roots of the equation w N w + ) L N = z L and the contours w N w + ) L N = z L with N = 8, L = 24 for three different values of z. The value of z increases from the left picture to the right picture. The middle picture is when z = ρ ρ ρ) ρ where ρ = N/L = /3. The periodic TASEP can be described if we keep track of N consecutive particles, say xt) < < xn t). If we focus only on these particles, they follow the usual TASEP rules plus the extra condition that xn t) < xt) + L for all t. Define the configuration space X N L) = {x, x 2,, x N ) Z N : x < x 2 < < x N < x + L}. 3.2) We call the process of the N particles TASEP in X N L). We use the same notations xit), i =,, N, to denote the particle locations in the TASEP in X N L). We state the result for the TASEP in X N L) first and then one for the periodic TASEP as a corollary. For z C, consider the polynomial of degree L given by Denote the set of the roots by q z w) = w N w + ) L N z L. 3.3) R z = {w C : q z w) = 0}. 3.4) The roots are on the level set w N w + ) L N = z L. It is straightforward to check the following properties of the level set. Set r0 := ρ ρ ρ) ρ 3.5) where, as before, ρ = N/L. The level set becomes larger as z increases, see Figure 6. If 0 < z < r0, the level set consists of two closed contours, one in Rw) < ρ enclosing the point w = and the other in Rw) > ρ. enclosing the point w = 0. When z = r0, the level set has a self-intersection at w = ρ. If z > r0, then the level set is a connected closed contour. Now consider the set of roots R z. Note that if z 0, then, 0 / R z. It is also easy to check that if a non-zero z satisfies z L r L 0, then the roots of q z w) are all simple. On the other hand, if z L = r L 0, then there is a double root at w = ρ and the rest L 2 roots are simple. For the results in this section, we take z to be any non-zero complex number. But in the next section, we restrict 0 < z < r0. Theorem 3. Joint distribution of TASEP in X N L) for general initial condition). Consider the TASEP in X N L). Let Y = y,, y N ) X N L) and assume that x0),, xn0)) = Y. Fix a positive integer m. Let k, t ),, k m, t m ) be m distinct points in {,, N} 0, ). Assume that 0 t t m. Let a i Z for i m. Then P Y xk t ) a,, xk m t m ) a m ) = Cz, k)d Y z, k, a, t) dz m dz 3.6) 2πiz m 2πiz 4

15 where the contours are nested circles in the complex plane satisfying 0 < z m < < z. Here z = z,, z m ), k = k,, k m ), a = a,, a m ), and t = t,, t m ). The functions in the integrand are m ) L N Cz, k) = ) m )+km )N+) z k )L zl ) 3.7) l=2 z k l k l )L l z l and D Y z, k, a, t) = det w R z w m R zm w i w + ) yi i w j m m l=2 w l w l ) m G l w l ) l= N i,j= 3.8) with where we set t 0 = k 0 = a 0 = 0. G l w) = ww + ) Lw + ρ) w k l w + ) a l+k l e t lw w k l w + ) a l +k l e t l 3.9) w Remark 3.2. The limiting joint distribution F in the previous section was not defined for all parameters: When τ i = τ i+, we need to put the restriction x i < x i+. See Property P4) in Subsubsection The finite-time joint distribution does not require such restrictions. The sums in the entries of the determinant D Y z, k, a, t) are over finite sets, and hence there is no issue with the convergence. Therefore, the right-hand side of 3.6) is well-defined for all real numbers t i and integers a i and k i. Corollary 3.3 Joint distribution of periodic TASEP for general initial condition). Consider the periodic TASEP with a general initial condition determined by Y = y,, y N ) X N L) and its periodic translations; xj+nn 0) = y j + nl for all n Z and j =,, N. Then 3.6) holds for all k i Z without the restriction that k i {,, N}. Proof. The particles in the periodic TASEP satisfies xj+nn t) = xjt) + nl for every integer n. Hence if k l is not between and N, we may translate it. This amounts to changing k l to k l + nn and a l to a l + nl for some integer n. Hence it is enough to show that the right-hand side of 3.6) is invariant under these changes. Under these changes, the term Cz, k) is multiplied by the factor z nnl l and by z nnl m z nnl if l = m. On the other hand, G l w l ) produces the multiplicative factor w nn l which is z nl l Cz, k). Similarly G l+ w l+ ) produces a factor which cancel out z nnl l+ if l m w l + ) nl+nn by 3.4). Taking this factor outside the determinant 3.8), we cancel out the factor zl nnl l+ if l m. Before we prove the theorem, let us comment on the analytic property of the integrand in the formula 3.6). The function Cz, k) is clearly analytic in each z l 0. Consider the function D Y z, k, a, t). Note that d dw q Lw + ρ) zw) = ww + ) wn w + ) L N. 3.0) Hence, if F w) is an analysis function of w in C \ {, 0}, and fw) = F w)w N w + ) L N, then ww + ) F w) Lw + ρ) = fw) 2πi w R w =r q z w) dw fw) 2πi w+ =ɛ q z w) dw fw) dw 3.) 2πi w =ɛ 2 q z w) z for any ɛ, ɛ 2 > 0 and r > max{ɛ +, ɛ 2 } such that all roots of q z w) lie in the region {w : ɛ 2 < w < r, w + > ɛ }. Note that q z w) is an entire function of z for each w. Since we may take r arbitrarily large from 5

16 and ɛ, ɛ 2 arbitrary small and positive, the right hand-side of 3.) defines an analytic function of z 0. Now the entries of the determinant in 3.8) are of the form w R z w m R zm F w,, w m ) m l= w l w l + ) Lw l + ρ) 3.2) for a function F w,, w m ) which is analytic in each variable in C \ {, 0} as long as w l w l for all l = 2,, m. The last condition is due to the factor m l=2 w l w l ) in the denominator. Note that if w l = w l, then z l = z l. Hence by using 3.) m times, each entry of 3.8), and hence D Y z, k, a, t), is an analytic function of each z l 0 in the region where all z l are distinct. When m =, the product in 3.7) is set to be and the formula 3.6) in this case was obtained in Proposition 6. in 4. For m 2, as we mentioned in Introduction, we prove 3.6) by taking a multiple sum of the transition probability. The main new technical result is a summation formula and we summarize it in Proposition 3.4 below. The transition probability was obtained in Proposition 5. of 4. Denoting by P X X ; t) the transition probability from X = x,, x N ) X N L) to X = x,, x N ) X NL) in time t, P X X ; t) = det wj i+ w + ) x L w + ρ w R z N i +xj+i j e tw i,j= dz 2πiz 3.3) where the integral is over any simple closed contour in z > 0 which contains 0 inside. The integrand is an analytic function of z for z 0 by using 3.). Proof of Theorem 3.. It is enough to consider m 2. It is also sufficient to consider the case when the times are distinct, t < < t m, because both sides of 3.6) are continuous functions of t,, t m. Note that 3.8) involve only finite sums. Denoting by X l) = x l),, xl) N ) the configuration of the particles at time t l, the joint distribution function on the left hand-side of 3.6) is equal to P Y X ) ; t )P X )X 2), t 2 t ) P X m )X m) ; t m t m ). 3.4) X l) X N L) {x l) k l a l } l=,,m Applying the Cauchy-Binnet formula to 3.3), we have P X X dz ; t) = L X W )R XW )QW ; t) 2πiz W R z) N 3.5) where for W = w,, w N ) C, L X W ) = det w j i w i + ) xj j N, R X W ) = det i,j= w j i w i + ) x j +j N i,j=, 3.6) and QW ; t) = N!L N N i= w i e twi w i + ρ. 3.7) 6

17 We insert 3.5) into 3.4) and interchange the order of the sums and the integrals. Assuming that the series converges absolutely so that the interchange is possible, the joint distribution is equal to dz 2πiz dz m 2πiz m W ) R z ) N W m) R zm ) N PW ),, W m ) m QW l) ; t l t l ) l= 3.8) where W l) = w l),, wl) N ) and PW ),, W m ) = L Y W ) ) m l= H kl,a l W l) ; W l+) ) R X W m) ). X X N L) x km a m 3.9) Here we set H k,a W ; W ) := X X N L) {x k a} R X W )L X W ) 3.20) for a pair of complex vectors W = w,, w N ) and W = w,, w N ). Let us now show that it is possible to exchange the sums and integrals if we take the z i -contours properly. We first consider the convergence of 3.20) and the sum in 3.9). Note that shifting the summation variable X to X b,, b), X X N L) {x k =b} R X W )L X W ) = Y X N X) {y k =0} N R Y W )L Y W w ) j b ) w j + The right hand side of 3.20) is the sum of the above formula over b a. Hence 3.20) converges absolutely and the convergence is uniform for w i, w i if N w j + is in a compact subset of 0, ). Similarly, the j= w j+ sum of R X W m) ) in 3.9) converges if N j= wm) j + >. Therefore, 3.9) converge absolutely if the intermediate variables W l) = w l),, wl) N ) satisfy N j= w ) j + > N j= w 2) j + > > N j= j= w m) j + >. 3.22) We now show that it is possible to choose the contours of z i s so that 3.22) is achieved. Since W l) R zl ) N, w l) j satisfies the equation w N w + ) L N = zl L. Hence wl) j = z j + O) as z j. Therefore, if we take the contours z l = r l where r > > r m > 0 and r l r l+ are large enough where r m+ := 0), then 3.22) is satisfied. Thus, 3.20) and the sum in 3.9) converge absolutely. It is easy to see that the convergences are uniform. Hence we can exchange the sums and integrals, and therefore, the joint distribution is indeed given by 3.8) if we take the contours of z i to be large nested circles. We simplify 3.8). The terms H kl,a l W l) ; W l+) ) are evaluated in Proposition 3.4 below. Note that since the z i -contours are the large nested circles, we have 3.22), and hence the assumptions in Proposition 3.4 are satisfied. On the other hand, the sum of R X W m) ) in 3.8) was computed in 4. Lemma 6. in 4 implies that for W = w,, w N ) R N z, N N R X W ) = ) k )N+) z k )L w j + ) w j + ) a+k+ det w i N. i,j= X X N L) x k =a j= j= w k j j 3.23) 7

18 Hence, from the geometric series, for W = w,, w N ) R N z, N R X W ) = ) k )N+) z k )L w k j w j + ) a+k+ det w i N j i,j= 3.24) X X N L) x k a if N j= w j + >. The last condition is satisfies for W = W m). We thus find that 3.9) is equal to an explicit factor times a product of m Cauchy determinants times a Vandermonde determinant. By using the Cauchy-Binet identity m times, we obtain 3.6) assuming that the z i -contours are large nested circles. Finally, using the analyticity of the integrand on the right-hand side of 3.6), which was discussed before the start of this proof, we can deform the contours of z i to any nested circles, not necessarily large circles. This completes the proof. The main technical part of this section is the following summation formula. We prove it in Section 5. Proposition 3.4. Let z and z be two non-zero complex numbers satisfying z L z ) L. Let W = w,, w N ) R z ) N and W = w,, w N ) R z )N. Suppose that N j= w j + < N j= w j +. Consider H k,a W ; W ) defined in 3.20). Then for any k N and integer a, z ) ) ) k )L z H k,a W ; W L N N w k ) = z j w j + ) a+k+ N z w j ) k w j + det ) a+k w i w i. 3.25) i,i = 4 Periodic step initial condition We now assume the following periodic step condition: j= j= xi+nn 0) = i N + nl for i N and n Z. 4.) In the previous section, we obtain a formula for general initial conditions. In this section, we find a simpler formula for the periodic step initial condition which is suitable for the asymptotic analysis. We express D Y z, k, a, t) as a Fredholm determinant times a simple factor. The result is described in terms of two functions Cz) and Dz). We first define them and then state the result. Throughout this section, we fix a positive integer m, and fix parameters k,, k m, a,, a m, t,, t m as in the previous section. 4. Definitions Recall the function q z w) = w N w + ) L N z L for complex z in 3.3) and the set of its roots R z = {w C : q z w) = 0} 4.2) in 3.4). Set r0 := ρ ρ ρ) ρ, ρ = N/L 4.3) as in 3.5). We discussed in the previous section that if 0 < z < r0, then the contour q z w) = 0 consists of two closed contours, one in Rw) < ρ enclosing the point w = and the other in Rw) > ρ enclosing the point w = 0. Now, for 0 < z < r0, set L z = {w R z : Rw) < ρ}, R z = {w R z : Rw) > ρ}. 4.4) 8

19 It is not difficult to check that L z = L N, R z = N. 4.5) See the left picture in Figure 6 in Section 3. Note that if z = 0, then the roots are w = with multiplicity L N and w = 0 with multiplicity N.) From the definitions, we have R z = L z R z. 4.6) In Theorem 3., we took the contours of z i as nested circles of arbitrary sizes. In this section, we assume that the circles satisfy 0 < z m < < z < r0. 4.7) Hence L zi and R zi are all well-defined. We define two functions Cz) and Dz) of z = z,, z m ), both of which depend on the parameters k i, t i, a i. The first one is the following. Recall the notational convention introduced in Definition 2.9. For example, R z ; L z ) = v R z u L z v u). Definition 4.. Define m E l z l ) m u L zl u) N v R zl v + ) L N Cz) = E l z l ) R zl ; L zl ) l= l= m zl L m 4.8) R zl ; L zl ) zl L zl l u L zl u) N v R zl v + ) L N l=2 l=2 where E i z) := for i =,, m, and E 0 z) :=. u R z,l u) ki N v R z,r v + ) ai+ki N e tiv 4.9) It is easy to see that all terms in Cz) other than m l=2 are analytic for z zl L,, z m within the zl l disk {z; z < r0}. Hence Cz) is analytic in the disk except the simple poles when zl L = zl l, l = 2,, m. z L l We now define Dz). It is given by a Fredholm determinant. Set F i w) := w ki+n+ w + ) ai+ki N e tiw for i =,, m, F 0 w) := ) Define Also, set Define, for 0 < z < r0, l z w) = F i w) F i w) f i w) = F i w) F i w) w + ) L N Jw) = for Rw) < ρ, for Rw) > ρ. 4.) ww + ) Lw + ρ). 4.2) w N u L z w u), r z w) = u R z w u). 4.3) 9

20 Figure 7: Example of S and S 2 when m = 3. The block dots are S and the white dots are S 2. The level sets are shown for visual convenience. Note that l z w)r z w) = q zw) w+) L N w N = wn w+) L N z L w N w+) L N. Set { lz w) for Rw) < ρ, H z w) := r z w) for Rw) > ρ, When z = 0, we define l z w) = r z w) = and hence H z w) =. Define two sets { Lzm if m is odd, S := L z R z2 L z3 if m is even, and S 2 := R z L z2 R z3 See Figure 7. We define two operators R zm { Rzm if m is odd, L zm if m is even. 4.4) 4.5) 4.6) K : l 2 S 2 ) l 2 S ), K 2 : l 2 S ) l 2 S 2 ) 4.7) by kernels. If w R zi S and w R zj S 2 for some i, j {,, m}, we set K w, w ) = δ i j) + δ i j + ) i Jw)f i w)h zi w)) 2 )) H zi ) i w)h zj ) j w )w w ) Q j). 4.8) Similarly, if w R zi S 2 and w R zj S for some i, j {,, m}, we set K 2 w, w ) = δ i j) + δ i j ) i Jw)f i w)h zi w)) 2 )) H zi+ ) i w)h zj+ ) j w )w w ) Q 2j). 4.9) Here we set z 0 = z m+ = 0. We also set Q j) := zj ) j z j ) L, Q 2 j) := zj+ ) j z j ) L. 4.20) Definition 4.2. Define Dz) = deti K K 2 ). 4.2) 20

21 Remark 4.3. The matrix kernels for K and K 2 have block structures similar to the infinite time case discussed in Subsection 2.3. The only change is that k is replaced by k which is given as follows. For u L zi, v R zi, u L zi+, v R zi+, 4.22) we have ku, v) ku, u Fiu) ) F kv, v) kv, u = i u) ) u v v v F iv ) F i+v ) u u v u uu+)r zi u) 2 Lu+ρ)r zi+ u) l zi+ v) r zi u ) v v +)l zi+ v ) 2 Lv +ρ)l zi v ) zl i+ z L i. zl i zi+ L 4.23) As in Subsection 2.4, the above Fredholm determinant also has two series formulas. The proof of the following lemma is given in Subsection 4.3. Lemma 4.4 Series formulas of Dz)). We have Dz) = n!) 2 D nz) 4.24) n Z 0 ) m with n! = m l= n l! for n = n,, n m ) and D n z) can be expressed in the following two ways. Here we set D n z) = 0 if one of n l is larger than N. a) We have D n z,, z m ) = ) n U l) L zl ) n l V l) R zl ) n l l=,,m det K w i, w j) n i,j= det K 2w i, w j ) n i,j= 4.25) where U = U ),, U m) ), V = V ),, V m) ) with U l) = u l),, ul) n l ), V l) = v l),, vl) n l ), and { l) u k if i = n + + n l + k for some k n l with odd integer l, and b) We have D n z) = w i = w i = v l) k if i = n + + n l + k for some k n l with even integer l, { v l) k if i = n + + n l + k for some k n l with odd integer l, U l) L zl ) n l V l) R zl ) n l l=,,m u l) k if i = n + + n l + k for some k n l with even integer l. m l= U l) )) 2 V l) )) 2 U l) ; V l) )) 2 ˆfl U l) ) ˆf l V l) ) m U l) ; V l ) ) V l) ; U l ) ) zl l ) n zl L l zl l ) n zl L l U l) ; U l ) ) V l) ; V l ) )r zl U l) )r zl U l ) )l zl V l) )l zl V l ) ) l=2 4.26) 4.27) 4.28) where ˆf l w) := Jw)f l w)h zl w)) 2 = { Jw) F l w) F l w) l z l w)) 2 for w L zl, Jw) F l w) F l w) r z l w)) 2 for w R zl. 4.29) 2

22 Remark 4.5. From 4.28), we can check that D n z) is analytic for each z l in 0 < z l < r0, l m just like Dz) of Section 2. The proof for Dz) is in Lemma 7.. The proof for D n z) is similar, and we skip it. 4.2 Result and proof Theorem 4.6 Joint distribution of TASEP in X N L) for step initial condition). Consider the TASEP in X N L) with the step initial condition xi0) = i N, i N. Set ρ = N/L. Fix a positive integer m. Let k, t ),, k m, t m ) be m distinct points in {,, N} 0, ). Assume that 0 t t m. Let a i Z for i m. Then P xk t ) a,, xk m t m ) a m ) = where z = z,, z m ) and the contours are nested circles satisfying Cz)Dz) dz m dz 4.30) 2πiz m 2πiz 0 < z m < < z < r0 4.3) with r0 = ρ ρ ρ) ρ. The functions Cz) and Dz) are defined in 4.8) and 4.2), respectively. Recall that Cz) is analytic in z l < r0 except for the poles when zl L = zl l, and Dz) is analytic in 0 < z l < r0. We point out that Remark 3.2 still applies to the above theorem; the Fredholm determinant expansion involves only finite sums. Corollary 4.7 Joint distribution of periodic TASEP for periodic step initial condition). Consider the periodic TASEP with periodic step initial condition, xi+nn 0) = i N + nl for i N and n Z. Then 4.30) holds for all integer indices k,, k m without the restriction that they are between and N. Proof. As in the proof of Corollary 3.3, it is enough to show that the formulas are invariant under the changes k i k i ± N and a i a i ± L for each i. This can be checked easily for Cz) using the identity u L z u) N = v R z v + ) L N, which is easy to prove; see 4.52) below. For Dz), we use the fact that u N u + ) L N = z L l for u L z l and v N i v i + ) L N = z L i for v R zl plus the special structure of K and K 2.) Proof of Theorem 4.6. When m =, the result was obtained in Theorem 7.4 of 4. We assume m 2. In Theorem 3., Cz, k) does not depend on the initial condition. Let us denote D Y z, k, a, t) by D step when Y = N,,, 0), the step initial condition. We need to show that D step = Dz) Cz) Cz,k). Inserting the initial condition y i = i N, re-writing G l in terms of F l and J in 4.0) and 4.2), and reversing the rows, w D step = ) NN )/2 i m det w j m m l=2 w l w l ) l= N Jw l ) F lw l ). 4.32) F l w l ) i,j= The sum is over all w R z,, w m R zm. Using the Cauchy-Binet identity m times, D step = )NN )/2 N!) m W l) R zl ) N l=,,m EW) m l= JW l) ) F lw l) ) F l W l) ) 4.33) where W l) = w l),, wl) N ) with wl) i R zl for each i, W = W ),, W m) ), and EW) = det w ) i ) j det det det w m) w 2) i w ) j w m) i w m ) i ) j. 4.34) j 22

23 Here all matrices are indexed by i, j N. Note that in 4.33), we use the notational convention such as F l W l) ) = N i= F lw l) i ) mentioned in Definition 2.9. Evaluating the Vandermonde determinants and the Cauchy determinants, D step is equal to recall the notations 2.46)) ) mnn )/2 m l= W l) ) 2 N m N!) m m l=2 W w ) l) ; W l ) i w m) i ) N JW l) ) F lw l) ) ) F l W l) ). 4.35) W l) R zl ) N l=,,m Note that for each l, we may assume that the coordinates of the vector W l) are all distinct since otherwise the summand is zero due to W l) ). Also note that the summand is a symmetric function of the coordinates of W l) for each l. Hence instead of taking the sum over the vectors W l) R zl ) N, we can take a sum over the subsets W l) R zl of size N: D step is equal to ) mnn )/2 W l) R zl W l) =N l=,,m m l= W l) ) 2 m l=2 W l) ; W l ) ) i= N w ) i w m) i i= l= ) N m l= J W l) ) F l W l) ) F l W l) ) 4.36) where w ) i are the elements of W ) and w m) i are the elements of W m). We now change the sum as follows. Since R zl is the disjoint union of L zl and R zl, some elements of the set W l) are in L zl and the rest in R zl. Recall that L zl = L N and R zl = N.) Let Ũ l) = W l) L zl and Ṽ l) = R zl \ W l). Observe that since W l) = R zl = N), we have Ũ l) = Ṽ l). Call this last number n l. We thus find that the sum in 4.36) can be replaced by the sums. 4.37) n l =0,,N l=,,m Ũ l) L zl,ṽ l) R zl Ũ l) = Ṽ l) =n l l=,,m We now express the summand in terms of Ũ l) and Ṽ l) instead of W l). First, for any function h, h W l) ) = hũ l) ) hr z l ) hṽ l) ). 4.38) Now consider W l) ) 2. We suppress the dependence on l in the next a few sentences to make the notations light. Setting tentatively S = R z \ Ṽ so that R z = Ṽ S. Note that W = Ũ S, a disjoint union. We thus have W ) 2 = Ũ)2 S) 2 Ũ; S) 2, R z ) 2 = Ṽ )2 S) 2 Ṽ ; S) ) Let Then, It is also direct to see that q z,r w) := v R z w v) = w N r z w). 4.40) q z,r Ũ) = Ũ; Ṽ ) Ũ; S), q z,rṽ ) = )NN )/2 Ṽ )2 Ṽ ; S). 4.4) R z ) 2 = ) NN )/2 q z,rr z ). 4.42) From these, after canceling out all terms involving S and inserting the dependence on l, we find that W l) ) 2 = ) NN )/2 Ũ l) ) 2 Ṽ l) ) 2 q zl,rũ l) )) 2 Ũ l) ; Ṽ l) ) 2 q z l,r Ṽ l) )) 2 q z l,rr zl ). 4.43) 23