Fluctuations of TASEP and LPP with general initial data

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1 Fluctuations of TASE and L with general initial data Ivan Corwin Zhipeng Liu and Dong Wang December 16, 2014 Abstract We prove Airy process variational formulas for the one-point probability distribution of discrete time parallel update TASE with general initial data, as well as last passage percolation from a general lattice path to a point. We also consider variants of last passage percolation with inhomogeneous weights and provide variational formulas of a similar nature. This proves one aspect of the conjectural description of the renormalization fixed point of the Kardar-arisi-Zhang universality class. 1 Introduction The totally asymmetric simple exclusion process TASE is a prototypical interacting particle system, or via integration random growth process. The theory of hydrodynamics describes the law of large number for the evolution of the system s particle density, or height function. In particular, if hx; t represents the height function, then ɛhɛ 1 x; ɛ 1 t converges as ɛ 0 as a space-time process to the deterministic solution to a Hamilton Jacobi equation with explicit model dependent flux [27]. The solution, of course, depends on the initial data and in particular on the limit as ɛ 0 of ɛh 0 ɛ 1 x. It is possible to consider initial data h 0;ɛ which depends on ɛ so that ɛh 0;ɛ ɛ 1 x has a limit. The aim of the present paper is to describe, in a similar spirit, how fluctuations around the law of large number evolves over time. Define Then it is conjectured in [18] that if we take h ɛ x; t = c 1 ɛ b hc 2 ɛ 1 x; c 3 ɛ z t h ɛ x; t. b = 1/2, and z = 3/2, Columbia University, Department of Mathematics, 2990 Broadway, ew York, Y 10027, USA, and Clay Mathematics Institute, 10 Memorial Blvd. Suite 902, rovidence, RI 02903, USA, and Institute Henri oincare, 11 Rue ierre et Marie Curie, aris, France, and Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Avenue, Cambridge, MA , USA. ivan.corwin@gmail.com Courant Institute of Mathematical Sciences, ew York University, 251 Mercer Street, ew York, Y 10012, USA zhipeng@cims.nyu.edu Department of Mathematics, ational University of Singapore, 10 Lower Kent Ridge Rd, Singapore matwd@nus.edu.sg 1

2 then for c 1, c 2, c 3 model dependent constant chosen in terms of microscopic dynamics in terms of the KZ scaling theory [35, 28] and suitable centering h ɛ x; t coming from the hydrodynamic theory the space-time process h ɛ ; will have a universal limit h ; which is independent of the underlying model. The class of all models which satisfy this is called the Kardar-arisi-Zhang universality class, and this limiting object is called the fixed point of this universality class. Much of the description and almost all of the universality of this fixed point remains a matter of conjecture. One of the main conjectures provided in [18] see also the review [33] about this fixed point is that its solution can be described via a variational problem in the spirit of the Lax- Oleinik formula for the inviscid Burgers equation involving a four-parameter random field called the space-time Airy sheet. A corollary of this conjectural description is that if h ɛ, 0 converges as a spatial process to some function h 0, then we have the following distributional equality, valid for any single pair of fixed x: hx, 1 r = Ay x y 2 h 0 y r. y R Here A is the Airy process Section 2.6 and by scaling properties of h, this implies a similar conjecture for general t. The main contribution of the present paper is a proof of this conjectured variational description for the limiting one-point distribution of TASE. In particular, consider the parameter q discrete time parallel update TASE height function Section 2.3 and define where h ɛ,tase x; t = ɛ1/2 h2c 0 ɛ 1 x; a 0 ɛ 3/2 t 2ɛ 1 t 2d 0 a 2 0 = 1 q, c 0 = 1 + q 2/3 q 1/6, d 0 = q1/6 1 + q 1/3. 2 Theorem 2.8 shows that if h ɛ,tase, 0 converges in distribution as a spatial process to some function h 0 then subject to certain growth hypotheses at infinity lim h ɛ,tase x, 1 r = Ay x y 2 h 0 y r. ɛ 0 y R In order to prove this we first relate the TASE height function one-point distribution to a discrete variational problem called point-to-curve geometric last passage percolation L. The particular curve in question encodes the height function initial data. L has been studied previously, and, in particular, Johansson [26] proved that the Airy process A minus a parabola describes the spatial fluctuations of point-to-point L as one point varies along an anti-diagonal line. In order to extend Johansson s result away from the anti-diagonal line and onto a general curve we prove a uniform slow decorrelation result, which shows that up to deterministic shift related to the given curve the fluctuations along the line and along a general curve agree. The final step in proving our result is to conclude that the resulting variational problem stays localized as ɛ goes to zero and this is achieved via a combination of large/moderate deviation bounds on TASE and a utilization of some regularity estimates coming from the Gibbs property of the associated multi-layer G line ensemble Section 6. In a similar manner we prove variational one-point distribution formulas for point to general curve L as well as L in which some of the weights have been perturbed. As a corollary of the TASE and L results we provide variational formulas for a number of known one-point distributions, such as arise in TASE with combinations of wedge, flat and stationary initial data. 2

3 Organization of the paper Section 2 introduces the models L and TASE as well as the main results Theorems 2.6, 2.8, 2.11 and 2.13 about them. The proofs of these theorems are applications of Theorem 2.18 on the uniform slow decorrelation and Theorem 2.19 on the Gibbs property, and they are given in Section 3. roofs of corollaries 2.9 and 2.14 are given in Section 4. The technical results, Theorems 2.18 and 2.19, are proved in Sections 5 and 6 respectively. Finally the appendix gives the proof of Lemma 2.2. Acknowledgements The authors thank to Jinho Baik, Jeremy Quastel and Daniel Remenik for fruitful discussions. Ivan Corwin was partially supported by the SF through DMS as well as by Microsoft Research and MIT through the Schramm Memorial Fellowship, by the Clay Mathematics Institute through the Clay Research Fellowship, by the Institute Henri oincare through the oincare Chair, and by the ackard Foundation through a ackard Foundation Fellowship. Zhipeng Liu greatfully acknowldeges the support from the department of mathematics, University of Michigan. Dong Wang was partially supported by the startup grant R Models and main results 2.1 oint-to-curve L Associate to each site i, j Z 2 an independent geometrically distributed random variable wi, j with parameter 1 q, such that wi, j = k = 1 qq k, k = 0, 1, 2, The point-to-point last passage time between two lattice points x, y and x, y is denoted by G x,y x, y and defined by { π G x,y x, y := i,j π wi, j } π x, y x, y if x x and y y, otherwise, where π stands for an up-right path such that π = π 0 = x, y, π 1, π 2,..., π x +y x y = x, y and π k+1 π k {1, 0, 0, 1}. More generally, if x, y is a lattice point, and x, y is on a line segment between two neighboring lattice points, then define { G G x,y x, y := the linear interpolation between x,y x, [y] and G x,y x, [y] + 1 if x Z, G x,y [x], y and G x,y [x] + 1, y if y Z. 3 If x, y and x, y are lattice points, we define the short-handed notations for the reversed last passage time as Ǧ x,y x, y = G x,y x, y and Ǧx, y := Ǧ0,0x, y = G x,y 0,

4 We also define Ǧx,y x, y by linear interpolation if x, y is on a line segment between two neighboring lattice points, analogous to 3. We will consider a more general point-to-curve last passage time, denoted by G x,y L in this paper. Let x, y be a lattice point and L be a lattice path in R 2 with L = Ls = {xs, ys s I}, for some interval I R. Here a lattice path means a directed path composed by line segments each of which connects two neighboring lattice points. Define { G x,y L = sup Gx,y xs, ys }. 5 s I Although s is a continuous parameter, it suffices to take the supremum among a discrete set of point-to-point last passage times. As preliminaries for our work, let us recall some important results about the asymptotic behavior of the point-to-point and point-to-curve last passage time. Focusing first on point-to-point last passage percolation, we state the law of large numbers, large/moderate deviations and the fluctuation limit theorems in the following proposition. ote that due to the symmetry of the lattice, we state our results in terms of Ǧx, y. roposition 2.1. Let γ be in a compact subset U of 0,. Then a Johansson [25] lim 1 Ǧγ, = a 0γ, almost surely, where a 0 γ = γ + 1q + 2 γq. 6 1 q b Baik-Deift-McLaughlin-Miller-Zhou [3] There exist a large constant M > 0 and a small constant δ > 0 such that for large, uniformly for all M x δ 1/3, there exists c > 0 such that Ǧγ, a0 γ x 1/3 e cx3. 7 c Johansson [25] where Ǧγ, lim a0 γ b 0 γ 1/3 x = F GUE x, 8 b 0 γ = q1/6 γ 1/6 q + γ 2/3 1 + γq 2/3, 9 1 q and F GUE x is the Tracy-Widom distribution for the limiting fluctuation of the largest eigenvalue in the Gaussian unitary ensemble GUE, see Section 2.6. In this paper, we need a counterpart of 7, which is stated below, and proved in Appendix A. Lemma 2.2. Let γ be in a compact subset U of 0,. There exist a large constant M > 0 and a small constant δ > 0 such that for large, uniformly for all M x δ 1/3, there exists c > 0 such that Ǧγ, a0 γ + x 1/3 < e cx. 10 4

5 We denote in this paper a 0 = a 0 1 = 2 q 1 q. 11 Define the limit shape curve see Figure 1 { } x Ľ := x, y 0, 0, ya 0 = a 0 y { rθ = cos θ, rθ sin θ θ 0, π 2 and rθ = 21 + } q cos θ + sin θ q + 2, cos θ sin θ and ote that 12 L := { x, y 1 y, 1 x Ľ}. 13 a 0 lim x 0 a 0 x = 2 + a 0 2q 1/2 and 1 lim x 0 a 0 x = 1 2q 1/2, 14 so Ľ L resp. is between 2 + 2q 1/2, 0 and 0, 2 + 2q 1/2 1 2q 1/2, 1 and 1, 1 2q 1/2 resp.. Then by roposition 2.1a we have that if x, y Ľ, then Ǧ[x], [y] = a 0 + o, or equivalently, if x, y L, then G, [x], [y] = a 0 + o q 1 2 c 3, 1 L 1, 1 1 2q 1 2 1, 1 Ľ 2 + 2q q 1 2 0, 0 1, c 3 1 L 1 2q 1 2 L 1 2q 1 2 Figure 1: The shapes of L and Ľ. Figure 2: Regions D shaded and D D together with the two corners enclosed by gray lines and an example of 1 L. The following result shows how the Airy process As see Section 2.6 arises in describing the spatial fluctuations of point-to-curve L. We define the step-like curve L 0 that is approximately a anti-diagonal straight line { L 0 = { l 0 s + s, l 0 s s s R}, where l 0 k s if s [k, k + 1 s = 2 ], s k 1 if s [k + 1 2, k + 1]. 15 5

6 roposition 2.3 Johansson [26]. Define the stochastic process H s := 1 Ǧ b 0 1/3 + l 0 c 0 2/3 s + sc 0 2/3, + l 0 c 0 2/3 s sc 0 2/3 a 0, 16 where a 0 is defined in 11 and b 0 = b 0 1 = q1/6 1 + q 1/3 1, and c 0 = 1 + q 2/3 q q 1/6. 17 Then on any interval [ M, M], we have the weak convergence as measures on C[ M, M], R as of H s As s The definition and some properties of the Airy process are provided in Section 2.6. This functional limit theorem for the fluctuations of all G, l 0 s+s, l 0 s s with s = O 2/3, together with a tightness argument for large s, yields roposition 2.4 Johansson [26]. As, the point-to-curve last passage time from, to L 0 satisfies G, lim L 0 a 0 b 0 1/3 x = As s R s2 x Main result on fluctuations in point-to-curve L Our main result, Theorem 2.6, provides a similar variational characterization as Johansson s results roposition 2.4 for point-to-curve L with a general class of the lattice paths. Before stating our theorem, we specify the class of lattice paths which we will consider. It is clear from 12, 13 and Figure 1 that the horizontal line y = 1, the vertical line x = 1 and the curve L enclose a region, which we denote by D. Then we define the region D as the main part of D with the two sharp corners cut off. To be precise, we define, as shown in Figure 2, D = D \ {x, y x < c 3 or y < c 3 }, where c 3 1 2q 1/2, The meaning of the constant 1 2q 1/2 is shown in 14 and Figure 2. Then we let C R, c 1 0, 1 and c 2 0, 1/3 be constants, let l : R R be a continuous function and {m } R + be a sequence of positive real numbers such that ls < C + c 1 s 2 and lim + m = We consider a sequence of lattice paths L. For each L, we denote its central part as { x, y L x y < 2c0 2/3+c 2 L central = Then we assume the following: Hypothesis 2.5. }. 22 6

7 There is an interval and such that L central = I = a, b, where c 2 a < b c 2, 23 a a { } R, b b {+ } R, 24 { sc 0 2/3 ls + l sd 0 1/3, sc 0 2/3 ls + l sd 0 1/3 s I }, where l s : I R is a continuous function with 25 l s < m, 26 s I and d 0 = b 0 = 1 + q 1/3 a 0 2q 1/3. 27 The other part of L satisfies { x, y x, y L \ L central as depicted in Figure 2, and { dist L \ L central, ], ], x, y x, y where the distance is the Euclidean distance. }, 1], 1] D, 28 } L > 1/3+2c 2, 29 ote that we have no requirement of L outside of the region, ], ], because G, L = G, L, ], ]. 30 Although it suffices to consider G, L, we state the result with an extra parameter σ to make it parallel to Theorem 2.8 stated later. Theorem 2.6. Fix ls and a sequence of lattice paths L satisfying Hypothesis 2.5 with constants C, c 1, c 2, c 3 and sequence {m } defined above Hypothesis 2.5, and also fix σ > 0. Then for all ɛ > 0 there exists 0 depending on C, c 1, c 2, c 3, {m } and σ but not ls or L such that for all > 0 and all x R G+[σc0 2/3 ], [σc 0 2/3 ] L a 0 b 0 1/3 x s a,b As s σ 2 + ls x < ɛ. 31 ote that s a,b As s σ 2 + ls is a well defined random variable, see Corollary

8 Remark 1. Theorem 2.6, as well as the subsequently stated results of Theorems 2.8 and 2.11 are stated for deterministic initial / boundary data. Here in the statement of the theorem, and later in the proof, we show that the convergence rate is independent of the particular formula of ls and the particular shape of L. This is because later we are going to use the result actually its analog in TASE model detailed below when ls is random, say, distributed as the path of random walk. Focusing on the above result, assume that ls is random and that for all ɛ > 0 there exist constants C R and c 1 0, 1 such that with probability at least 1 ɛ, ls < C + c 1 s 2 for all s. Then Theorem 2.6 holds for such a random ls. Instead of coupling all initial data L to a single possibly random ls it is also possible to consider L which satisfy all of the conditions of Hypothesis 2.5, except that 25 is replaced by L central = { sc 0 2/3 l s + l sd 0 1/3, sc 0 2/3 l s + l sd 0 1/3 s I }, 32 where l s converges as a spatial process to some possibly random ls satisfying the aforementioned bounds. The above theorem is proved in Section 3.1. There are two main ingredient in the proof. The first one is to show that the end of the longest path most likely lies in the vinicity of 0, 0, that is, a point x, y with x + y = O 1/3 and x y = O 2/3. The second one is to show that the theorem holds in the special case that L is of length 2/3, which is proved by the uniform slow decorrelation property of the L model, see Theorem TASE with general initial data For the analysis of the TASE model, we introduce a slightly different L model where the i.i.d. random variables w i, j associated to each site are geometrically distributed on Z >0 w i, j wi, j + 1, such that w i, j = k = 1 qq k 1, k = 1, 2, We similarly define the point-to-point L G x,y x, y, point-to-curve L G x,y L, and the reversed L Ǧ x,y x, y by 3, 5 and 4 with the weights changed from wi, j to w i, j. They have simple relations to the Ls G x,y x, y, G x,y L and Ǧ x,y x, y defined there, for example, if one of x, y and x, y is a lattice point, G x,y x, y = G x,y x, y + x + y x y The TASE model considered in our paper is that with discrete time and parallel updating dynamics [9], and is defined as follows. Let infinitely many particles be initially at time t = 0 placed on the integer lattice Z such that no lattice site is occupied by more than one particle, and there are infinitely many particles to the left of 0. At each integer time, the particles decide whether to jump to the right neighboring site simultaneously. For any particle x at time t = n, if its right neighboring site xn+1 was occupied at t = n, then it does not move and xn+1 = xn; otherwise it jumps to the right neighboring site xn + 1 = xn + 1 with probability 1 q, or does not move xn + 1 = xn with probability q. At any time t 0, we represent the positions of the particles by the height function h ; t : R R. We let h0; t = 2 t where t is the number of particles that have jumped from site 1 to 8

9 s 0, 0 hs; 0 2 L 0 Figure 3: The height function hs; t at the initial time t = 0. If at time t = 1 one particle jumps from 3 to 2, then hs; 1 is changed into the dashed shape. The polygonal chain L representing the initial state of the model is shown on the right. site 0 during the time interval [0, t. For any integer k, we define hk; t inductively from h0; t by hk + 1; t hk; t = ±1 where the sign is positive negative resp. if the site k is vacant occupied resp. by a particle at time t. At last, for non-integer s, we define hs; t by the linear interpolation between h[s]; t and h[s] + 1; t. See Figure 3 for an example. Also noting that hs; t = hs; [t] for all t R +, we have that hs; t is determined by the values of hk; n where k, n Z. Another observation is that the value of hk; t is an integer that has the same parity of k. To analyze the dynamics of the TASE model, or equivalently, the dynamics of the height function hs; t, we introduce the polygonal chain { s L = hs; 0, s hs; 0 s [K 1, K 2 ]}, 35 2 to represent the initial configuration of the model, as shown in Figure 3, where K 1 is the position of the leftmost unoccupied site at t = 0 if it exists, or otherwise, and K 2 is one plus the position of the rightmost occupied site at t = 0 if it exists, or + otherwise.. The TASE model can be coupled to the L model with weights w i, j defined in 33 see [25, 15] for example. The relation between the distribution of hj; t and the L is given by hj; t > k = G k+j 2, k j L t, 36 2 for any j, k Z with the same parity. Here L is the polygonal chain defined in 35. This coupling follows by defining the TASE height function at time t as the rotated envelop of all points which has last passage time less than or equal to t. The weights correspond with the probabilities of particle movement. 2.4 Main result on TASE with general initial data ow we consider the TASE model with general initial condition. Since the TASE model is mapped to the L model with weight function given in 33, the result for the TASE is analogous to that of the L model stated in Section 2.2. Below we set up the notations for the L model with weight 33, give technical conditions in terms of L, and then present the result in terms of the TASE model. 9

10 Analogous to roposition 2.1a, we have 1 Ǧ γ, = a 0γ almost surely, where a 0γ = a 0 γ + γ + 1 = γ γq q lim Then parallel to Ľ and L defined in 12 and 13, we define and then and Ľ := = a 0 = a 01 = 2 1 q = a 0 + 2, 38 { x, y 0, 0, } x ya 0 = a 0 y { rθ cos θ, rθ sin θ θ 0, π 2 and rθ = 21 + } q cos θ + sin θ + 2, cos θ sin θq 39 L := {x, y 1 y, 1 x Ľ }. 40 By Theorem 2.1a, b again, we have that if x, y Ľ, Ǧ x, y = a 0 + o, and equivalently if x, y L, G, x, y = a 0 + o. Then parallel to the regions D and D shown in Figure 2, we define the region D as the region enclosed by x = 1, y = 1 and L, and then D = D \ {x, y x < c 3 or y < c 3}, where c 3 1 2q 1/2, 0, 41 where the value 1 2q 1/2 is analogous to the value 1 2q 1/2 in 20. We also let C R, c 1 0, 1, c 2 0, 1/3, let l : R R be a continuous function and let {m } R + be a sequence of positive real numbers such that 21 is satisfied. We consider a sequence of lattice paths L analogous to L considered in Section 2.2. We assume that each L is defined by the initial condition of a TASE model, that is, for each index, we consider the TASE model represented by a height function h s; t, and then let L be the polygonal chain L that is defined by h s; 0 in 35. For each L, we denote its central part as { L,central = x, y L } x y 2c 0 2/3+c Then we assume the following Hypothesis 2.7. There is an interval where I = a, b, 43 c 2 a < b c 2, and a a { } R, b b {+ } R, 44 such that L,central = { sc 0 2/3 ls + l sd 0 1/3, sc 0 2/3 ls + l sd 0 1/3 } s I, 10 45

11 or equivalently, h 2sc 0 2/3 ; 0 = 2ls + l sd 0 1/3, s I, 46 where c 0 is defined in 17, l s : I R is a continuous function with l s < m, 47 s I and d 0 is defined, analogous to 27, as The other part of L satisfies { and dist d 0 = b 0 a 0 x, y x, y L \ L,central = q1/6 1 + q 1/ }, 1], 1] D, 49 L \ L,central, ], ] {, x, y x, y L } > 1/3+2c 2, 50 where the distance is the Euclidean distance. Theorem 2.8. Fix ls and a sequence of lattice paths L satisfying Hypothesis 2.7 with constants C, c 1, c 2, c 3 and sequence {m } defined above Hypothesis 2.7, and also fix σ > 0. Here for each, L is associated to the initial condition of a TASE model whose height function is denoted by h s; t via the relation 35. Then for all ɛ > 0 there exists 0 depending on C, c 1, c 2, c 3 and {m }, σ but not ls or L such that for all > 0 and all x R, the height function h s; 0 of the TASE model, as defined in 35, satisfies h 2σc ; a 0 2 2d 0 > x As s σ 2 + ls < x < ɛ s a,b Remark 2. By the relation 36, we have that under the assumption that 2d 0 1/3 x and 2σc 0 2/3 are integers with the same parity, h 2σc ; a 0 2 2d 0 > x = G 1 3 +σc d x, σc d x L [a 0]. 52 The above equation implies Theorem 2.8 on TASE is equivalent to an analog of Theorem 2.6 on L. Below we list several typical initial conditions of TASE, and their initial height functions. We characterize the initial height function hs; 0 only at integer-valued s. ote that all the initial conditions are -independent. Theorem 2.8 covers more general, -dependent initial conditions, for example, periodic initial conditions with period O 2/3. 11

12 Step initial condition Initially all negative sites are occupied and all non-negative sites are empty, i.e., h step s; 0 = s. 53 Flat initial condition Initially all even sites are occupied and all odd sites are empty, i.e., { h flat 0 if s = 0, ±2, ±4,..., s; 0 = 54 1 if s = ±1, ±3,..., Brownian/Bernoulli/stationary initial condition Initially all sites are independently occupied with probability 1 2 and empty with probability 1/2, i.e., 0 if s = 0. h Bern s; 0 = s 1 i=0 w i if s = 1, 2,..., 55 1 i= s w i if s = 1, 2,..., and w i are random variables in i.i.d. two-point distribution such that w i = 1 = 1/2 and w i = 1 = 1/2. Wedge-flat initial condition Initially all negative sites and all even sites are occupied, but all positive odd sites are empty, i.e., { h step/flat h flat s; 0 if s 0, s; 0 = h step 56 s; 0 if s < 0. Wedge-Bernoulli initial condition Initially all negative sites are occupied, and all nonnegative sites are independently occupied with probability 1/2 and empty with probability 1/2, i.e., { h step/bern h Bern s; 0 if s 0, s; 0 = h step 57 s; 0 if s < 0. Flat-Bernoulli initial condition Initially all even negative sites are occupied, all odd negative sites are empty, and all non-negative sites are independently occupied with probability 1/2 and empty with probability 1/2, i.e., { h flat/bern h Bern s; 0 if s 0, s; 0 = h flat 58 s; 0 if s < 0. As consequences of Theorem 2.8 we can prove variation formulas for one-point distributions of TASE started from initial data as in 54, 55 56, 57 and 58, we have the following results. To state the results in a uniform way, we denote the two-sided Brownian motion Bs by { B + s if s 0, Bs = 59 B s if s 0, where B + s and B s are independent standard Brownian motions starting at 0. 12

13 Corollary 2.9. Let σ be a real constant and hs; t be the height function of the TASE. a With the flat initial condition 54, lim h flat 2σc 0 2/3 ; a 0 2 2d 0 1/3 b With the Bernoulli initial condition 55, lim h Bern 2σc 0 2/3 ; a 0 2 2d 0 1/3 > x c With the Wedge-flat initial condition 56, lim h step/flat 2σc 0 2/3 ; a 0 2 2d 0 1/3 d With the Wedge-Bernoulli initial condition 57, lim h step/bern 2σc 0 2/3 ; a 0 2 2d 0 1/3 = e With the Flat-Bernoulli condition 58, s 0 lim h flat/bern 2σc 0 2/3 ; a 0 2 2d 0 1/3 s R > x = > x > x = s R s R As s 2 < x. 60 As s σ 2 + 2q 1/4 Bs < x. = s σ 61 As s 2 < x. 62 As s σ 2 + 2q 1/4 Bs < x. 63 > x = As s σ 2 + 2q 1/4 χ s 0 Bs < x. 64 Remark 3. The result for the flat initial condition 54 is obtained in [26] and is given, in an equivalent form, in roposition 2.4 in the case that σ = 0. Since the flat initial condition is translational invariant, the result holds for general σ. The step initial condition is singular in the sense that K 1 = K 2 = 0 in 35 and hence a = b = 0 in 43 and then the interval a, b is degenerate into a point {0}. The result, which is stated in roposition 2.1c, actually is used in the proof of Theorem 2.8, so we do not list it as a corollary. The situation is comparable to that explained in [16, Remark 1.6]. Comparing the results in Corollary 2.9 with the asymptotics of hs; t obtained in continuous TASE models see [4], [10], [2], [11] for details that corresponds to the q 1 limit of the 13

14 discrete TASE model considered in this paper, we obtain, modulo a change in order of taking limits q 1 and, which we do not justify it in this paper As s σ 2 < x = 2 1/3 A 1 2 2/3 σ < x, 65 t R As s σ 2 + 2Bs < x = A stat σ < x, 66 s R t R s 0 s 0 As s σ 2 < x = A 2 1 σ < x + σ 2 χ σ<0, 67 < x = A BM 2 σ < x + σ 2, 68 As s σ 2 + 2Bs As s σ 2 + 2χ s 0 Bs Below are explanations of notations: < x = A 2 1,1,0 σ < x + σ 2 χ σ>0. 69 In 65, A 1 stands for the Airy process with flat initial data, defined in [34] and [8, Formulas 1.4 and 1.5]. The A 1 process is stationary, and its 1-dimensional distribution is [21] where F GOE is the Tracy-Widom GOE distribution [36]. A 1 σ < x = F GOE 2x, 70 In 66, A stat stands for the Airy process with stationary initial data, defined in [4], and we follow the notation in [31, Section 1.11] and [33, Section 1.2]. The 1-dimensional distribution of A stat σ appears also in literature as see [4, Remark 1.3] and [22, Appendix A] A stat σ < x = F σ x = H x + σ 2 ; σ 2, σ, 71 2 where F σ x is defined in [22, Formula 1.20] and Hx; w +, w is defined in [5, Definition 3]. In 67, the transition process A 2 1 interpolating the A 2 and A 1 processes is introduced in [10, Definition 2.1] see also [32, Formula 1.7], where the notation for the right-hand side of 67 is G 2 1 σ x + σ 2 χ σ<0. In 68, the transition process A BM 2 interpolating the Brownian motion and A 2 process is introduced in [24, Formula 3.6], see also [14, Definition 2.13]. The 1-dimensional distribution of A BM 2 σ was conjectured in [29] and proved in [7] to be A BM 2 σ < x = F 1 x; σ, 72 where the distribution function F 1 is introduced in [2, Definition 1.3]. In formula 69, The transition process A 2 1,1,0 interpolating the Brownian motion and the A 1 process is introduced in [11, Definition 18]. It is defined from the TASE with one slow particle, and it is related to the TASE with flat-bernoulli initial condition via Burke s theorem, as explained in [11]. 14

15 Among formulas 65, 66, 67, 68 and 69, 65 is proved in [26], and then proved in a direct way in [19]. Formula 67 is proved in [32]. Formulas 66, 68 and 69 are conjectured in [33, Section 1.4]. ote that in [33, Section 1.4], the notations A 1 BM and A 2 BM are described but not precisely defined. From the context we figure out that A 2 BM σ = A BM 2 σ σ 2 χ σ>0, A 1 BM σ = A 2 1,1,0 σ σ 2 χ σ>0. 73 Formulas 66 and 68 are special cases of Corollary 2.15 c with w + = w = σ 2 and a with k = 1 in Section 2.5. And our argument in this paper is a strong support to the conjectural formula 69. Remark 4. The method in our study of the discrete time TASE, if applied on the continuous time TASE, that is, the q 1 limit of the discrete time one, yields the counterparts of 61, 62, 63 and 64 with q = 1, and then the formulas 65, 66, 67, 68 and 69 are derived directly. The only technical obstacle in the application of our method in the continuous time TASE is that the counterpart of roposition 2.4, where the discrete geometric distribution of wi, j is replaced by the continuous exponential distribution is not available in literature. We remark that the counterpart of roposition 2.4 can also be proved by the method in [26]. 2.5 L with an inhomogeneous weight distribution In this subsection we consider the point-to-point L on a Z 2 lattice where the weights on sites are in independent geometric distribution, but with nonidentical parameters. The strategy is to express the point-to-point L with respect to these weights by point-to-curve L with respect to homogeneous weight as considered in Section 2.1. Let L be the vertical path depending on which we suppress L := {0, y y D }, where D is an interval on R. 74 We are most interested in the case that D = R. But the L G, L is not well defined in this case, since G, 0, y + almost surely as y. We consider a modified L G f, L = y D G, 0, y f y 75 where f : D R is a function where D, the domain of f, is an interval. This modified L G f, L is well defined for D = R if f x + fast enough as x. By roposition 2.1, for y = c where c is in a compact subset of, 1, if f y = a 0 1 y/, then G, 0, y = o. So if f y is close to a 0 = a 0 1 for y around 0, and otherwise greater than a 0 1 y/ for all y <, then G f, L is o and the value of y such that G, 0, y f y attains its imum in the vicinity of 0. To make the idea above precise, we state a technical hypothesis for f analogous to Hypotheses 2.5 and 2.7. First let C R, c 1 0, 1, c 2 0, 1/3 and c 4 > 0 be constants, let l : R R be a continuous function and let {m } R + be a sequence of positive numbers such that 21 is satisfied. Hypothesis There is an interval I = a, b, 76 15

16 c 2 a < b c 2, and a a { } R, b b {+ } R, 77 such that f 2sc 0 2/3 = a 0 sa 0 c 0 2/3 ls + l sd 0 1/3, 78 for all s I, where l s : R R is any continuous function with s I l s < m, and ls and m are specified in 21. For all y D such that y 1/3 /2c 0, 1/3 /2c 0 ]\I, f y satisfies the inequality f y > a 0 a 0y y 2 2 c 1d 0, a 0 1 y + c 4 y. 79 2c 0 Theorem Fix ls and a sequence of functions f satisfying Hypothesis 2.10 with constants C, c 1, c 2, c 4, c 5, {m } defined above Hypothesis Then for all ɛ there exists 0 depending on C, c 1, c 2, c 4, c 5, m but not ls or f such that for all > 0 and all x R, Gf, L < x b s a,b Another similar question is to consider the -shaped path y 0 As s 2 + ls < x < ɛ. 80 L = { 0, y y 0} {x, 0 x 0 }. 81 The L G, L is well defined and equivalent to the point-to-point L G, 0, 0. If f : R R is a continuous function such that f x increases at a proper speed as x increases, then the modified L G f, L = G, 0, y f y, G, x, 0 f x 82 x 0 has nontrivial limiting property like that of G f, L stated in Theorem To make the idea above precise, we state a technical hypothesis for f analogous to Hypothesis First let C R, c 1 0, 1, c 2 0, 1/3 and c 4 > 0 be constants, let l : R R be a continuous function and let {m } R + be a sequence of positive numbers such that 21 is satisfied. Hypothesis There is an interval I = a, b, 83 c 2 a < b c 2, and a a { } R, b b {+ } R, 84 such that f 2sc 0 2/3 = a 0 s a 0 c 0 2/3 ls + l sd 0 1/3, 85 for all s I, where l s : R R is any continuous function with s I l s < m and ls and m are specified in

17 For all y 1/3 /2c 0 [ 1/3 /2c 0, 1/3 /2c 0 ] \ I, f y satisfies the inequality f y > a 0 a 0 y y 2 c 1 d 0, a 0 1 y + c 4 y c 0 Theorem Fix ls and a sequence of functions f satisfying Hypothesis 2.12 with constants C, c 1, c 2, c 4 and sequence {m } defined above Hypothesis Then for all ɛ there exists 0 depending on C, c 1, c 2, c 4, {m } but not ls or f such that for all > 0 and all x R, G f, L b < x s a,b As s 2 + ls < ɛ. 87 As applications of Theorems 2.11 and 2.13 or adaption of their proofs, see Remark 5, we have the following results for point-to-point L with inhomogeneous weight parameters. The weight parameters we will consider differ from the homogeneous ones considered in Section 2.1 in only finitely many columns and/or rows. So we use the same notation Ǧ, which is defined in 4 and 2, but the weights on some of the lattice points are defined differently. To state the following corollaries, we denote by A 1 and A 2 two independent Airy processes that are the A described in Section 2.6, and denote by B 1,..., B k independent two-sided Brownian motions that are the B defined in 59. Corollary In the Z 2 lattice we consider the point-to-point L Ǧ,, and denote where a 0 and b 0 are defined in 11 and 17 respectively. G = Ǧ, a 0 b 0 1/3, 88 a Suppose the weights wi, j are independent and geometrically distributed with parameter α i,j such that α i,j = 1 q if i / {1, 2,..., k} and α i,j = 1 q 1 2w i d 0 1/3 if i = 1,..., k, 89 where k Z + and w 1,..., w k R are constants. Then lim G x = 0=s 0 s 1 s k As k + 2 k B i s i B i s i 1 4 k w i s i s i 1 s 2 k x. b Suppose the weight w0, 0 is fixed to be 0, the weights wi, j are independent and geometrically distributed with parameter α i,j if i, j are not both 0, such that α i,j = 1 q if i, j are both nonzero, and 1 q α i,j = 1 q 1 2w + 1 2w d 0 1/3 d 0 1/3 17 if i 1 and j = 0, if i = 0 and j

18 where w +, w R are constants. Then lim G x = s R As + 2Bs + 4w + 1 s<0 w 1 s>0 s s 2 x. 92 c Suppose the weight wi, j are independent and geometrically distributed with parameter α i,j such that α i,j = 1 q if j [αn] or j > [αn] + k, and α i,j = 1 q 1 2w j [αn] d 0 1/3 if j = [αn] + 1,..., [αn] + k, 93 where α 0, 1, k Z + and w 1,..., w k R are constants. Then lim G x = α 1/3 A 1 α 2/3 s 0 + k 2 B i s i B i s i 1 s 0 s 1 s k + 1 α 1/3 A 2 1 β 2/3 s k 4 k w i s i s i 1 s2 0 α s2 k 1 α x. 94 Remark 5. arts a and b of Corollary 2.14 are direct conseqeunces of Theorems 2.11 and 2.13 respectively, but art c does not follow these theorems in a straightforward way, although the proofs of the theorems can be adapted to prove art c. The limits on the left-hand sides of 90, 92 and 94 have been analyzed previously in [5], [2] and [1], and the results were given in other forms by Fredholm determinants. Utilizing these earlier results we arrive at the following expressions for these statistics. Corollary For all x R, a for all parameters w 1,..., w k R, As k + k 2 B i s i B i s i 1 4 0=s 0 s 1 s k k w i s i s i 1 s 2 k x = F spiked k x; 2w 1,..., 2w k, 95 b for all parameters α 0, 1 and w 1,..., w k R, α 1/3 A 1 α 2/3 s 0 + k 2 B i s i B i s i α 1/3 A 2 1 β 2/3 s k s 0 s 1 s k 4 k c for all parameters w +, w R, s R w i s i s i 1 s2 0 α s2 k 1 α x As + 2Bs + 4w + 1 s<0 w 1 s>0 s s 2 x = F spiked k x; 2w 1,..., 2w k, 96 = Hx; w +, w, 97 where F spiked k x; w 1,..., w n is the distribution introduced in [2, Formula 54] and [1, Corollary 1.3], and Hx; w +, w is the distribution function introduced in [5]. 18

19 2.6 The Airy process The Airy process A [30] sometimes also denoted as A 2 and called the Airy 2 process, in contrast to the Airy 1 process A 1 considered in 65 is an important process appearing in the Kardar-arisi- Zhang universality class, see for example [13]. Its properties have been intensively studied, see for example [26], [17], [33]. The Airy process A is defined through its finite-dimensional distributions which are given by a Fredholm determinant formula. For x 0,..., x n R and t 0 <... < t n in R, At 0 x 0,..., At n x n = deti f 1/2 K ext f 1/2 L 2 {t 0,...,t n} R, 98 where we have counting measure on {t 0,..., t n } and Lebesgue measure on R, f is defined on {t 0,..., t n } R by ft j, x = 1 x xj,, and the extended Airy kernel [30] is defined by { K ext t, ξ; t, ξ = 0 dλ e λt t Aiξ + λ Aiξ + λ, if t t 0 dλ e λt t Aiξ + λ Aiξ + λ, if t < t, where Ai is the Airy function. It is readily seen that the Airy process is stationary. The one point distribution of A is the F GUE distribution i.e., the GUE Tracy-Widom distribution [36]. Since our main results appear as variational problems involving the Airy process, it is important to know that these problems are well-posed with finite answers. It was proved in [30, Theorem 4.3] and [26, Theorem 1.2] that there exists a measure on CR, R continuous functions from R R endowed with the topology of uniform convergence on compact subsets whose finite dimensional distributions coincide with those of the Airy process i.e., there exists a continuous version of the Airy process. Further properties of the Airy process were demonstrated in [17]. We summarize those properties which we will appeal to. art a of roposition 2.16 is a special case of [17, roposition 4.1], our At is their A 1 t, while art b is a generalization of [17, roposition 4.4] where the parameter c is taken as 1, and the proof can be used for our generalized case with little modification. roposition a Local Brownian absolute continuity For any s, t R, t > 0, the measure on functions from [0, t] R given by A + s As is absolutely continuous with respect to Brownian motion of diffusion parameter 2. b For all positive constants α and c such that α < c, there exists ɛ > 0 and Cα, c > 0 such that for all t Cα, c > 0 and x αt 2, sup As cs 2 > x e ɛct2 +x 3/2. 99 s/ [ t,t] One direct consequence of roposition 2.16 is the well-definedness of the limit distributions in Theorems 2.6, 2.8, 2.11 and Corollary Let l : R R be a continuous function that satisfies 21 and a, b be an interval such that a < b +. Then s a,b As s σ 2 + ls is a well defined random variable. The definition of the Airy process given by 98 is not well adapted to studying variational problems as it only deals with finite dimensional distributions. Let us note that [19, Theorem 2] provides a concise Fredholm determinant formula for As gs for s [l, r], for any interval [l, r] and any g H 1 [l, r] i.e. both g and its derivative are in L 2 [l, r]. As we do not utilize this formula, we do not restate it here. 19

20 2.7 Main technical tools The main technical tools in this paper are results stemming from the uniform slow decorrelation property that allows us to generalize roposition 2.3 by Johansson, and the Gibbs property of a multilayer line ensemble extension of the L model. As this will require some explanation, we delay a discussion of it until Section 6. Recall the stochastic process H s defined in 16. We define more generally H s = 1 b 0 1/3 Ǧ + l s α + sc 0 2/3, + l s α sc 0 2/3 a 0 + l s α 100 where α [0, 1 is a parameter and l s is a sequence of continuous functions such that the curve L = l s α + sc 0 2/3, l s α sc 0 2/3, s R is a lattice path. If α = 0 and l s = l 0 s/c 0 2/3 where l 0 s is defined in 15, then Hs is equal to H s defined in 16. Theorem Let H s be defined in 100 with α 0, 1 and l t continuous on [ M, M] and s [ M,M] l s < C for all large enough. Then H s H s converges in probability to 0 in C[ M, M], R, that is, given ɛ, δ > 0, there is an integer 0 that depends only on M, α and C such that H s H s δ < ɛ 101 s [ M,M] if > 0. The slow decorrelation property is a common feature in many models in the KZ universality, including the L model, and equivalently the TASE model, considered in this paper. As a pointwise property, it is studied first in [20] and then comprehensively in [15]. Let M 0 +, then we have the result that as, 1/3 Ǧ, a 0 is equal to 1/3 Ǧ +l 0 α, + l 0 α a 0 + l 0 α in probability. This is a special case of the slow decorrelation result obtained in [15], where the charasteristic line is the π/4 radial line. Theorem 2.18 generalizes the pointwise slow decorrelation to be uniform on an interval. Theorem 2.18 gives control of H s in any fixed interval [ M, M]. Outside this fixed interval we need the following lemma to control the point-to-curve Ls by point-to-point Ls as shown in Figure 4. The lemma is a special consequence of the Gibbs property see Section 6, but it suffices for our paper. Lemma Suppose > 0, K 1 < K 2 < K 3 are integers between and, and M 1, M 2, M 3 are real numbers such that K 1, M 1, K 2, M 2, K 3, M 3 are colinear, i.e.,, M 1 M 2 K 1 K 2 = M 2 M 3 K 2 K Let c 0, 1 be a constant and let l 0 s be defined in 15. Then Ǧ + l 0 s + s, + l 0 s s M 0 K 1 s K 2 ck 2 K ɛ minck2 K 1,K 3 K 2 Ǧ + K 2, K 2 M 2 + Ǧ + K 3, K 3 M 3, where for all t > 0, ɛ t is a positive constant such that ɛ t 0 as t

21 +K 1, K 1 +K 1 ck 2 K 1, K 1 +ck 2 K 1 +K 2, K 2 +K 3, K 3 0,0 Figure 4: The points + K 1, K 1, + K 2, K 2 and + K 3, K 3 are on the same diagonal lattice path. The left-hand side of 103 is the L between the point 0, 0 and the lattice path between + K 1, K 1 and + K 1 ck 2 K 1, K 1 + ck 2 K 1, shown in solid. 3 roof of Theorems 2.6, 2.8, 2.11 and 2.13 In this section, we give the detail of the proof of Theorem 2.6 in Section 3.1, and show briefly that Theorem 2.8 can be proved by the method the same as the proof of Theorem 2.6 in Section 3.2. The proofs of Theorems 2.11 and 2.13, as well as the proof of Corollary 2.14b are by the same method with some adaptions, and we discuss it in Section roof of Theorem 2.6 By the translational invariance of the lattice, we can shift the point + [σc 0 2/3 ], [σc 0 2/3 ] into,, and thus if we can prove Theorem 2.6 in the special case that σ = 0, the general case is proved by shifting the lattice. Therefore, we only prove the σ = 0 case of Theorem 2.6 for notational simplicity. In the proof of Theorem 2.6, we suppose x is a fixed real number and the constants C, c 1, c 2, c 3 defined in are fixed. Without loss of generality, we prove only for the case that I, the interval defined in Hypothesis 2.5, is [ c 2, c 2 ]. By 30, we only need to consider the curve L,, where L is defined in 21 29, and we divide it into into parts L micro M, L meso,l M, L meso,r M, and L macro, where the first three depending on a constant M > 0, such that, recalling that L central defined in 22, L micro M = {x, y L central x y 2Mc 0 2/3 }, 104 L meso,l M = {x, y L central \ L micro M x < 0}, 105 L meso,r M = {x, y L central \ L micro M x > 0}, 106 L macro = L,, \ L central. 107 In Subsection 3.1.1, we show that for any fixed M > 0 and ɛ > 0, G, L micro M a 0 d 0 1/3 x As s [ M,M] s2 + ls x < ɛ

22 for all large enough, independent of the particular formula of ls. In Subsection 3.1.2, we show that for any fixed ɛ > 0, there is an M such that for all large enough, independent of the particular formula of ls, G, L meso, M a 0 d 0 1/3 > x < ɛ, for = L or R, 109 and for any fixed ɛ > 0, for all large enough, independent of the particular formula of ls, G, L macro a 0 d 0 1/3 > x < ɛ. 110 Thus by the three inequalities 108, 109 and 110, and the limit identity lim As s 2 + ls < x = As s 2 + ls < x M s [ M,M] s R 111 that is a consequence of roposition 2.16b, we prove the inequality 31 of Theorem Microscopic estimate In this subsection we prove that the inequality 108 holds for large enough, where M > 0 and ɛ > 0 is a constant. The main technique to prove 108 is Theorem Since Theorem 2.18 requires a boundedness of l s, we first prove 108 under the condition ls < c s [ M,M] Recall the stochastic processes H s defined in 16, and H s defined in 100. By the symmetry of the lattice, we have that G, L micro M a 0 d 0 1/3 x = H s + ls + l s x, 113 s [ M,M] where H s is defined with l s = ls + l s and the parameter α = 1/3. Since H s, as a stochastic process in s [ M, M], converges weakly to As s 2 and l s uniformly converges to 0, with the help of Skorohod s representation theorem, we have that for any ɛ > 0 there is a δ > 0 such that for large enough independent of ls s [ M,M] H s + ls + l s x + δ 2 s [ M,M] < As s 2 + ls x δ ɛ3 < s [ M,M] As s 2 + ls x + δ + ɛ 3, 114 H s + ls + l s x δ. s [ M,M]

23 By roposition 2.16a, the Airy process is locally like the Brownian motion [17], so if δ is small enough, then As s 2 + ls x + δ As s 2 + ls x < ɛ s [ M,M] s [ M,M] 3, 116 As s 2 + ls x As s 2 + ls x δ < ɛ s [ M,M] s [ M,M] The uniform slow decorrelation of L given in Theorem 2.18 implies that H s H s > δ < ɛ s [ M,M] for large enough, independent of the particular formula of ls and l s, as long as ls satisfies 112. The inequalities 116, 117, 114, 114 and 118 yield 108 for all ls satisfying both 21 and 112. For large enough independent of ls, G, L micro M a 0 d x = s [ M,M] H s + ls + l s x < H s + ls + l s x + δ s [ M,M] 2 < As s 2 + ls x + δ + 2ɛ s [ M,M] 3 < As s 2 + ls x + ɛ. s [ M,M] + ɛ Thus one direction of inequality 108 is proved under the condition 112. The proof of the other direction of 108 under the condition 112 is similar. Finally note that the condition 21 implies 112 for large. Therefore 108 holds for all ls that satisfy Macroscopic and mesoscopic estimates Macroscopic estimate Inequality 110 is a direct consequence of Lemma 2.2. For any x, y on L macro. Since 1 x, 1 y D by 28 where D is defined in 20, and L satisfies the relation 29, by Lemma 2.2, we have that for all large enough, G, x, y a 0 d 0 1/3 > x < e c 2c2, 120 where c > 0 depends on c 3 in 29 but not the shape of L macro. ote that G, x, y is a constant for x, y in a lattice square, and there are fewer than 41 + q 1/2 2 2 lattice squares whose image under the scaling transform x, y 1 x, 1 y is in D. Then we can pick x i, y i on L macro 23

24 where i = 1,..., [41 + q 1/2 2 2 ], such that for all x, y L macro, G, x, y is equal to at least one G, x i, y i. Thus G, L macro a 0 d 0 1/3 > x < [41+q 1/2 2 2 ] < 41 + q 1/2 2 2 e c 2c2, G, x i, y i a 0 d 0 1/3 > x 121 and obtain inequality 110 if is large enough. Mesoscopic estimate By the symmetry of the lattice model, we only need to prove 109 with = R. Before giving the proof, we remark that the simple approach in the macroscopic estimate fails in this case, since summing up all the point-to-point L between, and lattice points on L meso,r M gives a too large upper bound of the point-to-curve L G, L meso,r. Before giving the technical proof, we explain the idea. We divide L meso,r into segments according to the intervals Ik in 125. Then on each segment, we estimate the point-to-curve L actually the upper bound k defined in 127 by the point-to-point Ls between 0, 0 and the two points in 131b and 131c. We estimate the point-to-point Ls by Lemma 2.2, and the relation between point-to-point Ls and the point-to-curve L is established by Lemma Recall that L meso,r M L central is defined in 25 by a continuous function ls + l s for s [M, c 2 ], where ls is bounded below by C + c 1 s 2 and l s converges uniformly to 0 as. By the inequality 21, we have that ls < c 1s 2 for all s [ M, c 2 ], where c 1 c 1, 1 and M = C/c 1 c Then we take c 2 1 1, 1 + c Since x is a constant, it suffices to prove the inequality G, L meso,r M > a 0 c 1c 1 2 M 2 d 0 1/3 < ɛ 124 for all M > M and large enough. For all k = 0, 1, 2,... we denote ck = c 1 k, C k = c 1ckM 2, and the interval Ik = [ck 1M, ckm], 125 and define the lattice paths Lk = {sc 0 2/3 l 0 sc 0 2/3 [C k d 0 1/3 ], sc 0 2/3 l 0 sc 0 2/3 [C k d 0 1/3 ] s Ik}. 126 Since on each Ik, ls < C k as long as ls is defined, and c 1 c 1 2 M 2 < C k for all k, it is clear that if we denote k = G, Lk a 0 C k d 0 1/3,

25 then as is large enough, G, L meso,r M > a 0 c 1c 1 2 M 2 d 0 1/3 G,Lk a 0 C k d 0 1/3 1 k [log c 2 / log c 1 ] [log c 2 / log c 1 ] k=1 k. 128 To estimate k, we note that by the choice of c 1 in 123, there exist δ 1, δ 2, δ 3, δ 4 > 0 such that δ 2 < δ 3 and the points 1, c 1c 1 2, c 1 + δ 1, 1 δ 3 c 1 + δ 1 2, c 1 + δ 2, 1 + δ 4 c 1 + δ are collinear. Then by a simple affine transformation, the points + ck 1Mc 0 2/3, a 0 C k d 0 1/3, + ck + δ 1 ck 1Mc 0 2/3, a 0 1 δ δ 2 /c 1 2 C k d 0 1/3, + ck + δ 2 ck 1Mc 0 2/3, a δ δ 2 /c 1 2 C k d 0 1/3 130 are collinear, as well as the three points + [ck 1Mc 0 2/3 ], a 0 C k d 0 1/3, 131a + [ck + δ 1 ck 1Mc 0 2/3 ], a 0 1 δ 3,,k +δ 1 /c 2 2 C k d 0 1/3, 131b + [ck + δ 2 ck 1Mc 0 2/3 ], a δ 4 ck + δ 2 2 M 2 d 0 1/3 131c collinear, where δ 3,,k δ 3 as uniformly in k. We only need that for large enough δ 3,,k > δ Then by using the symmetry of the lattice and applying Lemma 2.19, we have [ckmc0 k 2/3 +1] Ǧ + [C k d 0 1/3 ] + s, + [C k d 0 1/3 ] s a 0 c 1ckM 2 d 0 1/3 s=[ck 1Mc 0 2/3 ] 2 + ɛ minδ1,δ 2 δ 1 ck 1Mc 0 Ǧ 2/3 + [C k d 0 1/3 ] + [ck + δ 1 ck 1Mc 0 2/3 ], + [C k d 0 1/3 ] + [ck + δ 1 ck 1Mc 0 2/3 ] + Ǧ + [C k d 0 1/3 ] + [ck + δ 2 ck 1Mc 0 2/3 ], a 0 1 δ 3,,k 1 + δ 1 /c 2 2 C k d 0 1/3 + [C k d 0 1/3 ] + [ck + δ 2 ck 1Mc 0 2/3 ] a δ 4 ck + δ 2 /c 2 2 C k d 0, 1/

26 where the term ɛ minδ1,δ 2 δ 1 ck 1Mc application of Lemma 2.2, shows that is defined in Lemma 2.19 and vanishes as. An k < e Mk 134 for large enough M. Thus 124 is proved by taking the sum of k in roof of Theorem 2.8 Using the representation of TASE by the L model, and Remark 2 in particular, to prove Theorem 2.8 we need only to compute the limit of G +σc 0 2/3 d 0 1/3 x, σc 0 2/3 d 0 1/3 x L [a 0]. 135 Thus Theorem 2.8 can be proved by the same method as the proof of Theorem 2.6, since the estimate of the L associated to the TASE differs from the L model in Theorem 2.6 only by a constant shift at each lattice site. In fact, the theorem follows as a corollary of Theorem roof of Theorems 2.11 and 2.13 For the proof of Theorem 2.11, we assume D = R without loss of generality. We express G f, L = G f,micro, L, G f,meso, L, G f,macro, L, 136 where G f,,l = G, 0, y f y, = micro, meso or macro, 137 y I and, letting M > 0, [ M2c 0 2/3, M2c 0 2/3 ] I = [ 2c 0 2/3+c 2, 2c 0 2/3+c 2 ] \ I micro, ] \ I micro I meso for = micro, for = meso, for = macro. 138 Similar to the proof of Theorem 2.6 in Section 3.1, we show that for any fixed M and ɛ > 0, Gf,micro, L a 0 b 0 1/3 x As s 2 + ls x s [ M,M] < ɛ 139 for all large enought, independent of the particular formula of f. Then we show that for any fixed ɛ > 0, for all large enough, independent of the particular formula of f, Gf,meso, L a 0 b 0 1/3 > x < ɛ, 140 and at last show that for any fixed ɛ > 0, for all large enough, independent of the particular formula of f, Gf,macro, L a 0 b 0 1/3 > x < ɛ