Universality of the GOE Tracy-Widom distribution for TASEP with arbitrary particle density

We consider TASEP in continuous time with non-random initial conditions and arbitrary fixed density of particles rho. We show GOE Tracy-Widom universality of the one-point fluctuations of the associated height function. The result phrased in last passage percolation language is the universality for the point-to-line problem where the line has an arbitrary slope.


Introduction
We consider the totally asymmetric simple exclusion process (TASEP) in continuous time on Z.It is an interacting particle system with the constraint that there is at most one particle per site.Particles jump to their right-neighboring site with rate 1, provided the arrival site is empty.A very natural and important observable is the integrated current at (for example) the origin, that is, J(t) = # particles which jumped from site 0 to site 1 during time [0, t].(1.1) TASEP is a model in the Kardar-Parisi-Zhang (KPZ) universality class and thus one expects that for some model-dependent constants, c 1 , c 2 , has in the t → ∞ limit a non-trivial distribution function, say D. It is well-known that for KPZ models the distribution D depends on classes of initial conditions [8,9,35] (see also the reviews [19,24]).In particular, consider the case of non-random initial condition with density ρ = 1/2, realized by placing at time 0 particles on every even sites.The joint distribution of the current at different points has been studied [13,42].As a particular case, the one-point distribution is given by the Fredholm determinant, which is shown to be equal to the GOE Tracy-Widom distribution in [26], where F GOE denotes the GOE Tracy-Widom distribution function discovered first in random matrix theory [45].The analogue result was previously known for discrete time TASEP with parallel update and for a combinatorial model of longest increasing subsequences with involutions [8,9].This latter model was brought in connection to the KPZ world in [35], where it was reinterpreted as a stochastic growth model (the so-called polynuclear growth model).
From [32] we also have the variational formula where A 2 is called the Airy 2 process [31,36].There are many more variational formulas related with the Airy 2 process, see e.g.[5] and the review [37].By universality one expects that the GOE Tracy-Widom distribution describes the fluctuations of J(t) in the large time limit for any non-random initial condition with density ρ ∈ (0, 1).Beyond the case of ρ = 1/2, this was proven for densities ρ = 1/d, d = 2, 3, 4, . . . in [12], and for the low-density limit of reflecting Brownian motions in [28] (in these works also the joint distribution of the current have been analyzed).In these papers, the results are achieved by exact formulas for a correlation kernel which describes the system.However, beyond the d = 2 case, the asymptotic analysis in these special cases turned out to be quite involved.An exact formula has very recently been derived for arbitrary initial condition as well [34].Formulas for the system with periodic boundary condition are also know only for densities 1/2, 1/3, . . .[6,7].
In this paper we prove that for any ρ ∈ (0, 1), comparethis with Theorem 2.6.The proof of our result is in his core probabilistic, where we use results which are known from the study of some exactly solvable cases just as inputs.We prove the convergence to the variational problem (1.4).The proof follows the idea firstly presented in [23].In that paper, for initial conditions (possibly random) they obtained universal results showing that the distribution converges to a variational problem (which depends on how the initial condition scales under diffusive scaling), for cases which are macroscopically at density 1/2.In the continuous time setting, this was studied in [18].In particular, if the initial condition "scales subdiffusively", then for ρ = 1/2 one still sees F GOE fluctuations.This fact was predicted in the context of the KPZ equation in [39].
To prove the convergence to the variational problem, we work in the last passage percolation (LPP) framework (see Section 2.1 for definitions and details).In that language we need to study a "line-to-point" problem with the line having arbitrary slopes.Using a tightness result for the "point-to-point" problem (see Theorem 2.3) and a slow-decorrelation result (see Theorem B.1) (which is then extended to a functional slow-decorrelation theorem (see Theorem 4.1)) we can show, analogously to [23], that the convergence of a restricted "line-to-point" LPP problem converges to the variational problem (1.4) with |u| ≤ M. The second step of the proof consists in showing that the original LPP is localized.For that, we obtain Gaussian decay (in M) of the probability that the LPP is not localized on a O(Mt 2/3 ) region (see Lemma 4.3).
The main technical novelty of our proof concerns the localization.All we need is a good control on the point-to-point process along a horizontal line.This is achieved by using a comparison with two stationary initial conditions, with densities slightly higher/lower than ρ.This comparison was used first in [17] to show tightness for the Hammersley process.In our paper we use the inequality to show tightness but also to control the fluctuations of the process over large distances.This simplifies considerably the approach of [23], where they based their result on an inequality which comes from a Gibbs Brownian property of an associated non-intersecting line ensembles.Also, our method is completely universal, while the one in [18] was designed to work for density 1/2.The approach of the recent paper on KPZ fixed point by Matetski, Quastel and Remenik [34], which relies on an exact formula for the correlation kernel, could be used to obtain our result as well (it was used mainly for other purposes only at density ρ = 1/2).
Outline.In Section 2 we define TASEP, LPP and present the main results.Section 3 contains the proof of tightness and the derivation of a bound needed to control localization as well.Finally, we prove the main theorem for LPP and TASEP in Section 4.

LPP and TASEP
A last passage percolation (LPP) model on Z 2 with independent random variables {ω i,j , i, j ∈ Z} is the following.An up-right path π = (π(0), π(1), . . ., π(n)) on Z 2 from a point A to a point E is a sequence of points in Z 2 with π(k + 1) − π(k) ∈ {(0, 1), (1, 0)}, with π(0) = A and π(n) = E, and where n is called the length ℓ(π) of π.Now, given a set of points S A and E, one defines the last passage time L S A →E as (2.1) Finally, we denote by π max S A →E any maximizer of the last passage time L S A →E .For continuous random variables, the maximizer is a.s.unique.
TASEP is an interacting particle system on Z with state space Ω = {0, 1} Z .For a configuration η ∈ Ω, η = (η j , j ∈ Z), η j is the occupation variable at site j, which is 1 if and only if j is occupied by a particle.TASEP has generator L given by [33] Lf where f are local functions (depending only on finitely many sites) and η j,j+1 denotes the configuration η with the occupations at sites j and j + 1 interchanged.Notice that for the TASEP the ordering of particles is preserved.That is, if initially one orders from right to left as then for all times t ≥ 0 also x n+1 (t) < x n (t), n ∈ Z.
TASEP can be also though as a growth process by introducing the height function h(j, t) as for j ∈ Z, t ≥ 0, where J(t) counts the number of jumps from site 0 to site 1 during the time-span [0, t].
The connection between TASEP and LPP is as follows.Take ω i,j to be the waiting time of particle j to jump from site i − j − 1 to site i − j.Then ω i,j are exp(1) i.i.d.random variables.Further, setting the set (2.4) Figure 1: The last passage percolation setting considered in Theorem 2.1.
The maximizer π from L ρ to (N, N) starts in a O(N2/3 )-neighborhood of

Universality for LPP
For any fixed ρ ∈ (0, 1), we consider the LPP model with S A corresponding to TASEP with initial condition x k (0) = −⌊k/ρ⌋, k ∈ Z.We denote this initial set by and we are interested in the LPP from L ρ to (N, N) in the limit N → ∞ illustrated in Figure 1.This by (2.4) will give us the distribution of the height function at the origin1 .We show the following universality result.
In [4] the distribution of the position where the maximum of A 2 (u) − u 2 is attained has been derived.Due to the quadratic term it is localized and bounds can be found in [22,38].These bounds can be compared with our Lemma 4.3, where we obtain a Gaussian bound in M of the probability that the maximizers is not in a main region of order O(MN 2/3 ) (uniformly for all N large enough).
Remark 2.2.From the work on KPZ equation of Remenik and Quastel [39] it is conjectured that for KPZ growth models, if the initial configuration is flat with subdiffusive scaling, then the limiting distribution is the same as for the flat case (see Theorem 1.5 and subsequent remarks in [39]).In the LPP framework this corresponds to have L ρ replaced by a (possibly random) down-right line, which at distance X from the origin has fluctuations at most O(|X| δ ) for some δ < 1/2.Theorem 2.1 could be easily extended to this case as well, compare with [18,23], but for simplicity of the exposition we decided not to do it.
The proof of the main theorem (Theorem 2.1) is in his core probabilistic and it is based on the comparison of the LPP problem from a horizontal line to (N, N), where the line is around the region where the LPP from L ρ to (N, N) is achieved.If we look the maximizers from the (N, N) position backwards, this is equivalent to consider the LPP from (0, 0) to a horizontal line crossing (γ2 n, n) for some γ ∈ (0, ∞) with n proportional to N. Therefore consider the following LPP setting: for i, j ≥ 1, let ω i,j be i.i.d.exp(1) random variables, ω i,j = 0 for i ≤ 0 or j ≤ 0.
The estimate from law of large numbers for the LPP from the origin to (M, N) is given by ( √ M + √ N ) 2 (as shown by Rost [41] in the TASEP setting).Due to KPZ scaling we define the rescaled last passage time 2 where we set The coefficient β 2 is chosen to have the one-point distribution given by the GUE Tracy-Widom distribution [44], as shown by Johansson in Theorem 1.6 of [30].The coefficient β 1 is chosen such that the limit process converges to the Airy 2 process [36], A 2 .The finite-dimensional convergence to the Airy 2 process is a special case of [11,15,29].Note that since we can replace in (2.7) also the approximation of the LLN until the order n 1/3 only without any relevant changes.
As a direct consequence of the convergence of finite-dimensional distributions and tightness we have: The next result which is in itself interesting is a bound of the exit point probability for the stationary situation, which can be achieved (see more details in Section 3.1) if we consider the LPP as before but with extra random variables if i = 0 or j = 0, namely with (2.9) Here Exp(a) denotes exponential random variables with parameter a (thus average 1/a).For the LPP with boundary conditions (2.9) we define the exit point as the last point of a path π (0,0)→(m,n) on the x-axis or the y-axis.Since we need to distinguish whether the exit point is on the x-or on the y-axis, we introduce a random variable Lemma 2.5 (Exit point probability).Let κ > 0 be given and set Then there exists a n 0 such that for all n ≥ n 0 , for some constants C, c independent of κ.

Universality for TASEP
The LPP result studied in Theorem 2.1 corresponds to TASEP in continuous time with initial condition x k (0) = −⌊k/ρ⌋, k ∈ Z.We have the following universality result for the one-point fluctuations for TASEP with flat initial conditions for any density ρ ∈ (0, 1).
Remark 2.7.In this paper we consider only a non-random line.The same method can be easily extended to random initial conditions as stationary as it was made in [18,23].For instance, considering stationary initial conditions with density ρ would lead to the variational formula which is known to give the Baik-Rains distribution (see Corollary 2.3 of [18]).We did not pursue this small extension in this paper, as the Baik-Rains distribution of stationary initial condition with arbitrary density was already obtained using non-intersecting line ensembles in [27].
3 Comparison with stationary LPP and proof of Theorem 2.3 In this section we will prove tightness of the process L resc,h n .This mainly follows the approach of Cator and Pimentel [17].The key observation in [17] is that the increments of the LPP with end-points on a horizontal line can be bounded by the increments of the LPP for the stationary case on the set of events where the "exit point" is on the right or the left of the origin.Then the idea is to consider stationary LPP with slightly higher/lower density to have the given exit point events highly probable and at the same time the increments of the LPP are controlled by the ones in the stationary LPPs.In [17] the case of the Hammersley process was studied in details and it was stated the result for the exponential random variable along the diagonal only, i.e. γ = 1.The proof of the latter is left to the reader as it was mentioned that it is similar to the case of the Hammersley.
We have a few reasons to present the details for the result with generic densities: (a) here we consider the space of continuous functions instead of the càdlàg functions and there are some minor twists which have to be taken into account for generic density ρ = 1/2; (b) we get a much stronger bound for the exit point distributions with respect to [17] (see Lemma 2.5); (c) we derive an estimate on the increments, which is not needed for proving tightness, but it is key for the control of the probability that the maximizer of the LPP from L ρ to (N, N) is localized.This latter estimate will allow us to simplify noticeably that part with respect to the previous papers [23] (they made use of a Brownian-Gibbs property) and [18] (an ad-hoc comparison with half-line problem with slope −1 was used).

Stationary LPP and exit points
Let us now explain what we mean with stationary LPP with density ρ ∈ (0, 1) and report a result of Balázs, Cator and Seppäläinen [10].Consider the LPP as given by (2.9).We denote by L ρ (m, n) the last passage percolation from (0, 0) to (m, n) in this setting, while we use L(m, n) for the last passage percolation from (0, 0) to (m, n) if we set ω i,0 = ω 0,j = 0.
The boundary conditions (2.9) correspond to a TASEP starting from the stationary Bernoulli(ρ) measure, conditioned on η 0 (0) = 0 and η 1 (0) = 1.Let P 0 (t) be the position at time t of the particle which started at 1 at time 0, and H 0 (t) be the position at time t of the hole which started at 0 at time 0. It was shown in Corollary 3.2 of [10] (as a corollary of Burke's theorem [16]) that P 0 (t) − 1 and −H 0 (t) are two independent Poisson processes with jump rates 1 − ρ and ρ.They extended the result to get independent increments also in the bulk of the system.
The result we will use is the following: Lemma 3.1 (Special case of Lemma 4.2 of [10]).Fix any n ≥ 1.Then the increments are are i.i.d.exponential random variables with parameter 1 − ρ.
With this definition we have the following lower and upper bounds in the increments of the process m → L(m, n) that we want to study: while, if Z ρ ≤ 0, then we have From the law of large number results one easily obtains that Z ρ is typically around 0 (it will fluctuates over a n 2/3 scale) if one chooses ρ = 1/(γ +1).Therefore we set The choice of n −1/3 is due to the fact that the increments of the scaled process are just increased/decreased by a finite amount (proportional to κ), but on the other hand P(Z ρ + > 0) and P(Z ρ − < 0) goes to 1 as κ → ∞.The first step is to get an estimate on these probabilities.

Bounds on exit points
Now we want to derive a bound on P(Z ρ + > 0) and on P(Z ρ − < 0).The last passage time L ρ is the maximum between the last passage time from (0, 1) and the one from (1, 0), since any up-right path from (0, 0) has to go through one of these points.These LPP are denoted by In terms of these two random variables, we have Now we are ready to prove Lemma 2.5.
).Then, uniformly for n large enough, we have for some κ-independent constants C, c ∈ (0, ∞) (c is depending on a).
Proof.Denoting L ρ 0 ,resc := , the first inequality becomes an estimate on 1 − P(L ρ 0 ,resc ≤ aκ 2 ).The distribution of L ρ 0 ,resc has been studied in [1] in the framework of sample covariance matrices.One can use the connection of this LPP to a rank-one problem in sample covariance matrices (see Section 6 of [1]) to recover the result.Let us explain how it goes.
From (62) of [1] we have that where K n is a trace-class operator acting on L 2 (R + ).The integral kernel of K n , can be expressed as where H n (u, v) = H(ξ + u + v) and J n (u, v) = J (ξ + u + v) with H, J given in (93)-(96) of [1].Using the triangular inequality and a standard inequality on Fredholm determinants (see e.g.Theorem 3.4 of [43]) we have The limits of H and J are denoted by H ∞ and J ∞ and they are given in (120) and (122) of [1] for k = 1.H ∞ (u) = e −εu R + Ai(ξ +λ+u)dλ and J ∞ (u) = e εu Ai ′ (ξ +u) with ε > 0 being any small constant.Using triangular inequalities and the identity AB 1 ≤ A HS B HS (see e.g.Theorem VI.22 of [40]) we can bound each of the norms in (3.12) by a finite sum of product of two of the following Hilbert-Schmidt norms, As a function of ξ, the latter two have exponential bounds (see Proposition 3.1 of [1]) uniformly for n large enough, while the first two have (super-)exponential decay from the known asymptotics of the Airy functions (e.g., |Ai(x)| ≤ e −x and |Ai ′ (x)| ≤ e −x , for all x ∈ R).
To prove the second inequality, it is enough to have a bound on the probability for a lower bound for L ρ + − .For any choice of ξ 0 > 0, we have where the L without ρ + means the LPP with all ω's to be Exp(1).Then Let us see what is a good choice for ξ 0 .The estimate from the law of large numbers gives and (3.17)The sum of (3.16) and (3.17) (up to O(n 1/3 )) is maximal for ξ 0 = 2γ(1 + γ) 2 κ, which is the value that we choose.Let us define the rescaled LPP by For any a ∈ (0, (1 + γ) 8/3 γ −2/3 ) we have s > 0.Then, uniformly for n large enough, by Proposition A.1(c) we have 3 for some constants C, c, c ∈ (0, ∞).

Tightness
Now we prove tightness of the rescaled process L resc,h n (see (2.7)).Following the ideas in [17] we prove it using the bounds of Lemma 3.2 together with the estimates of Lemma 2.5 and of the fluctuations of sums of i.i.d.random variables.
First let us see what Lemma 3.2 becomes for the rescaled processes.This bounds will be used for showing tightness, but also to control the fluctuations beyond the central region of the maximisation problem (see Lemma 4.3).Let us shortly recall the scaling (2.7) under which L resc,h n converges in the sense of finite-dimensional distributions [11,15,29] to the Airy 2 process, A 2 , Here O(n −1/3 ) is uniformly for u, v, κ in a bounded set.
Proof.By Lemma 3.2 and the definition of the scalings (3.23) and (3.24) we have Using the explicit expression for β 1 , β 2 , and ρ + we get (3.25).
Let us denote the modulus of continuity for the rescaled process L resc,h n in the interval [−M, M] by ̟ n (δ): Proof of Theorem 2.3.First of all, notice that the random variable L resc,h n (0) is tight, see the upper and lower tail estimates in Proposition A.1.Thus to show tightness it remains to control the modulus of continuity, namely we need to prove that for any ε, ε > 0, there exists a δ > 0 and a n 0 such that for all n ≥ n 0 .

Proof of Theorem 2.1
In this section we prove the main theorem of LPP.The proof consists in showing that the LPP converges to a variational process.One essentially shows that (a) the LPP from L ρ to (N, N) is with high probability the same as the LPP from a subset of L ρ of size O(MN 2/3 ), and (b) that in that region the LPP converges to the variational process of the theorem restricted to |u| ≤ M. The most important novelty of our proof, with respect to the works in [18,23], is part (a).In [23] they first needed to prove a Brownian-Gibbs property for an associated non-intersecting lines.In [18] one bounded a Fredholm determinant of a half-line problem corresponding to density ρ = 1/2 for TASEP (and this approach can not be extended to the generic ρ ∈ (0, 1) case).
Proof of Theorem 2.1.Let us recall that we study the LPP from L ρ to (N, N) which was defined in (2.5) by From the law of large numbers of the point-to-point LPP, see Proposition A.1(a), by optimizing over the positions on L ρ we obtain that the maximizer starts around ((2ρ − 1)/ρ, −(2ρ − 1)/(1 − ρ))N.Therefore we define the points on L ρ by For a fixed M > 0, define the following LPP problems: According to (2.6) we need to determine the N → ∞ limit of For large M (as we will show) one expects that L M > L M c with high probability.
Thus we define the events With these definitions we have In Lemma 4.3 we show that, P(R M ∩ G M ) ≤ Ce −cM 2 uniformly in N.This implies that lim Thus it remains to determine lim M →∞ lim N →∞ P(G M ).
The limit is obtained by first considering the last passage percolation problem from points on the horizontal line crossing A(0), see Figure 2, for which the finitedimensional distribution is known, and then using the functional slow-decorrelation result of Theorem 4.1 we transport the fluctuations to the line L ρ .We define and (4.9)As mentioned above, the process v → L A(v)→(N,N ) properly rescaled is known to converge to the Airy 2 process, A 2 .In [15] it is shown4 the convergence of finite dimensional distributions of the rescaled process: as N → ∞.In terms of the rescaled processes, we have In Theorem 2.3 we show that as a process v → L resc N (v) is tight in the set of continuous functions with supremum norm, C([−M, M]), extending the sense of convergence to the weak*-convergence.In particular, this implies that lim (4.13) Finally, using that the distribution of the position of the maximum of A 2 (v) − v 2 is tight (see Proposition 4.4 of [22], reported also in Proposition 2.13 of [23]), we have that lim where the last equality was proven in [31].This ends the proof of Theorem 2.1.
Theorem 4.1 (Functional slow-decorrelation).Let A(v) and A(v) be defined as in (4.2) and (4.8).Consider the rescaled processes (defined for any v ∈ R through linear interpolation) and L resc N (v) defined in (4.10).Then L resc N − L resc N converges in probability to 0 in C([−M, M]) as N → ∞.More precisely, for any ε, ε > 0 there is a N 0 such that for all N ≥ N 0 , Proof.The proof is almost identical to the one of Theorem 2.11 in [18], see also Theorem 2.15 of [23] (which is two pages long) and therefore we do not repeat it.Let us just mention the strategy and on the way the inputs which are needed.Using Theorem 2.3 one knows that the processes along the horizontal lines L ± crossing A(±M) are tight.One defines the rescaled processes L resc,± N (v) to be the analogues of L resc N (v) but with starting points on L ± , which we call A ± (v), see for some constants C, c > 0 which are uniform in N. Furthermore, where we used the lower tail estimate of the point-to-point LPP from Proposition A.1.

Thus we consider below any
with ν ∈ (0, 1/2) (ν < 1/2 is needed only in the last estimate of this lemma), For the point-to-point estimates we can use the bounds of Proposition A.1, which are uniform for the slopes η in a bounded set of (0, ∞).To avoid slopes which are close to 0 or ∞, we need to restrict the use of the point-to-point estimates for the LPP from A(v) and add the LPP from the starting points C ± as well.
1st bound.First of all, using the notations for L resc ℓ and σ with η = 16/ρ 2 , ℓ = ρN/16 as in Proposition A.1, we have 3rd bound.Finally we need a bound for P(L D k →(N,N ) > a 0 N − 1 4 a 1 M 2 N 1/3 ) uniform in N, which is summable in k and such that its sum is going to zero as M → ∞.The bound for P(L D ℓ →(N,N ) > a 0 N − 1 4 a 1 M 2 N 1/3 ) is completely analogue and thus we present in details only the first one.

Figure 2 :
Figure 2: Zoom of the LPP around the line relevant region of L ρ where the maximizers starts.For a given v, A ± (v), A(v), and A(v) are on the same line, the line parallel to A(0), (N, N).

Figure 2 .Corollary 4 . 2 .
Using tightness of L resc N (see Theorem 2.3) and one-point slow-decorrelation (see Theorem B.1) one bounds max |v|≤M | L resc,± N (v) − L resc N (v)|.Finally one needs to control for example the increments of L resc,+ N(v) − L resc N (v).For this one employs use of the subadditivity property of LPP,L A + (v)→(N,N ) ≥ L A + (v)→A(v) +L A(v)→(N,N ) ,and the bound on the left tail of L A + (v)→A(v) provided in Proposition A.1.A direct consequence of tightness of L rescN and the functional slow-decorrelation result (Theorem 4.1) is the following.Fix any M ∈ (0, ∞).Then the rescaled LPP process from L ρ to (N, N), v → L resc N (v) defined in(4.15), is tight in the space of continuous functions on [−M, M], C([−M, M]).It converges weakly to an Airy 2 process u → A 2 (u).

Figure 3 :
Figure 3: The setting used to control the LPP outside the central part.
20) for any choice of L such that each point in L ρ ∩ {A(v), |v| > M} can be reached by an up-right paths from a point in L. Our choice for L, illustrated in Figure 3, is the following: define the segments D k = A(kM)A((k + 1)M) and D ℓ = A(−ℓM)A(−(ℓ + 1)M), then we set