Current fluctuations for TASEP: A proof of the Prähofer--Spohn conjecture

Introduction and results

We study the fluctuations of the height function for the Totally Asymmetric Simple Exclusion Process (TASEP)—a stochastic process of great interest due to its wide applicability and mathematical accessibility. Under hydrodynamic scaling, this height function is the integrated solution to the deterministic Burgers equation TL1999s . This hydrodynamic limit is sensitive to the initial conditions of TASEP. It is of great interest to determine how the initial conditions of TASEP affect the random fluctuations of the height function. Ultimately, one would like to have a dictionary between initial conditions of TASEP and the resulting orders of the fluctuations of the height function, along with the scaling functions and correlation structures. This paper serves to lay some groundwork for understanding the phenomena which figure into this dictionary. The two phenomena which must be considered in TASEP are shocks and rarefaction fans. We study the simplest family of initial conditions which give rise to both of these phenomena. These initial conditions are simply Bernoulli independent at each site $x$, with density $\rho_{-}$ for $x\leq 0$ and $\rho_{+}$ for $x>0$. We study the fluctuations of the height function or equivalently the current for these two-sided initial conditions. We solve an important conjecture of Prähofer and Spohn PS2002c (see also PLFS2006s ). Understanding the fluctuation theory for two-sided TASEP provides the logical link between the well-developed theory for equilibrium initial conditions ($\rho_{-}=\rho_{+}$) PAFF1994c , PLFS2006s and step initial conditions ($\rho_{-}=1,\rho_{+}=0$) KJ2000s . Two-sided

TASEP interpolates between systems which are in equilibrium and systems which are entirely out of equilibrium. Our analysis shows how this interpolation occurs. The main result, Theorem 1.1, was first conjectured in PS2002c based on a scaling theory and analogous results for the PNG model and discrete TASEP BR2000l . Figure 1 illustrates the main result of this paper—it shows how the order of and scaling functions for the fluctuations of the height function for TASEP depend on the observation location with respect to shocks and rarefaction fans.

The proof of our main results makes use of the aforementioned result of PLFS2006s for the critical point (equilibrium $\rho_{-}=\rho_{+}=\rho$ and $y=1-2\rho$). For every other set of initial conditions, the proof relies on the main result of BBP2005p , a paper about the largest eigenvalue of finite rank perturbations of Wishart (sample covariance) random matrix ensembles. The connection between these two, seemingly disparate mathematical construction (TASEP and Wishart ensembles) is due to KJ2000s and is facilitated through an intermediate random process known as directed last passage percolation (LPP). Our results about fluctuations of currents (or height functions) for TASEP follow from equivalent results for fluctuations of the last passage time for a directed last passage percolation model with two-sided boundary conditions (given in Theorem 1.3).

A natural and important quantity to study in TASEP is the current of particles past an observer moving with speed $y$. It is defined as $J_{yt,t}=$ number of particles to the left of the origin at time zero and to the right of $yt$ at time $t$ minus number of particles to the right of the origin at time zero and to the left of $yt$ at time $t$. The current encodes the same information as the height function $h_{t}(j)$ [which we will define in (6)]:

For equilibrium TASEP with density $\rho$, the law of large numbers and central limit theorem PAFF1994c states that

where $N(0,D_{J})$ is a normal with variance

For every velocity aside from $y=1-2\rho$, current (and height function) fluctuations are Gaussian of order $t^{1/2}$. However, for a single critical velocity the central limit theorem of PAFF1994c is degenerate as the fluctuations are of a lower order than $t^{1/2}$. In terms of the hydrodynamic limit, this velocity corresponds to the slope of the characteristic line for Burgers equation. Heuristically this is the speed at which the initial condition fluctuations travel. Therefore, at any other speed, the current will depend on more initial conditions than just that localized to the origin—it is this that ensures the $t^{1/2}$ fluctuations and Gaussian scaling function for other velocities. At the critical speed, the initial environment’s fluctuations are of lesser order, and only the dynamic fluctuations (those due to the actual TASEP process) are felt. These dynamic fluctuations are of central importance to understanding KPZ universality. At the critical speed $y=1-2\rho$ the fluctuations are of order $t^{1/3}$ and converge, under suitable centering and scaling to a distribution function related to the Tracy–Widom $\operatorname{GUE}$ distribution PS2002c , PLFS2006s . Rewriting expression (1.14) of PLFS2006s in terms of the current $J_{yt,t}$, with $y=1-2\rho$, their $w=0$, and $\chi=\rho(1-\rho)$, the result shows that

where $F_{1,1}(x;0;0)=\frac{\partial}{\partial x}(F_{0}(s)g(x,0))$. The $g(x,0)$ is a scaling function given in their equation (1.18). See Section 1.2 for an overview of how our notation translates into the notation used in PS2002c , PLFS2006s .

In TASEP starting with step initial conditions, there are no fluctuations in the initial environment and consequently for every velocity $y\in(-1,1)$ the current has fluctuations from the dynamics of order $t^{1/3}$ and with scaling function which corresponds to the Tracy–Widom $\operatorname{GUE}$ distribution KJ2000s (which we write as $F_{0}$ so as to be in line with the notation of BBP2005p ). In terms of the hydrodynamic limit, the range of speeds $y\in(-1,1)$ corresponds to the entire rarefaction fan, and the fluctuations are entirely due to the dynamics of TASEP. Ranges of speed bounded away from the fan correspond to regions which are, in the allotted time, unchanged by the dynamics of TASEP.

Drawing on the heuristics about the fluctuations along flat and fanned regions in the hydrodynamic limit, as well as based on a scaling theory and previous work of BR2000l for the PNG model, Prähofer and Spohn PS2002c conjectured that these two fluctuation theorems (for equilibrium and step initial conditions) arise as cases of a complete fluctuation theory for two-sided TASEP (see Figure 1). In their Conjecture 7.1, Prähofer and Spohn claimed that the critical point in PAFF1994c of $t^{1/3}$ fluctuations for equilibrium TASEP becomes a critical window (representing the region of the rarefaction fan) as $\rho_{-}$ is increased and $\rho_{+}$ decreased. Ultimately, as $\rho_{-}=1$ and $\rho_{+}=0$ the critical window of velocities equals the interval $(-1,1)$ as showed in KJ2000s . Likewise, they conjectured Gaussian behavior outside of this window, as well as in the case where $\rho_{-}<\rho_{+}$.

Previous to this paper, part of the Prähofer–Spohn conjecture had been proved via random matrix techniques in both the papers of Nagao and Sasamoto NS2004a and Baik, Ben Arous and Péché BBP2005p . Both papers essentially dealt with the case of $\rho_{+}=0$ and any $\rho_{-}\in$. Our results are dependent on coupling arguments which allow us to bootstrap these boundary cases into every type of two-sided initial condition except for the critical equilibrium case (which is dealt with via the result of PLFS2006s ). The methods of TS2002d , FR2002c prove the part of the conjecture corresponding to the shock ($\rho_{-}<\rho_{+}$) by means of a microscopic Hopf–Lax–Oleinik formula. In these papers, the entire one time fluctuation process is characterized in the case of the shock. The scaling conjectured for the rarefaction fan was proved in BCS2006c (in terms of the corresponding corner growth/LPP model discussed below), though the scaling functions were not addressed therein.

Beyond giving a complete proof of the Prähofer–Spohn conjecture, we believe that our coupling methods are very natural and provide a highly intuitive explanation for the transition between Gaussian and Tracy–Widom scalings. These methods are also useful in studying last passage percolation models with more general weights and more general boundary conditions. In proving the Prähofer–Spohn conjecture, one may alternatively follow the approach of PLFS2006s which is necessary in the critical case $\rho_{-}=\rho_{+}=\rho$ and $y=1-2\rho$. That argument is very strong and widely applicable. It is based on the idea of the Schur measure and involves a shift argument and a necessary analytic continuation argument. Coupling completely avoids these technical issues and replaces the complex analysis and asymptotic analysis with simple and intuitive probability. It also seems to be applicable in certain cases where the Schur measure argument cannot be applied.

Much effort has been devoted to understanding the analogous picture for ASEP, where particles may move to either the left of the right but are still subject to the exclusion rule. Progress in this direction was made in PAFKS1991m , PAF1992s , PAFF1994c , PAFF1994s , PAFK1995s in the early 1990s. The work of Baik and Rains BR2000l , Prähofer and Spohn PS2002s and Imamura and Sasamoto IS2004f in the context of the closely related PNG model and last passage percolation with geometric weights was very important in formulating and understanding the theory of fluctuations. Very recently, due to the efforts of Tracy and Widom TW2008i , TW2008f , TW2008a , TW2009t , TW2009o ), Derrida and Gerschenfeld DG2009c , Balázs and Seppäläinen BS2008o , BS2008f , Quastel and Valkó QV2007t , Mountford and Guiol MG2005t significant progress has been made in answering this question in the general ASEP. In particular, in TW2009o , Tracy and Widom extend their step initial condition integrable system approach to ASEP with one-sided Bernoulli initial conditions. In that case, they observe the exact same fluctuation regimes as for TASEP. At present, the Prähofer–Spohn conjecture has not been proved for ASEP. It is tempting to try to use coupling methods to extend the one-sided picture for ASEP to the two-sided initial condition case. It is not clear if this is possible, as ASEP is not related to a last passage percolation model and the coupling occurs at the level of such a model.

The main result of this paper is a complete proof of PS2002c Conjecture 7.1—our Theorem 1.1.

Following PS2002c , assign to a TASEP configuration $\eta_{t}(j)$ the height function

Recall that $J_{0,t}$ is defined as the number of particles which have crossed the bond $(0,1)$ up to time $t$. For $|y|<1$ denote

which exists almost surely due to the law of large numbers established via the hydrodynamic theory HR1981n , TS1998h . This limit $\bar{h}$ depends not just on $y$, but also on $\rho_{-}$ and $\rho_{+}$ as follows.

with $y_{c}=(\rho_{+}(1-\rho_{+})-\rho_{-}(1-\rho_{-}))/(\rho_{+}-\rho_{-})=1-(\rho_{-}+\rho_{+})$.

We prove the following (note that in parenthesis we record the distribution names as used in PS2002c ). For an illustration of the results below, see Figure 1.

($F_{G}$). Let either $\rho_{-}<\rho_{+}$, $y>y_{c}$ and $y<1-\rho_{+}$, or $\rho_{-}>\rho_{+}$, $y>1-2\rho_{+}$ and $y<1-\rho_{+}$. Then

Let either $\rho_{-}<\rho_{+}$, $y<y_{c}$ and $-\rho_{-}<y$, or $\rho_{-}>\rho_{+}$, $y<1-2\rho_{-}$ and $-\rho_{-}<y$. Then

($F_{G}^{2}$). Let $\rho_{-}<\rho_{+}$ and $y=y_{c}$, then

($F_{\operatorname{GUE}}$). Let $\rho_{-}>\rho_{+}$ and $1-2\rho_{-}<y<1-2\rho_{+}$. Then

($F_{\operatorname{GOE}}^{2}$). Let $\rho_{-}>\rho_{+}$ and either $y=1-2\rho_{-}$ or $y=1-2\rho_{+}$. Then

($F_{0}$). Let $\rho_{-}=\rho=\rho_{+}$ and $y=1-2\rho$. Then

There is one difference between what we prove in Theorem 1.1 and what is stated in Conjecture 7.1 of PS2002c which is that for the case of the Gaussian scaling limit, by virtue of the fact that our proof goes by way of a mapping with directed last passage percolation, there are certain parts of the Gaussian region (with respect to $y,\rho_{-},\rho_{+}$) for which our methods do not apply.

The statistics of $L(N,M)$ are dependent on the choice of distribution for the random weights, and in certain cases related to eigenvalue statistics for random matrices.

The current fluctuations for TASEP with step initial conditions were determined by identifying the height function with a corner growth model whose growth times correspond with the last passage times for a specific LPP model with independent rate one exponential random weights $w_{i,j}$ for $i,j>0$ and boundary weights $w_{i,j}=0$, for $i=0$ or $j=0$ KJ2000s . Theorem 1.6 of KJ2000s shows that as $M,N\rightarrow\infty$ such that $M/N$ is in a compact subset of $(0,\infty)$, $L(N,M)$ (as defined by the above weights) is approximated in distribution by

where $\chi_{0}$ is distributed as a Tracy–Widom $\operatorname{GUE}$ distribution. The first term gives the asymptotic average for $L(N,M)$ and the second term shows that the fluctuations scale like $M^{1/3}$ and have a well understood scaling function. Via the height functions mapping, these results translate back into the current fluctuations for TASEP with step initial conditions. The corner growth model height function is exactly the random interface bounding the growth region.

This theorem was proved by using the tools of generalized permutations, the RSK correspondence and Young Tableaux, to relate the distribution of the last passage time to the distribution of the largest eigenvalue of a Wishart ensemble, whose statistics are known to follow the Tracy–Widom $\operatorname{GUE}$ distribution KJ2000s .

By analogy, our method of proof is to first relate two-sided TASEP to a LPP model, which we appropriately call LPP with two-sided boundary conditions [see (18) for a definition], and then to relate the statistics of the last passage time for that model to the statistics of eigenvalues of already studied random matrices. The first mapping is already found in PS2002c and relies on Burke’s theorem (we review this mapping in Section 3). LPP with two-sided boundary conditions is not directly connected to a random matrix ensemble, however we can realize its last passage time as the maximum of last passage time for a pair of coupled LPP with one-sided boundary conditions [see (26) for a definition]. The last passage time in such one-sided LPP models is related (see Section 6 of BBP2005p ) to the largest eigenvalue of Wishart matrices with finite rank perturbations. In fact, the phase transitions, with respect to the magnitude of the finite perturbation, which are discussed in BBP2005p correspond exactly to the transitions between different orders of and scaling functions for the height function of two-sided TASEP. By a set of coupling arguments, and using the results of BBP2005p and PLFS2006s we provide a proof of Theorem 1.1. In proving Theorem 1.1, we reprove the fluctuation results for step initial conditions as well as for equilibrium initial conditions (except at the critical point). We also show that these two results arise from a much more complete picture (see Figure 1 for an illustration of this).

As noted above, the proof of this theorem relies on understanding the fluctuations of the last passage time in a LPP model with two-sided boundary conditions. The specific LPP with two-sided boundary conditions which we will devote much of this paper to studying has three different types of independent exponential weights $w_{i,j}$:

In the later part of Section 2, we will allow for more general boundary condition where a finite number of columns and rows can have different (though uniform within the column or row) rates. The LPP with one-sided boundary conditions is defined similarly using the weights in (26). At this point, it is worth remarking that changing the distribution of a finite number of weights does not have any affect on the asymptotic fluctuations of the last passage time (see Lemma 3.1).

With this connection in mind, we completely characterize both the order and the scaling functions for the fluctuations of the last passage time of LPP with two-sided boundary conditions in terms of the three parameters $\pi$, $\eta$ and $\gamma$. As noted before, the main result we appeal to in this paper is from BBP2005p (extended to the case $\gamma<1$ in AO2008t ) which classifies the fluctuations of the largest eigenvalue of complex Wishart ensembles with finite rank perturbations. There is a single critical point which does not yield to our method of argument, but this corresponds exactly with the critical point considered in PLFS2006s . Using these

two results and coupling arguments, we prove our LPP with two-sided boundary conditions classification theorem (see Figures 2–4).

For $\gamma\in(0,\infty)$ and $M/N\rightarrow\gamma^{2}$, then for $\pi,\eta$ such that $\pi>\frac{1}{1+\gamma}$ and $\eta>\frac{\gamma}{1+\gamma}$ (the $\operatorname{GUE}$ region)

where $F_{0}(x)$ is the Tracy–Widom $\operatorname{GUE}$ distribution function.

For $\gamma\in(0,\infty)$ and $M/N\rightarrow\gamma^{2}$, then for $\pi,\eta$ such that $\pi>\frac{1}{1+\gamma}$ and $\eta=\frac{\gamma}{1+\gamma}$ or $\pi=\frac{1}{1+\gamma}$ and $\eta>\frac{\gamma}{1+\gamma}$ (the $\operatorname{GOE}^{2}$ region),

where $F_{1}(x)$ is the square of the Tracy–Widom $\operatorname{GOE}$ distribution function.

For $\gamma\in(0,\infty)$ and $M/N\rightarrow\gamma^{2}$, then for $\pi=1/(1+\gamma)$ and $\eta=\gamma/(1+\gamma)$,

where $F_{1,1}(x;0;0)$ is the same distribution as what PLFS2006s refer to as $F_{0}(x)$.

For $\gamma\in(0,\infty)$ and $M/N\rightarrow\gamma^{2}$, then for $\pi,\eta$ such that $\pi<1/(1+\gamma)$ and $\eta>\frac{\pi}{\pi(1-\gamma^{-2})+\gamma^{-2}}$ [the $G$ ($\pi$ controlled) region],

Likewise for $\pi,\eta$ such that $\eta<\gamma/(1+\gamma)$ and $\eta<\frac{\pi}{\pi(1-\gamma^{-2})+\gamma^{-2}}$ [the $G$ ($\eta$ controlled) region],

For $\gamma\in(0,\infty)$ and $M/N\rightarrow\gamma^{2}$, then for $\pi,\eta$ such that $\pi+\eta<1$ and $\eta=\frac{\pi}{\pi(1-\gamma^{-2})+\gamma^{-2}}$ (the $G^{2}$ line),

It is worth noting that there are many ways to write the expressions above, and our choices are to facilitate the greatest ease in our proofs.

This type of fluctuation classification picture has been previously discussed in BR2000l and Sa2007f . In fact, in BR2000l Baik and Rains provide a proof of an analogous fluctuation classification result for two closely related particle system models: LPP with geometric weights, and the polynuclear growth model. Recently, BPS2009t studied two-speed (different though related to two-sided initial conditions, half flat and half Bernoulli) TASEP and proved a fluctuation classification theorem for that model. As noted before, BCS2006c previously provided the order of fluctuations for LPP models with two-sided boundary condition corresponding to the rarefaction fan. With an even more general type of boundary condition, the paper establishes $t^{1/3}$ scaling for the fluctuations of the last passage time.

2 Notation

In a paper such as this which connects two different lines of thought, it is easy to become lost in the disparity between notations. We will adopt notation in the style of BBP2005p throughout, and when making connections with distributions as found in papers such as PS2002c , PLFS2006s , BR2000l , we will take care to make note of the alternative notation used in those contexts. In this section, we define all of the distributions which we will encounter herein and provide references for their previous use and definition.

$G_{k}(x)$ is a family of distributions defined in BBP2005p , Definition 1.2 and Lemma 1.1. It represents the distribution of the largest eigenvalue of a $k\times k$ $\operatorname{GUE}$. From this representation, it is clear that $G_{1}(x)=\operatorname{erf}(x)$, the standard Gaussian distribution function.

$F_{J}(x;x_{1},\ldots,x_{J})$ is a family of distributions defined in BBP2005p , Definition 1.3. In the case when the $x_{j}=0$ for all $j$, these distributions coincide with those from BBP2005p , Definition 1.1. Of note is $F_{0}(x)$ which is often written as $F_{\operatorname{GUE}}$, the $\operatorname{GUE}$ Tracy–Widom distribution function, and $F_{1}(x;0)$ which is often written as $F_{\operatorname{GOE}}(x)^{2}$, where $F_{\operatorname{GOE}}$ is the $\operatorname{GOE}$ Tracy–Widom distribution function.

$F_{J,I}(x;x_{1},\ldots,x_{J};y_{1},\ldots,y_{I})$ is a family of distributions which we conjecture come up in LPP with thick two-sided boundary conditions. The only member of this family for which we know the correct definition is $F_{1,1}(x;0;0)$ which corresponds to the distribution denoted by $F_{0}$ in PLFS2006s . As of yet, we do not know how the other distributions should be defined.

3 Outline

The main theorems (Theorems 1.1 and 1.3) have already been recorded above in this section. Section 2 provides an intuitive sketch of the proof for Theorem 1.3. Section 3 explains the connection between the LPP with two-sided boundary conditions and the TASEP with two-sided initial conditions as well as briefly sketches how to translate the result of Theorem 1.3 into a proof of Theorem 1.1. Section 4 gives the full proof of the two main theorems, complete with the necessary technical lemmas for the coupling arguments.

Fluctuations in last passage percolation with boundary conditions

We start this section by reviewing the result of BBP2005p which relates directed last passage percolation with boundary conditions to finite rank perturbations of Wishart ensembles. We then apply these results to prove Theorem 1.3 which fully characterizes the fluctuations of last passage times in terms of boundary conditions and the ratio $M/N=\gamma^{2}$. Using coupling arguments, supplemented in one case by the result of PLFS2006s , we provide both the order and the scaling function for these fluctuations. Using the exact same arguments but fully taking advantage of the scope of the results of BBP2005p , we prove almost all of the cases in Partial Theorem 2.1. In this section, we will only sketch our proofs, which can be found in entirety in Section 4.

Consider a directed last passage percolation model with one-sided boundary conditions defined as follows:

Let $L_{1}(N,M)$ denote the last passage time from $(0,0)$ to $(N,M)$ (for this LPP model with one-sided boundary conditions, but also for any LPP model with thick one-sided boundary conditions). Then the distribution of $L_{1}(N,M)$ is related to the distribution of the largest eigenvalue of the normalized covariance matrix $\frac{1}{M}XX^{\prime}$ where $X$ is $N\times M$ and each column is drawn (independent of other columns) from a complex $N$-dimensional Gaussian distribution with covariance matrix $\Sigma$. The matrix $\Sigma$ has eigenvalues all equal to one aside from a single one, which is $l_{1}=\eta^{-1}$. Depending on the value of $\eta^{-1}$, $L_{1}(N,M)$ behaves differently.

The following theorem is adapted from Theorem 1.1 of BBP2005p and the extension to all $\gamma\in(0,\infty)$ given in AO2008t , as applied to the one-sided boundary condition LPP. The connection between the largest eigenvalue and the LPP with one-sided boundary conditions given above is explained in Section 6 of BBP2005p and is briefly rehashed in Remark 2.2.

With $L_{1}(N,M)$ defined as above, as $M,N\rightarrow\infty$ while $M/N=\gamma^{2}$ is in a compact subset of $(0,\infty)$, the following hold for any real $x$ in a compact set.

The connection between last passage time in LPP with one-sided boundary conditions and the largest eigenvalue of the spikedWishart ensemble was observed in BBP2005p . The connection is not via an exact map but rather an equality of distributions. Proposition 6.1 of BBP2005p records this fact and explains how a modification of the argument in KJ2000s can be used to prove this.

An intuitive explanation for the cutoff of $\eta^{-1}=1+\gamma^{-1}$ in terms of a simple calculus problem of maximizing the law of large numbers for LPP paths forced to travel a specific fraction of the way along the left column can be found in Section 6 of BBP2005p .

2 LPP with two-sided boundary conditions

Presently, we turn our attention to the LPP with two-sided boundary conditions as defined in (18): on the left-most column there are exponential weights of rate $\eta$, on the bottom-most row there are exponential weights of rate $\pi$ at the origin there is a weight of zero, and for all strictly positive lattice points the weight is of rate one. Define, respectively, $X(N,M)$ and $Y(N,M)$ as the coupled last passage times of paths which have taken the first step to the right and the first step up (resp.). One should be careful to note that we are not conditioning on the location of the optimal path, but rather, for each configuration of weights, defining $X$ to be the length of the optimal path which first goes right, and $Y$ the length of the optimal path which first goes up. It is clear then that $X$ and $Y$ are coupled, dependent and that

Consider now the marginals of $X$ and $Y$ and observe that each of these marginals is of the type of the last passage time for a LPP model with one-sided boundary conditions. The boundary conditions for $Y$ are exactly as above ($\eta$ weights and an $N$ by $M$ region). However, for $X$, the boundary conditions are $\pi$ weights and an $M$ by $N$ region (note that the region has been flipped in order to conform with the setup for Proposition 2.1). From this observation, we can apply Proposition 2.1 to completely characterize the marginals of the joint distribution for the pair $(X,Y)$. Note that while $X$ and $Y$ are not exactly of the form of a last passage time for a LPP with one-sided boundary conditions, they only differ by a finite number of weights and therefore have the exact same asymptotic statistics via Lemma 3.1.

With $X(N,M)$ defined as above, as $M,N\rightarrow\infty$ while $M/N=\gamma^{2}$ is in a compact subset of $(0,\infty)$, the following hold for any real $x$ in a compact set:

With $Y(N,M)$ defined as above, as $M,N\rightarrow\infty$ while $M/N=\gamma^{2}$ is in a compact subset of $(0,\infty)$, the following hold for any real $x$ in a compact set:

We assume that both $\eta$ and $\pi$ are between zero and one. In fact, it is clear from our proofs that the order and fluctuation of our two-sided last passage time $L_{2}(N,M)$ for parameters $\eta$ and $\pi$ is that same as that for parameters $\eta\wedge 1,\pi\wedge 1$. Thus, it suffices to consider only $\eta,\pi\in^{2}$.

For each of $X$ and $Y$, there are two regions of different fluctuation orders, and one critical point which has $1/3$ order fluctuations. We call the point $(\pi,\eta)=(1/(1+\gamma),\gamma/(1+\gamma))$ the critical point for the pair $\pi,\eta$. If $\pi<\frac{1}{1+\gamma}$, then the $X$ fluctuations are of order $1/2$, and likewise if $\eta<\frac{\gamma}{1+\gamma}$ then the $Y$ fluctuations are of order $1/2$, whereas in the complementary cases, the fluctuations are of order $1/3$. By comparing leading (law of large number) terms in Propositions 2.3 and 2.4 we see that if either of these two inequalities hold, then the fluctuations must be of order $1/2$. In this case, either the leading term for $X$ or $Y$ clearly wins, in which case $L_{2}(N,M)$ has the leading order behavior and Gaussian fluctuations of the winner random variable, or the two random variables have the same leading terms. The second case, or equal leading terms, occurs when

In this case, the fluctuations will remain of order $1/2$, but will behave as the fluctuations of two independent normal random variables (what we call $G^{2}$).

If $\gamma=1$, there are two solutions to (37). One is $\eta=\pi$ and the other is $\eta=1-\pi$. Since we are only considering the Gaussian region, the anti-diagonal solution is of no interest, and we find that we have $G^{2}$ density for our fluctuations if $\eta=\pi$ and $\eta<1/2$.

For $\gamma\neq 1$, the solution set is a little harder. Recall $M=\gamma^{2}N$ and using this we can factor out $N$ from both sides giving

Applying the change of variable $\pi\rightarrow 1-\pi$, we find that it suffices to solve

and change the solution back to our original variables.

This again has the solution $\eta=\pi$. Solving for the other solution and then changing variables back we get

By plugging in the critical point $(1/(1+\gamma),\gamma/(1+\gamma))$ it is easy to see that this $G^{2}$ curve is continuous between the origin and the critical point, though only linear for $\gamma=1$.

Determining the scaling function at the critical point is a harder problem. One may identify it as the maximum of two $F_{1}$ distributions, coupled as $X$ and $Y$ are coupled. This characterization, a priori, yields a tight family of random variables. However, how to prove that it converges on more than just a subsequence is not immediately clear, and more over it is not clear to what it convergences. A posteriori, this characterization is justified since the result of PLFS2006s can readily be translated into a proof that the scaling function at the critical point is $F_{1,1}(x;0;0)$ (what they call $F_{0}$).

The results from BBP2005p used in the proof of Theorem 1.3 yield, in fact, a much more general result via essentially the same argument. We now define what we call the LPP model with thick two-sided boundary conditions in terms of boundary row and column thickness integer parameters $J,I\geq 0$; two vectors of row and column weight rates $\pi=(\pi_{1},\ldots,\pi_{J})$, $\eta=(\eta_{1},\ldots,\eta_{I})$; two vectors of row and column convergence rates $X=(x_{1},\ldots,x_{J})$, $Y=(y_{1},\ldots,y_{I})$. With these parameters, our model is defined in terms of the following LPP weights (which implicitly depend on $M$ and $N$):

To see that this model is a broad generalization of our previously considered two-sided boundary condition model, take $J,I=1$, $\pi_{1}=\pi$, $\eta_{1}=\eta$ and $x_{1}$, $y_{1}=0$. Corresponding to this model, we now provide a complete characterization of its asymptotic fluctuations. A number of distributions not previously discussed are introduced in this theorem. A full discussion of these distributions can be found in Section 1.2.

The coupling arguments given to prove Theorem 1.3 can be easily adopted to this new setting. Given the above parameters define, again, two coupled random variables $X(N,M)$ and $Y(N,M)$ as follows. $X(N,M)$ is the last passage time from $(0,0)$ to $(N,M)$ of the set of up/right paths which cross through at least one vertex from the set $\{(i,j)\dvtx i=I,j\in\{0,\ldots,J-1\}\}$. Likewise $Y(N,M)$ is the last passage time from $(0,0)$ to $(N,M)$ of the set of up/right paths which cross through at least one vertex from the set $\{(i,j)\dvtx i\in\{0,\ldots,I-1\},j=J\}$. Clearly, any up/right path from $(0,0)$ to $(N,M)$ must go through one and only one of these two regions. Furthermore, by virtue of the definition of the last passage time, the maximizing path for $X(N,M)$ and $Y(N,M)$ will necessarily go through the points $(0,J)$ and $(I,0)$ (resp.). Thus we may refine our definitions of $X(N,M)$ and $Y(N,M)$ to require passing through these two points. Again, we see that $L_{2}(N,M)=\max(X(N,M),Y(N,M))$ and just as before BBP2005p provides an immediate proof of the following.

For the vector $\pi$, fix the set $K_{1}\subset\{1,\ldots,J\}$ by

Then with $X(N,M)$ defined as above, as $M,N\rightarrow\infty$ while $M/N-\gamma^{2}$ is in a compact subset of $(0,\infty)$, the following holds for any real $x$ in a compact set:

A similar proposition exists for $Y(N,M)$. Using these two results, the same types of coupling arguments then apply and give both the orders and the scaling functions for $L_{2}(N,M)$. As before, these coupling arguments break down when both boundary conditions are critical. With single width boundary conditions, we appealed to PLFS2006s , however in this case no existing argument provides a characterization of the behavior in this case. The following partial theorem therefore contains a single conjectured equation ( ‣ (1)) whose study seems very difficult.

For vectors $\pi,\eta$ fix the sets $K_{1}\subset\{1,\ldots,J\}$ and $K_{2}\subset\{1,\ldots,I\}$ by

Then define $X_{K_{1}}$ and $Y_{K_{2}}$ as the elements of $X$ and $Y$ which correspond to indices in $K_{1}$ and $K_{2}$, respectively.

For $\gamma\in(0,\infty)$ and $M/N\rightarrow\gamma^{2}$, then for vectors $\pi,\eta$ such that $\pi_{j}\geq\frac{1}{1+\gamma}$ for all $i\in\{1,\ldots,J\}$ and $\eta_{i}\geq\frac{\gamma}{1+\gamma}$ for all $i\in\{1,\ldots,I\}$ then if:

For $\gamma\in(0,\infty)$ and $M/N\rightarrow\gamma^{2}$, then for $\pi,\eta$ such that:

Finally, let us note two applications of LPP with thick one-sided boundary conditions which can be found in JB2006p . The first application deals with what Baik called traffic of slow start from stop in which particles start in the step initial condition of TASEP and have a start-up profile—that is, every particle moves slower for its first few jumps, and then returns to jumping at rate one. The second application is dual to the first one and is called traffic with a few slow cars in which particles always move at a slower rate. In both cases, Baik identifies the fluctuation scaling limits by using the BBP2005p type results which we have made use of herein.

In the next section, we will give an important application for the LPP with two-sided boundary conditions model to two-sided TASEP. It is unclear whether the thick two-sided boundary conditions model has any similar application to TASEP or related models.

Mapping TASEP to last passage percolation with boundary conditions

In this section, we explain the connections between the last passage time in LPP with two-sided boundary conditions and the fluctuations of the height function for the two-sided TASEP model. Making use of this mapping, we explain how the results of Theorem 1.3 imply the results of PAFF1994c stated in the Introduction. Furthermore, we briefly explain how this theorem translates into a proof of Conjecture 7.1 of PS2002c (full proof is given in Section 4).

We start with a lemma which states that finite perturbations of our LPP model, have no affect on the asymptotic behavior of the last passage time.

Fix some LPP model with weights $w_{i,j}$ (independent but not necessarily identically distributed) such that

for $M/N\rightarrow\gamma^{2}\in(0,\infty)$, and for $F$ a nondegenerate probability distribution. Randomly, independent of the values of $w_{i,j}$, change a set of these weights to a new set of weights $w^{\prime}_{i,j}$ and let $L^{\prime}(N,M)$ denote the last passage time with respect to the original weight with the newly updated weights. Call $A$ the set of changed indices $(i,j)$ and $W_{A}=\sum_{(i,j)\in A}w_{i,j}+w^{\prime}_{i,j}$. Then if $E[W_{A}]<\infty$ and if $b_{N}\rightarrow\infty$, we also have

Below is an outline of the proof. The full level of details is suppressed since a similar style of proof is given for Lemma 4.1 in full detail. {pf*}Proof of Lemma 3.1 Since the total effect of the change of weights corresponding to $A$ has finite expectations, the Markov inequality shows that for any $\varepsilon$ we can find $l$ large enough so that $P(W_{A}\geq l)\leq\varepsilon$. If we restrict ourselves to this region of our statespace, then since the $b_{N}$ goes to infinity, the effect of the change of weights is negligible in the limit. Since this is true on all but an $\varepsilon$ region of the state space, we have that the distribution functions are within $\varepsilon$ of each other in the limit, but taking $\varepsilon$ to zero gives equality.

Recall our definition of the two-sided TASEP model given by the initial conditions of Bernoulli with parameter $\rho_{-}$ on the left of zero and with parameter $\rho_{+}$ on the right. Corresponding to the TASEP process started with this random initial condition, we consider the height function $h_{t}(j)$ defined in (6). Theorem 2.1 of PS2002c relates the joint distributions for this height function to those of the height function for a particular growth model associated with a variant on the LPP with two-sided boundary conditions. The weights for this variant LPP are defined with respect to two independent geometric random variables $\zeta_{+}$ and $\zeta_{-}$. Let $\zeta_{+}$ be geometric with parameter $1-\rho_{+}$ [i.e., $P(\zeta_{+}=n)=\rho_{+}(1-\rho_{+})^{n}$] and $\zeta_{-}$ be geometric with parameter $\rho_{-}$ [i.e., $P(\zeta_{-}=n)=(1-\rho_{-})\rho_{+}^{n}$]. The weights are then defined as independent random variables with:

In the sense of joint distributions, we have

This theorem essentially says that for the height profile which lies above the boundary of the rotated upper corner, the two profiles have the same joint distribution. It is worthwhile to recall that there is a similar map between the TASEP height function for TASEP with step initial conditions and the height function for standard (no boundary condition) LPP KJ2000s . The proof of Theorem 3.2 can be found in PS2002c and essentially amounts to a study of the dynamics of the right most particle to the left of the origin, as well as the dynamics of the left most hole to the right of the origin. Tagging this particle and this hole, we observe that their initial location is geometric and using Burke’s theorem we find that their waiting times between successive moves is exponential with rate relating to the densities $\rho_{-}$ and $\rho_{+}$. Between the tagged particle on the left and the tagged hole on the right, particles move according to normal TASEP rules, and hence the two height functions evolve with the same dynamics. The dynamics of the boundary of the part of the TASEP height function lying in the rotated upper corner is matched by the effect of the LPP boundary conditions, and the theorem follows.

From this theorem, we see that the following equality:

While the boundary conditions (18) differ from those (58) used by Prähofer and Spohn, they are much simpler and also describe the TASEP with two-sided initial conditions, as described in BCS2006c .

It is important to note that Lemma 3.1 only applies if both $\rho_{-}<1$ and $\rho_{+}>0$. If either of these inequalities is violated, then the geometric number of zeros on the boundary will in fact, almost surely be infinite. However, in any of these cases, the classification of one-sided LPP then readily applies.

This corresponds to an equilibrium measure on TASEP with density $\frac{\gamma}{1+\gamma}$.

In the next section, we will show how Theorem 1.3 implies an almost complete (all but a few regions of the claimed Gaussian region are fully proved) proof of PS2002c , Conjecture 7.1 (our Theorem 1.1). From this result, we may easily deduce the results of PAFF1994c stated in the Introduction.

For simplicity assume $r\in$, as the case $r\in$ follows similarly. We wish to prove that

where as defined before $G_{1}(x)$ is the standard Gaussian distribution function.

In the case of $\rho_{-}=\rho_{+}=\rho$ and $r\geq 0$, we can conclude from (1.1) that

where $\bar{h}(r)=(1-2\rho)r+2\rho(1-\rho)$. Substituting the relationship in (1) and rearranging terms, we arrive at the exact result of PAFF1994c desired.

We now briefly explain the approach to proving Conjecture 7.1 from our Theorem 1.3. The conjecture deals with height functions. We have provided above the relationship between height function distributions and LPP distributions. From (60), we see that if one is to consider $P(h_{t}(j)\geq x)$ in terms of LPP, you must solve for $N=\frac{x+j}{2}$ and $M=\frac{x-j}{2}$. The variables $j$ and $x$ both are functions of time $t$ and a speed $y$. If $M/N=\frac{x-j}{x+j}$ has a limit, we call that $\gamma^{2}$. This allows us to asymptotically write $M$ (or $N$) just as a function of time (thus the $y$ dependence goes into $\gamma$). In the cases we consider, we can invert the expression for $M$ in terms of $t$ and get an expression for $t$ in terms of $M$, thus putting us in the form of the limit theorems we proved in Theorem 1.3.

Proof of fluctuation theorems

In this section, we provide a proof of Theorem 1.3 (which easily generalizes to prove Partial Theorem 2.1) and a proof of Theorem 1.1.

The following two technical lemmas provide the basis for the coupling arguments necessary in our proof of Theorem 1.3.

We now partition the interval $[-M,M]$ into $\varepsilon$ size blocks and define deterministic numbers $a_{j}(n)$ for $j\in\{1,\ldots,\lceil\frac{2M}{\varepsilon}\rceil\}$ by

Multiplying both sides by $P(A)$ and rewriting without conditioning gives

Adding these two inequalities and using the fact that $P(A)>\delta/2$ gives, for all $n$ large enough

If $t$ is any continuity point for $D$, then we can take $\varepsilon$ to zero and we find that

Let us redefine our variables by properly shifting and scaling them, so that they have a limiting distribution. Our new $X(N,M)$ is

where $E_{1}$ and $E_{2}$ are, respectively, the events

2 Proof of Theorem 1.1 (Conjecture 7.1 from PS2002c )

The following elementary lemma will find repeated use in what follows.

Likewise if $M=at+bt^{1/3}$ then for large $t$,

Proof of Theorem 1.1 We provide proofs of only the $G^{2}$ and $F_{0}$ cases of this theorem as the $G$ case is analogous to $G^{2}$ and the $F_{1}$ case is analogous to $F_{0}$. The $F_{1,1}$ case of the theorem already is proved in PLFS2006s . The proofs are based on the fact that if the height at a given time value exceeds a point $(N,M)$, then the last passage time of that point is less than the above time value. Then Theorem 1.3 applies and gives asymptotic height distribution results. As noted before, the mapping between the TASEP height function and the last passage time for our model of LPP with two-sided boundary conditions is not exact (as the exact LPP model has geometric numbers of boundary zeros) however Lemma 3.1 ensures that asymptotically all results in our LPP model correspond to results for the two-sided TASEP height function.

$G^{2}$ case: recall that in LPP with two-sided boundary conditions $\pi=1-\rho_{+}$ and $\eta=\rho_{-}$. We presently assume that $\eta+\pi<1$ ($\rho_{-}<\rho_{+}$) and $y=y_{c}$. Recalling that $\bar{h}(y)=(1-2\eta)y+2\eta(1-\eta)$, we wish to determine the asymptotic (large $t$) value of

We will reduce this probability to a probability in the related LPP with two-sided boundary conditions, and then use Theorem 1.3 to conclude that this probability is the correct product of Gaussian probability functions.

The first step in translating to a LPP problem is to relate the speed to $\gamma$. We may solve for the asymptotic value of $\gamma$ as a function of the speed $y$. As shown in Figure 5, the height function event corresponds to the LPP event where $N-M=yt$ and $N+M=((1-2\eta)y+2\eta(1-\eta))t-\sqrt{(1-\pi-\eta)}t^{1/2}x)$. From that, we find that

From this equation, we can solve for $y$ as a function of $\gamma$:

Since we have assume that $y=y_{c}$, we may use these two expressions for $y$ to relate $\eta,\pi$ and $\gamma$ to find that

This is exactly the curve along which the $G^{2}$ part of Theorem 1.3 applies.

Finally, we may use Lemma 4.3 to invert our expression for $N$ in terms of $t$. Asymptotically

where $t$ is as above. It then follows after a little algebra that Theorem 1.3 applies and gives that these probabilities asymptotically equal

as desired to prove this part of Theorem 1.1.

$F_{0}$ case: similarly to the previous case, we rehash the height function event in terms of the LPP with two-sided boundary conditions event and show that the desired asymptotic probabilities arise from Theorem 1.3. The region of $\rho_{-},\rho_{+}$ for which we wish to prove $F_{0}$ fluctuations corresponds to $\pi+\eta>1$. We wish to prove $F_{0}$ fluctuations for all $y$ such that $1-2\rho_{-}<y<1-2\rho_{+}$. Translating this region into $\eta,\pi,\gamma$ variables exactly corresponds to the region in which Theorem 1.3 implies $F_{0}$ fluctuations.

We wish to compute the asymptotic formula for

where we have used the fact that $\bar{h}(y)=(y^{2}+1)/2$ in the region of $y,\rho_{-},\rho_{+}$ which we are considering. Without loss of generality, let us assume that $y\geq 0$ (the other case follows similarly). As before, set

The height function event we are considering has the same probability as

Using the equations for $N-M$ and $N+M$, we can solve for

These expressions allow us to express $\gamma$ asymptotically as

We may use Lemma 4.3 to invert our expression for $M$ in terms of $t$. Asymptotically

This implies that $t^{-2/3}=\frac{4^{-2/3}}{(1-y)^{-4/3}}M^{-2/3}+o(M^{-2/3})$. This can be plugged into our expression for $\gamma^{2}$ and gives a new expression for $\gamma^{2}$ in term of $M$ now:

This can then be substituted into (110) which gives

Plugging this into $P(L_{2}(N,M)\leq t)$, we find that our height function probability is asymptotically equal to

As it was already noted, the $y,\rho_{-},\rho_{+}$ which we are considered maps exactly onto the range of $\eta,\pi,\gamma$ which the LPP probability above is asymptotically equal to $F_{0}(x)$ and hence the same holds for the the height function probability.

Acknowledgments

This paper came out of discussions at the Courant ASEP seminar. The authors would like to thank Percy Deift for jointly organizing this seminar with them, as well as thank all of the participants. The authors appreciate helpful discussion of this material with Antonio Auffinger. The authors are grateful to Tomohiro Sasamoto for catching an important calculation error in an early reading of this paper, and to Timo Seppäläinen for orienting them to what has been previously proved.

References