Influence of the initial condition in equilibrium last-passage percolation models

In this paper we consider an equilibrium last-passage percolation model on an environment given by a compound two-dimensional Poisson process. We prove an $\LL^2$-formula relating the initial measure with the last-passage percolation time. This formula turns out to be a useful tool to analyze the fluctuations of the last-passage times along non-characteristic directions.


Introduction and the main result
1.1. The last-passage percolation model. Let P ⊆ R 2 be a two-dimensional Poisson random set of intensity one. On each point p ∈ P we put a random positive weight ω p and we assume that {ω p : p ∈ P} is a collection of i.i.d. random variables, distributed according to a distribution function F , which are also independent of P. Throughout this paper we will make the following assumption on the distribution function F of the weights: (1) ∞ 0 e ax dF (x) < +∞ , for some a > 0 .
This condition was used in [7] to prove the existence of invariant measures for the Hammersley's interacting fluid process we will introduce below. For each p, q ∈ R 2 , with p < q (inequality in each coordinate, p = q), let Π(p, q) denote the set of all increasing (or up-right) paths, consisting of points in P, from p to q, where we exclude all points that share (at least) one coordinate with p. So we consider the points in the rectangle ]p, q], where we leave out the south and the west side of the rectangle. The last-passage time between p ≤ q is defined by L(p, q) := max When F is the Dirac distribution concentrated on 1 (each point has weight 1 and we will denote this F by δ 1 ), then we refer to this model as the classical Hammersley model [1,9]. A crucial result is the following shape theorem (see Theorem 1.1 in [8], p.164): set 0 = (0, 0), n = (n, n), (2) γ = γ(F ) = sup n≥1 E(L(0, n)) n > 0 and f (x, t) := γ √ xt .
1.2. The interacting fluid system formulation. It is well known that the classical Hammersley model has a representation as an interacting particle system [1,9]. The general model has a similar description, although a better name might be an interacting fluid system. We start by restricting the compound Poisson process {ω p : p ∈ P} to R × R + . To each measure ν on R we associate a non-decreasing process ν(·) defined by ν(x) = ν((0, x]) for x ≥ 0 −ν((x, 0]) for x < 0.
Let N be the set of all positive, locally finite measures ν such that lim inf We need this condition to define the evolution of the process, since otherwise all mass will be pulled to minus infinity. The Hammersley interacting fluid system (M ν t : t ≥ 0) will be defined as a Markov process with values in N , as was done in [7]. Its evolution is defined as follows: if there is a Poisson point with weight ω at a point ( Here, M ν t− is the "mass distribution" of the fluid at time t if the Poisson point at (x 0 , t) would be removed. To the left of x 0 the measure does not change. In words, the Poisson point at (x 0 , t) moves a total mass ω to the left, to the point x 0 , taking the mass from the first available fluid to the right of x 0 . (See Figure 1 for a visualization, in case of atomic measures, of the process inside a space-time box.) In this paper we follow the Aldous and Diaconis [1] graphical representation in the last-passage model (compare to the result in the classical case, found in their paper): For each ν ∈ N , x ∈ R and t ≥ 0 let The measure M ν t defined by M ν t ((x, y]) := L ν (y, t) − L ν (x, t) for x < y , defines a Markov process on N and it evolves according to the Hammersley interacting fluid system [7].
We now make the following important observation for a random initial condition ν, which basically follows from translation invariance. Theorem 1. Suppose ν ∈ N is a random initial measure on R independent of the Poisson process in R × R + , whose distribution is translation invariant. For any speed V ∈ R and any x ∈ R, we have The relevance of this result is most clear when we consider equilibrium measures of the Hammersley's interacting fluid process. Assume that we have a probability measure defined on N and consider ν ∈ N as a realization of this probability measure. We say that ν is time invariant for the Hammersley interacting fluid process (in law) if In this case, we also say that the underlying probability measure on N is an equilibrium measure. It is known that there is only one family of ergodic equilibrium measures for the Hammersley interacting fluid system [7]. Let us denote it by {ν λ : λ > 0}, where (5) λ := Eν λ (1) .
For simple notation, put L λ := L ν λ . The main result of this paper is the following formula:     Here, V λ is the characteristic speed corresponding to L λ and ψ λ is the growth rate of L λ (V λ t, t). Then Cator and Groeneboom [6] proved that the variance of L λ grows sub-linearly along the characteristic speed λ −2 . Together with Corollary 1, this implies Corollary 2. Let (z t ) t≥0 be a deterministic path. Then Proof of Corollary 2: Formula (7), applied to the classical model, gives us On the other hand, [6] shows that Then where N is a standard Gaussian random variable.
Proof of Corollary 3: Corollary 2 shows that in the L 2 sense. Since ν λ is a one-dimensional Poisson process of intensity λ, this implies (9) and (10). 2 Remark 1. Cator and Groeneboom [6] proved that Var L λ (λ −2 t, t) is of order t 1/3 , which gives us the same order for the L 2 -distance between L λ (z t , t) and ν λ (x − λ −2 t) + 2λ −1 t . Remark 2. The central limit theorem for L λ (along any direction) was proved by Baik and Rains [5]. Their method was based on very particular combinatorial properties of the classical model that do not seem to hold for the general set-up. Our approach reveals the strong relationship with the initial configuration.
Remark 3. In the general set-up, Corollary 1 implies: If the variance of L λ along the characteristic speed V λ is sub-linear, and the equilibrium measure has Gaussian fluctuations, then L λ will also have Gaussian fluctuations along non-characteristic directions.
Remark 4. For the classical Hammersley process an important formula for the variance of L λ (x, t) was derived in [6], Theorem 2.1: where X λ (t) is the position at time t of a second class particle starting at zero. This formula was pivotal in deriving the cube-root behavior of L λ in [6], and later corresponding formulas were used to prove cube-root behavior for TASEP [3] and for ASEP [4]. However, this formula does not directly show the relationship with the initial configuration. Also, there seems to be no direct way to deduce (7) from this formula, even if we reformulate it, as was done in Equation (3.6) of [6], in terms of the exit-point of the longest path from (0, 0) to (x, t), which is the right-most z for which the supremum in (4) is attained.

Proof of Theorem 1 and Corollary 1
Recall that Clearly, L((z, 0), (V t, t)) D = L((z + x − V t, 0), (x, t)). By assumption, ν has a translation invariant distribution, independent of L. This implies that {z → ν(z)} This proves Theorem 1. 2 Corollary 1 now follows from results in [7]: there it is shown that for any speed V , the stationarity of L λ leads to This follows from the fact that the Hammersley fluid process has intensity λ on the bottom side of the rectangle between (0, 0) and (x, t), and intensity γ 2 /(4λ) on the left side (this refers to the expected mass of the fluid leaving the interval [0, x] through 0 per time unit). When we define the characteristic speed V λ = γ 2 /(4λ 2 ), then EL λ (V λ t, t) = ψ λ t.

The lattice last-passage percolation model
In the lattice last-passage percolation model one considers i.i.d. weights {ω p : p ∈ Z 2 }, distributed according to a distribution function F . For F (x) = 1 − e −x (exponential weights), we have a similar shape theorem as (3) with limit shape given by We know from [3] that the invariant measures are given by where {X z : z ∈ Z} is a collection of i.i.d. exponential random variables with parameter ρ. The analog to formula (7) is where V ρ := ρ 2 (1 − ρ) 2 and ψ ρ := Together with the cube-root asymptotics [3], this implies that where N is a standard Gaussian random variable.
Remark 5. Ferrari and Fontes [10] determined the dependence on the initial condition for the totally asymmetric exclusion process, which is isomorphic to the lattice last-passage percolation model with exponential weights. The method developed in this paper resembles the ideas in their paper. Balázs [2] used a different method to get a generalization of the Ferrari-Fontes result for certain types of deposition models. It is not clear to us whether our methods would work for these more general deposition models.

Remark 6.
In the general lattice model, the shape theorem (3) holds. However, not much is known about the limit shape f . If this function would not be strictly curved (we know it is convex, so this would mean that there are "flat" pieces), then the methods used in [8] to prove the existence and uniqueness of semi-infinite geodesics in a fixed direction do not apply, and we are not able to prove the existence of equilibrium measures.