On the local fluctuations of last-passage percolation models
aa r X i v : . [ m a t h . P R ] N ov On the local fluctuations of last-passage percolation models
Eric Cator and Leandro P. R. PimentelOctober 16, 2018
Abstract
Using the fact that the Airy process describes the limiting fluctuations of the Hammersleylast-passage percolation model, we prove that it behaves locally like a Brownian motion. Ourmethod is quite straightforward, and it is based on a certain monotonicity and good controlover the equilibrium measures of the Hammersley model (local comparison).
In recent years there has been a lot of research on the Airy process and related processes such asthe Airy sheet [10]. Most papers use analytic methods and exact formulas given by Fredholm de-terminants to prove properties of these processes, but some papers use the fact that these processesare limiting processes of last-passage percolation models or random polymer models, and they useproperties of these well studied models to prove the corresponding property of the limiting process.A nice example of these different approaches can be found in two recent papers, one by H¨agg [11]and one by Corwin and Hammond [9]. H¨agg proves in his paper that the Airy process behaveslocally like a Brownian motion, at least in terms of convergence of finite dimensional distributions.He uses the Fredholm determinant description of the Airy process to obtain his result. Corwinand Hammond on the other hand, use the fact that the Airy line process, of which the top linecorresponds to the Airy process, can be seen as a limit of Brownian bridges, conditioned to benon-intersecting. They show that a particular resampling procedure, which they call the BrownianGibbs property, holds for the system of Brownian bridges and also in the limit for the Airy lineprocess. As a consequence, it follows that the local Brownian behavior of the Airy process holdsin a stronger functional limit sense. Our paper will prove the same theorem, also using the factthat the Airy process is a limiting process, but in a much more direct way: we will consider theHammersley last-passage percolation model, and show that we can control local fluctuations of thismodel by precisely chosen equilibrium versions of this model, which are simply Poisson processes.Then we show that in the limit this control suffices to prove the local Brownian motion behaviorof the Airy process, as well as tightness of the models approaching the Airy process. We also ex-tend the control of local fluctuations of the Hammersley process to scales smaller than the typicalcube-root scale.Our method is quite straightforward, yet rather powerful, mainly because we have a certainmonotonicity and good control over the equilibrium measures. In fact, we think that we can extendour result to the more illustrious Airy sheet, the two dimensional version of the Airy process. Weaddress the reader to [10], for a description of this process in terms of the renormalization fixed1oint of the KPZ universality class. However, here we run in to much more technical problems,and this will still require a lot more work, beyond the scope of this paper.We will continue the introduction by developing notation, introducing all relevant processes andstating the three main theorems. In Section 2 we introduce the local comparison technique and ineach of the following three sections one theorem is proved.
The Hammersley last-passage percolation model [1] is constructed from a two-dimensional ho-mogeneous Poisson point process of intensity 1. Denote [ x ] t := ( x, t ) ∈ R and call a sequence[ x ] t , [ x ] t , . . . , [ x k ] t k of planar points increasing if x j < x j +1 and t j < t j +1 for all j = 1 , . . . , k − L ([ x ] s , [ y ] t ) between [ x ] s < [ y ] t is the maximal number of Poisson pointsamong all increasing sequences of Poisson points lying in the rectangle ( x, y ] × ( s, t ]. Denote L [ x ] t := L ([0] , [ x ] t ) and define A n by u ∈ R
7→ A n ( u ) := L [ n + 2 un / ] n − (2 n + 2 un / ) + u n / n / . Pr¨ahofer and Spohn [13] proved that lim n →∞ A n ( · ) dist. = A ( · ) , (1.1)in the sense of finite dimensional distributions, where A ≡ ( A ( u )) u ∈ R is the so-called Airy process.This process is a one-dimensional stationary process with continuous paths and finite dimensionaldistributions given by a Fredholm determinant [12]: P ( A ( u ) ≤ ξ , . . . , A ( u m ) ≤ ξ m ) := det (cid:16) I − f / Af / (cid:17) L ( { u ,...,u m }× R ) . The function A denotes the extended Airy kernel, which is defined as A s,t ( x, y ) := R ∞ e − z ( s − t ) Ai( x + z )Ai( y + z ) dz , if s ≥ t , − R −∞ e − z ( t − s ) Ai( x + z )Ai( y + z ) dz , if s < t , where Ai is the Airy function, and for ξ , . . . , ξ m ∈ R and u < · · · < u m in R , f : { u , . . . , u m } × R → R ( u i , x ) ( ξ i , ∞ ) ( x ) . The main contribution of this paper is the development of a local comparison technique tostudy the local fluctuations of last-passage times and its scaling limit. The ideas parallel the workof Cator and Groeneboom [5, 6], where they studied local convergence to equilibrium and thecube-root asymptotic behavior of L . This technique consists of bounding from below and fromabove the local differences of L by the local differences of the equilibrium regime (Lemma 1), withsuitable parameters that will allow us to handle the local fluctuations in the right scale (Lemma2). For the Hammersley model the equilibrium regime is given by a Poisson process. We havestrong indications that the technique can be applied to a broad class of models, as soon as one hasGaussian fluctuations for the equilibrium regime. Although this is a very natural assumption, onecan only check that for a few models. As a first application, we will prove tightness of A n .2 heorem 1 The collection {A n } is tight in the space of cadlag functions on [ a, b ] . Furthermore,any weak limit of A n lives on the space of continuous functions. The local comparison technique can be used to study local fluctuations of last-passage timesfor lengths of size n γ , with γ ∈ (0 , /
3) (so smaller than the typical scale n / ). Let B ≡ ( B ( u )) u ≥ denote the standard two-sided Brownian motion process. Theorem 2
Fix γ ∈ (0 , / and s > and define ∆ n by u ∈ R ∆ n ( u ) := L [ sn + un γ ] n − L [ sn ] n − µun γ σn γ/ , where µ := s − / and σ := s − / . Then lim n →∞ ∆ n ( · ) dist. = B ( · ) , in the sense of weak convergence of probability measures in the space of cadlag functions. As we mentioned in the previous section, the Airy process locally behaves like Brownian motion[9, 11]. By applying the local comparison technique again, we will present an alternative proof ofthe functional limit theorem for this local behavior.
Theorem 3
Define A ǫ by u ∈ R
7→ A ǫ ( u ) := ǫ − / ( A ( ǫu ) − A (0)) . Then lim ǫ → A ǫ ( · ) dist. = √ B ( · ) , in the sense of weak convergence of probability measures in the space of continuous functions. Consider a collection { ω [ x ] t : [ x ] t ∈ Z } of i.i.d. non negative random variables with an exponentialdistribution of parameter one. Let Π([ x ] t , [ y ] u ) denote the collection of all lattice paths ̟ =([ z ] v j ) j =1 ,...,k such that: • [ z ] v ∈ { [ x ] t + [1] , [ x ] t + [0] } and [ z ] v k = [ y ] u ; • [ z ] v j +1 − [ z ] v j ∈ { [1] , [0] } for j = 0 , . . . , , k − x ] t < [ y ] u is defined by L l ([ x ] t , [ y ] u ) := max ̟ ∈ Π([ x ] t , [ y ] u ) (cid:8) X [ z ] v ∈ ̟ ω [ z ] v (cid:9) . Denote L l [ x ] t := L l ([0] , [ x ] t ) and define A ln by u ∈ R
7→ A ln ( u ) := L l [ n + 2 / un / ] n − (4 n + 2 / un / ) + 2 / u n / / n / . n →∞ A ln ( · ) dist. = A ( · ) , (1.2)in the sense of finite dimensional distributions. The local comparison method can be used in thiscontext as well. (The lattice version of Lemma 1 is straightforward. For exponential weights, theanalog to Lemma 2 was proved in [2].) Theorem 4
The collection {A ln } is tight in the space of cadlag functions on [ a, b ] . Furthermore,any weak limit of A n lives on the space of continuous functions. Theorem 5
Fix γ ∈ (0 , / and s > and define ∆ n by u ∈ R ∆ ln ( u ) := L l [ sn + un γ ] n − L l [ sn ] n − µun γ σn γ/ , where µ = σ := s − / (1 + s / ) . Then lim n →∞ ∆ ln ( · ) dist. = B ( · ) , in the sense of weak convergence of probability measures in the space of cadlag functions. To avoid repetitions, we will not present a proof of the lattice results. We hope that the readercan convince his (or her) self that the method that we will describe in detail for the Hammersley last-passage percolation model can be easily adapted to the lattice models with exponentials weights.
Remark 1
We also expect that the local comparison method can be used in the log-gamma polymermodel, introduced by Sepp¨al¨ainen [14]. The polymer versions of Lemma 1 and Lemma 2 were provedin [14].
The Hammersley last-passage percolation model has a representation as an interacting particlesystem, called the Hammersley process [1, 5]. We will use notations used in [7]. In the positivetime half plane we have the same planar Poisson point process as before. On the x -axis we takea Poisson process of intensity λ >
0. The two Poisson process are assumed to be independent ofeach other. For x ∈ R and t > L λ [ x ] t ≡ L ν λ [ x ] t := sup z ∈ ( −∞ ,x ] { ν λ ( z ) + L ([ z ] , [ x ] t ) } , where, for z ≤ x , ν λ ( z ) = (cid:26) the number of Poisson points in (0 , z ] × { } for z > z, × { } for z ≤ . The process ( M tλ ) t ≥ , given by M tν λ ( x, y ] ≡ M tλ ( x, y ] := L λ [ y ] t − L λ [ x ] t for x < y ,
4s a Markov process on the space of locally finite counting measures on R . The Poisson process isthe invariant measure of this particle system in the sense that M tλ dist. = Poisson process of intensity λ for all t ≥ . (2.1)Notice that the last-passage time L = L ν can be recovered in the positive quadrant by choosinga measure ν on the axis that has no points to the right of 0, and an infinite amount of points inevery interval ( − ε, ∀ ε > L λ ≡ L ν λ ( P ) and L ≡ L ν ( P )are coupled by the same two-dimensional Poisson point process P , which corresponds to the basiccoupling between M tν λ and M tν .Define the exit points Z λ [ x ] t := sup { z ∈ ( −∞ , x ] : L λ [ x ] t = ν λ ( z ) + L ([ z ] , [ x ] t ) } , and Z ′ λ [ x ] t := inf { z ∈ ( −∞ , x ] : L λ [ x ] t = ν λ ( z ) + L ([ z ] , [ x ] t ) } . By using translation invariance and invariance under the map ( x, t ) ( λx, t/λ ), we have that Z λ [ x + h ] t dist. = Z λ [ x ] t + h and Z λ [ x ] t dist. = λZ [ λx ] t/λ . (2.2)We need to use one more symmetry. In [5], the Hammersley process was set up as a process in thefirst quadrant with sources on the x -axis and sinks on the t -axis. In our notation this means thatthe process t L λ [0] t is a Poisson process with intensity 1 /λ which is independent of the Poissonprocess ν λ restricted to the positive x -axis and independent of the Poisson process in the positivequadrant. We can now use reflection in the diagonal to see that the following equality holds: P (cid:0) Z ′ λ [ x ] t < (cid:1) = P (cid:0) Z /λ [ t ] x > (cid:1) . (2.3)We use that Z ′ λ [ x ] t < t -axis, and not the positive x -axis.The local comparison technique consists of bounding from above and from below the localdifferences of L by the local differences of L λ . These bounds depend on the position of the exitpoints. It is precisely summarized by the following lemma. Lemma 1
Let ≤ x ≤ y and t ≥ . If Z ′ λ [ x ] t ≥ then L [ y ] t − L [ x ] t ≤ L λ [ y ] t − L λ [ x ] t , and if Z λ [ y ] t ≤ then L [ y ] t − L [ x ] t ≥ L λ [ y ] t − L λ [ x ] t . Proof
When we consider a path ̟ from [ x ] s to [ y ] t consisting of increasing points, we will view ̟ as the lowest increasing continuous path connecting all the points, starting at [ x ] s and ending at[ y ] t . In this way we can talk about crossings with other paths or with lines. The geodesic between5 x ] s and [ y ] t is given by the lowest path (in the sense we just described) that attains the maximumin the definition of L ([ x ] s , [ y ] t ). We will denote this geodesic by ̟ ([ x ] s , [ y ] t ). Notice that L ([ x ] s , [ y ] t ) = L ([ x ] s , [ z ] r ) + L ([ z ] r , [ y ] t ) , for any [ z ] r ∈ ̟ ([ x ] s , [ y ] t ).Assume that Z ′ λ [ x ] t ≥ c be a crossing between the two geodesics ̟ ([0] , [ y ] t ) and ̟ ([ z ′ ] , [ x ] t ), where z ′ := Z ′ λ [ x ] t . Such a crossing always exists because x ≤ y and z ′ = Z ′ λ [ x ] t ≥ L λ [ y ] t ≥ ν λ ( z ′ ) + L ([ z ′ ] , [ y ] t ) ≥ ν λ ( z ′ ) + L ([ z ′ ] , c ) + L ( c , [ y ] t ) . We use this, and that (since c ∈ ̟ ([ z ′ ] , [ x ] t )) ν λ ( z ′ ) + L ([ z ′ ] , c ) − L λ [ x ] t = − L ( c , [ x ] t ) , in the following inequality: L λ [ y ] t − L λ [ x ] t ≥ ν λ (cid:0) z ′ (cid:1) + L ([ z ′ ] , c ) + L ( c , [ y ] t ) − L λ [ x ] t = L ( c , [ y ] t ) − L ( c , [ x ] t ) . By superaddivity, − L ( c , [ x ] t ) ≥ L ([0] , c ) − L [ x ] t , and hence (since c ∈ ̟ ([0] , [ y ] t )) L λ [ y ] t − L λ [ x ] t ≥ L ( c , [ y ] t ) − L ( c , [ x ] t ) ≥ L ( c , [ y ] t ) + L ([0] , c ) − L ([0] , [ x ] t )= L [ y ] t − L [ x ] t . The proof of the second inequality is very similar. Indeed, denote z := Z λ [ y ] t and let c be a crossingbetween ̟ ([0] , [ x ] t ) and ̟ ([ z ] , [ y ] t ). By superaddivity, L λ [ x ] t ≥ ν λ ( z ) + L ([ z ] , [ x ] t ) ≥ ν λ ( z ) + L ([ z ] , c ) + L ( c , [ x ] t ) . Since c ∈ ̟ ([ z ] , [ y ] t ) we have that L λ [ y ] t − ν λ ( z ) − L ([ z ] , c ) = L ( c , [ y ] t ) , which implies that L λ [ y ] t − L λ [ x ] t ≤ L λ [ y ] t − ν λ ( z ) − L ([ z ] , c ) − L ( c , [ x ] t )= L ( c , [ y ] t ) − L ( c , [ x ] t ) ≤ L [ y ] t − L ([0] , c ) − L ( c , [ x ] t )= L [ y ] t − L [ x ] t , where we have used that c ∈ ̟ ([0] , [ x ] t ) in the last step. ✷ emark 2 In fact the first statement of the lemma is also true when Z λ [ x ] t ≥ and the secondstatement is true when Z ′ λ [ y ] t ≤ , both without any change to the given proof. This is a strongerstatement, since Z ′ λ [ x ] t ≤ Z λ [ x ] t , but we will only need the lemma as it is formulated. In order to apply Lemma 1 and extract good bounds for the local differences one needs tocontrol the position of exit points. This is given by the next lemma.
Lemma 2
There exist constant
C > such that, P (cid:16) Z [ n ] n > rn / (cid:17) ≤ Cr , for all r ≥ and all n ≥ . Proof
See Corollary 4.4 in [6]. ✷ For simple notation, and without loss of generality, we will restrict our proof to [ a, b ] = [0 , Lemma 3
Fix β ∈ (1 / , and for each δ ∈ (0 , and n ≥ set λ ± = λ ± ( n, δ ) := 1 ± δ − β n / . Define the event E n ( δ ) := n Z ′ λ + [ n ] n ≥ and Z λ − [ n + 2 n / ] n ≤ o . Then there exists a constant
C > such that, for sufficiently small δ > , lim sup n →∞ P ( E n ( δ ) c ) ≤ Cδ β . Proof
Denote r := δ − β and let n + := λ + n < n and h + := (cid:18) λ + − λ + (cid:19) n > rn / > rn / / n ). By (2.2) and (2.3), P (cid:16) Z ′ λ + [ n ] n < (cid:17) = P (cid:0) Z /λ + [ n ] n > (cid:1) = P (cid:0) Z [ n/λ + ] λ + n > (cid:1) = P (cid:0) Z [ λ + n − h + ] λ + n > (cid:1) = P (cid:0) Z [ λ + n ] λ + n > h + (cid:1) ≤ P (cid:16) Z [ n + ] n + > rn / / (cid:17) . n − := nλ − < n and h − := (cid:18) λ − − λ − (cid:19) n > rn / > rn / − / , we have that P (cid:16) Z λ − [ n + 2 n / ] n > (cid:17) = P (cid:16) Z λ − [ n ] n > − n / (cid:17) = P (cid:16) λ − Z [ λ − n ] n/λ − > − n / (cid:17) ≤ P (cid:16) Z [ n − − h − ] n/λ − > − n / (cid:17) = P (cid:16) Z [ n − ] n − > h − − n / (cid:17) ≤ P (cid:16) Z [ n − ] n − > ( r − n / − / (cid:17) . Now one can use Lemma 2 to finish the proof. ✷ Lemma 4
Let δ ∈ (0 , and u ∈ [0 , − δ ) . Then, on the event E n ( δ ) , for all v ∈ [ u, u + δ ] wehave that B n, − ( v ) − B n, − ( u ) − δ − β ≤ A n ( v ) − A n ( u ) ≤ B n, + ( v ) − B n, + ( u ) + 4 δ − β , where B n, ± ( u ) : L λ ± [ n + 2 un / ] n − L λ ± [ n ] n − λ ± un / n / . Proof
For fixed t , Z ′ λ [ x ] t and Z λ [ x ] t are non-decreasing functions of x . Thus, on the event E n ( δ ), Z ′ λ + [ n + 2 un / ] n ≥ Z λ − [ n + 2( u + δ ) n / ] n ≤ . By Lemma 1, this implies that, for all v ∈ [ u, u + δ ], L [ n + vn / ] n − L [ n + un / ] n ≤ L λ + [ n + vn / ] n − L λ + [ n + un / ] n , and L [ n + vn / ] n − L [ n + un / ] n ≥ L λ − [ n + vn / ] n − L λ − [ n + un / ] n . Since ( λ + − v − u ) n / + v − u ≤ δ − β + 2 δ ≤ δ − β , and ( λ − − v − u ) n / + v − u ≥ − δ − β , we have that, on the event E n ( δ ), A n ( v ) − A n ( u ) ≤ B n, + ( v ) − B n, + ( u ) + 4 δ − β . A n ( v ) − A n ( u ) ≥ B n, − ( v ) − B n, − ( u ) − δ − β , for all v ∈ [ u, u + δ ]. ✷ Proof of Theorem 1
For fixed u ∈ [0 ,
1) take δ > u + δ ≤
1. By Lemma 4,sup v ∈ [ u,u + δ ] |A n ( v ) − A n ( u ) | ≤ max ( sup v ∈ [ u,u + δ ] |B n, ± ( v ) − B n, ± ( u ) | ) + 4 δ − β , on the event E n ( δ ). Hence, for any η > P sup v ∈ [ u,u + δ ] |A n ( v ) − A n ( u ) | > η ! ≤ P ( E n ( δ ) c )+ P sup v ∈ [ u,u + δ ] |B n, + ( v ) − B n, + ( u ) | > η − δ − β ! + P sup v ∈ [ u,u + δ ] |B n, − ( v ) − B n, − ( u ) | > η − δ − β ! . By (2.1), P n ( x ) := L λ (cid:0) [ n + x ] n (cid:1) − L λ (cid:0) [ n ] n (cid:1) , for x ≥ , is a Poisson process of intensity λ . Since λ ± → n → ∞ , B n, − ( u/
2) and B n, + ( u/
2) converge indistribution to a standard Brownian motion B . Thus, by Lemma 3, for δ < ( η/ / (1 − β ) ,lim sup n →∞ P sup v ∈ [ u,u + δ ] |A n ( v ) − A n ( u ) | > η ! ≤ Cδ β + 2 P sup v ∈ [ u,u + δ ] |B (2 v ) − B (2 u ) | > η − δ − β ! ≤ Cδ β + 2 P sup v ∈ [0 , |B ( v ) | > η √ δ ! , which implies that (recall that β ∈ (1 / , δ → + δ lim sup n →∞ P sup v ∈ [ u,u + δ ] |A n ( v ) − A n ( u ) | > η !! = 0 . (3.1)Since [6] lim sup n →∞ E | L [ n ] n − n | n / < ∞ , we have that {A n (0) , n ≥ } is tight. Together with (3.1), this shows tightness of the collection {A n , n ≥ } in the space of cadlag functions on [0 , ✷ Proof of Theorem 2
For simple notation, we will prove the statement for s = 1 and restrict our selves to [0 , s >
0, since L [ sn ] n dist. = L [ s / n ] s / n . Lemma 5
Fix γ ′ ∈ ( γ, / and let λ ± = λ ± ( n ) := 1 ± n γ ′ / . Define the event E n := n Z ′ λ + [ n ] n ≥ and Z λ − [ n + n γ ] n ≤ o . There exists a constant
C > such that P ( E cn ) ≤ Cn − γ ′ / for all sufficiently large n . Proof
Denote r := n / − γ ′ / and let n + := λ + n < n and h + := (cid:18) λ + − λ + (cid:19) n > rn / > rn / / , (for all sufficiently large n ). By (2.2) and (2.3), P (cid:16) Z ′ λ + [ n ] n < (cid:17) = P (cid:0) Z /λ + [ n ] n > (cid:1) = P (cid:0) Z [ n/λ + ] λ + n > (cid:1) = P (cid:0) Z [ λ + n − h n ] λ + n > (cid:1) = P (cid:0) Z [ λ + n ] λ + n > h + (cid:1) ≤ P (cid:16) Z [ n + ] n + > rn / / (cid:17) . Analogously, for n − := nλ − < n and h − := (cid:18) λ − − λ − (cid:19) n > rn / > rn / − / , we have that P (cid:0) Z λ − [ n + n γ ] n > (cid:1) = P (cid:0) Z λ − [ n ] n > − n γ (cid:1) = P (cid:0) λ − Z [ λ − n ] n/λ − > − n γ (cid:1) ≤ P (cid:0) Z [ n − ] n − > h − − n γ (cid:1) ≤ P (cid:16) Z [ n − ] n − > ( r − n γ − / ) n / − / (cid:17) , Now one can use Lemma 2 to finish the proof. ✷ emma 6 On the event E n , for all u < v in [0 , , Γ − n ( v ) − Γ − n ( u ) − n ( γ ′ − γ ) / ≤ ∆ n ( v ) − ∆ n ( u ) ≤ Γ + n ( v ) − Γ + n ( u ) + 1 n ( γ ′ − γ ) / , where Γ ± n ( u ) := L λ ± [ n + un γ ] n − L λ ± [ n ] n − λ ± un γ n γ/ . Proof
By Lemma 1, if Z ′ λ + [ n ] n ≥ L [ n + vn γ ] n − L [ n + un γ ] n ≤ L λ + [ n + vn γ ] n − L λ + [ n + un γ ] n , and if Z λ − [ n + n γ ] n ≤ L [ n + vn γ ] n − L [ n + un γ ] n ≥ L λ − [ n + vn γ ] n − L λ − [ n + un γ ] n . Using that λ ± := 1 ± n − γ ′ / , one can finish the proof of the lemma. ✷ Proof of Theorem 2
By Lemma 6, on the event E cn , | ∆ n ( v ) − ∆ n ( u ) | ≤ max (cid:8) | Γ ± n ( v ) − Γ ± n ( u ) | (cid:9) + 1 n ( γ ′ − γ ) / . Thus, by Lemma 5, P sup v ∈ [ u,u + δ ] | ∆ n ( v ) − ∆ n ( u ) | > η ! ≤ P sup v ∈ [ u,u + δ ] | Γ + n ( v ) − Γ + n ( u ) | + 1 n ( γ ′ − γ ) / > η ! + P sup v ∈ [ u,u + δ ] | Γ − n ( v ) − Γ − n ( u ) | + 1 n ( γ ′ − γ ) / > η ! + P ( E cn ) . As before, λ ± → n → ∞ , which implies thatlim sup n →∞ P sup v ∈ [ u,u + δ ] | ∆ n ( v ) − ∆ n ( u ) | > η ! ≤ P sup v ∈ [0 ,δ ] | B ( v ) | > η ! = 2 P sup v ∈ [0 , | B ( v ) | > η √ δ ! , and hence lim sup δ → + δ lim sup n →∞ P sup v ∈ [ u,u + δ ] | ∆ n ( v ) − ∆ n ( u ) | > η !! = 0 . (4.1)Since ∆ n (0) = 0, (4.1) implies tightness of the collection { ∆ n , n ≥ } in the space of cadlagfunctions on [0 , u , . . . , u k ∈ [0 ,
1] and a , . . . , a k , ∈ R , P (cid:16) ∩ ki =1 { ∆ n ( u i ) ≤ a i } (cid:17) ≥ P (cid:18) ∩ ki =1 (cid:26) Γ + n ( u i ) ≤ a i − n ( γ ′ − γ ) / (cid:27)(cid:19) − P ( E cn ) , P (cid:16) ∩ ki =1 { ∆ n ( u i ) ≤ a i } (cid:17) ≤ P (cid:18) ∩ ki =1 (cid:26) Γ − n ( u i ) ≤ a i + 1 n ( γ ′ − γ ) / (cid:27)(cid:19) + P ( E cn ) , which shows that the finite dimensional distributions of ∆ n converge to the finite dimensionaldistributions of the standard Brownian motion process. ✷ Lemma 7
Fix β ∈ (0 , / and for ǫ ∈ (0 , let λ ± = λ ± ( n, ǫ ) := 1 ± ǫ − β n / . Define the event E n ( ǫ ) := n Z ′ λ + ([ n ] n ) ≥ and Z λ − ([ n + n / ] n ) ≤ o . There exists a constant
C > such that, for all sufficiently small ǫ > , lim sup n →∞ P ( E n ( ǫ ) c ) ≤ Cǫ β . Proof
The same proof as in Lemma 3 applies. ✷ Lemma 8
On the event E n ( ǫ ) , for all u ∈ [0 , − δ ) and v ∈ [ u, u + δ ] , we have that B n, − ( ǫv ) − B n, − ( ǫu ) − δǫ − β ≤ A n ( ǫv ) − A n ( ǫu ) ≤ B n, + ( ǫv ) − B n, + ( ǫu ) + 4 δǫ − β . Proof
The same proof as in Lemma 4 applies. Note that in this case we have( λ + − εv − εu ) ≤ ε − β δ. ✷ Proof of Theorem 3
For u ∈ [0 , A ǫn ( u ) := ǫ − / ( A n ( ǫu ) − A n (0)) and B ǫn, ± ( u ) := ǫ − / B n, ± ( ǫu ) . By Lemma 8, on the event E n ( ǫ ), for all v ∈ [ u, u + δ ], B ǫn, − ( v ) − B ǫn, − ( u ) − δǫ / − β ≤ A ǫn ( v ) − A ǫn ( u ) ≤ B ǫn, + ( v ) − B ǫn, + ( u ) + 4 δǫ / − β , which shows thatsup v ∈ [ u,u + δ ] |A ǫn ( v ) − A ǫn ( u ) | ≤ max ( sup v ∈ [ u,u + δ ] |B ǫn, ± ( v ) − B ǫn, ± ( u ) | ) + 4 δǫ / − β . P sup v ∈ [ u,u + δ ] |A ǫn ( v ) − A ǫn ( u ) | > η ! ≤ P ( E n ( ǫ ) c )+ P sup v ∈ [ u,u + δ ] |B ǫn, + ( v ) − B ǫn, + ( u ) | > η − δǫ / − β ! + P sup v ∈ [ u,u + δ ] |B ǫn, − ( v ) − B ǫn, − ( u ) | > η − δǫ / − β ! . Since A ǫn is converging to A ǫ , and B ǫn, ± is converging to a Brownian motion B , the precedinginequality implies that P sup v ∈ [ u,u + δ ] |A ǫ ( v ) − A ǫ ( u ) | > η ! ≤ Cǫ β + 2 P sup v ∈ [ u,u + δ ] |B (2 v ) − B (2 u ) | > η − δǫ / − β ! . Hence lim sup ǫ → + P sup v ∈ [ u,u + δ ] |A ǫ ( v ) − A ǫ ( u ) | > η ! ≤ P sup v ∈ [0 , |B ( v ) | > η √ δ ! , which shows that lim sup δ → + δ lim sup ǫ → + P sup v ∈ [ u,u + δ ] |A ǫ ( v ) − A ǫ ( u ) | > η !! = 0 . (5.1)Since A ǫ (0) = 0, by (5.1) we have that {A ǫ , ǫ ∈ (0 , } is tight [4].The finite dimensional distributions of the limiting process can be obtained in the same way.Indeed, by Lemma 8, for u , . . . , u k ∈ [0 ,
1] and a , . . . , a k , ∈ R , P (cid:16) ∩ ki =1 {A ǫn ( u i ) ≤ a i } (cid:17) ≤ P (cid:16) ∩ ki =1 n B ǫ − ,n ( u i ) ≤ a i + 4 ǫ / − β o(cid:17) + P ( E n ( ǫ ) c ) , and P (cid:16) ∩ ki =1 {A ǫn ( u i ) ≤ a i } (cid:17) ≥ P (cid:16) ∩ ki =1 n B ǫ + ,n ( u i ) ≤ a i − ǫ / − β o(cid:17) − P ( E n ( ǫ ) c ) . Thus, by Lemma 7, P (cid:16) ∩ ki =1 {A ǫ ( u i ) ≤ a i } (cid:17) ≤ P (cid:16) ∩ ki =1 n B (2 u i ) ≤ a i + 4 ǫ / − β o(cid:17) + Cǫ β , and P (cid:16) ∩ ki =1 {A ǫ ( u i ) ≤ a i } (cid:17) ≥ P (cid:16) ∩ ki =1 n B (2 u i ) ≤ a i − ǫ / − β o(cid:17) − Cǫ β , which proves that, lim ǫ → + P (cid:16) ∩ ki =1 {A ǫ ( u i ) ≤ a i } (cid:17) = P (cid:16) ∩ ki =1 n √ B ( u i ) ≤ a i o(cid:17) . (5.2) ✷ eferences [1] Aldous, D. and Diaconis, P. (1995). Hammersley’s interacting particle system and longestincreasing subsequences.
Probab. Theory Rel. Fields :199–213.[2]
Bal´azs, M., Cator, E. A. and Sepp¨al¨ainen, T. (2006). Cube root fluctuations for thecorner growth model associated to the exclusion process.
Elect. J. Probab. :1094–1132.[3] Baik, J., Ferrari, P .L., P´ech´e, S. (2010). Limit process of stationary TASEP near thecharacteristic line.
Commun. Pure Appl. Math. :1017–1070.[4] Billingsley, P. (1968). Convergence of probability measures. John Wiley & Sons, New York.[5]
Cator, E. A. and Groeneboom, P. (2005). Hammersley’s process with sources and sinks.
Ann. Probab. :879–903.[6] Cator, E. A. and Groeneboom, P. (2006). Second class particles and cube root asymp-totics for Hammersley’s process.
Ann. Probab. :1273–1295.[7] Cator, E. A. and Pimentel, L. P. R. (2012). Busemann functions and equilibrium mea-sures in last passage percolation models.
Probab. Theory Rel. Fields :89–125.[8]
Corwin, I., Ferrari, P. L., P´ech´e, S. (2010). Limit processes for TASEP with shocks andrarefaction fans.
J. Stat. Phys. :232–267.[9]
Corwin, I., Hammond, A. (2011). Brownian Gibbs property for Airy line ensembles. Avail-able from arXiv:1108.2291.[10]
Corwin, I., Quastel, J. (2011). Renormalization fixed point of the KPZ universality class.Available from arXiv:1103.3422.[11]
H¨agg, J. (2008). Local fluctuations in the Airy and discrete PNG process.
Ann. Probab. :1059–1092.[12] Johansson, K. (2003). Discrete polynuclear growth and determinantal processes.
Comm.Math. Phys. :277–329.[13]
Pr¨ahofer, M. and Spohn, H. (2002). Scale invariance of the PNG droplet and the Airyprocess.
J. Statist. Phys. :1076–1106.[14]
Sepp¨al¨ainen, T. (2012). Scaling for a one-dimensional directed polymer with boundary con-ditions.
Ann. Probab.40