Local stationarity of exponential last passage percolation

We consider point to point last passage times to every vertex in a neighbourhood of size $\delta N^{\frac{2}{3}}$, distance $N$ away from the starting point. The increments of these last passage times in this neighbourhood are shown to be \emph{jointly equal} to their stationary versions with high probability that depends on $\delta$ only. With the help of this result we show that 1) the tree of point to point geodesics starting from every vertex in a box of side length $\delta N^{\frac{2}{3}}$ going to a point at distance $N$ agree inside the box with the tree of infinite geodesics going in the same direction; 2) two geodesics starting from $N^{\frac{2}{3}}$ away from each other, to a point at distance $N$ will not coalesce too close to either endpoints on the macroscopic scale.

behaviour under this scaling. The LPP with exponential weights belongs to the set of models in the KPZ universality class that are exactly solvable, or integrable. For models in this group, one can obtain closed form expressions for their prelimiting statistics. Coupling this with techniques from representation and random matrix theory, combinatorics, one can take the limits of the prelimiting expression to obtain the statistics of the limiting object. By the KPZ universality conjecture, this should be the limit of all models in the KPZ universality class.
LPP can be viewed as a 1`1 growing surface, and also as a Markov process that takes values in the space of continuous functions. Using the 1 : 2 : 3 KPZ scaling, the conjectural limit of this Markov process is believed to be the KPZ-fixed point [18]. An extension of this limiting object was shown to exist recently in [5]. In [16] Johansson showed the convergence of the spatial fluctuations to the Airy 2 process minus a parabola and that the limit is continuous. The fact that LPP has stationary counterparts whose spatial fluctuations are that of a simple random walk suggests that locally, the Airy 2 process should have Brownian behaviour around a fixed point. In [21] Pimentel, in the LPP setup, showed that locally the Airy 2 process converges to a Brownian motion in the Skorohord topology. The proof relied on a technique called 'Comparison Lemma' or 'Crossing Lemma'. The idea is that the spatial increments of the last passage time can be compared with high probability to stationary increments with a small drift. In [3] Corwin and Hammond showed that the Airy line ensemble minus parabola, conditioned on its values at the boundaries has the distribution of Brownian bridges conditioned not to meet. Building on these ideas, Hammond obtained through the Brownian LPP [12], among other things, a control on the moment of the Radon-Nykodim derivative of the law of the Airy line ensemble with respect to the Brownian bridge and a modulus of continuity of the Airy 2 process. In LPP on the lattice, control on the modulus of continuity of the prelimiting spatial fluctuations was obtained in [1] by Basu and Ganguly. In [18], Matetski, Quastel and Remenik showed that the Airy 2 process has Brownian regularity and converges to the two-sided brownian motion in finite dimensional distributions. In [6] Dauvergne and Virág, using better insight on the sampled Airy line ensemble, managed to show that the Airy line ensemble can be approximated, in total variation, by Brownian bridges without the conditioning on the lower boundary that appears in the Brownian Gibbs property. In [22] Pimentel shows Brownian regularity for the KPZ-fixed point.
The local Brownian behaviour of the limiting objects translates to the local prelimiting fluctuations of the LPP being close to the stationary LPP or the Busemann cocycle. To make this more concrete, let G x be the last passage time between the points p0, 0q and x. Define L N px,yq " G pN,N q`y´GpN,N q`x .
The local Brownian behaviour stems from L N being close to the stationary LPP increments for x and y of order δN 2 3 . In the case where |x|, |y| " Op1q, L N essentially converges to the so called Busemann function associated with the 45˝direction. Busemann functions can be thought of as the extention of the stationary LPP to the whole lattice and play a major role in the study of infinite geodesics.
The main contribution of this paper is to show the convergence in total variation of L N to the Busemann function when |x|, |y| ď δN 2 3 and δ goes to 0. Moreover, the results are quantitative. We stress that this cannot be simply obtained by using the 'Crossing Lemma'. Indeed, in order to compare the LPP increments to those of the stationary LPP one must tweak the intensity of the stationary LPP by order of N´1 3 such that the error of the approximation along each edge is of the order of N´1 3 as well. Therefore, a simple union bound on the different N 2 3 edges will give N 2 3 N´1 3 " N 1 3 and will not work. In a recent work, Fan and Seppäläinen [7] obtained a coupling of different Busemann functions using queueing mapping. We use new insights on this coupling to obtain the result which we refer to as local stationarity.
The rest of our results are applications of local stationarity to questions about geodesics. We study two aspects of the behaviour of geodesics, their behaviour close to the end points which we refer to as stabilization, and the coalescence of point to point geodesics starting from two points whose distance scales with N . Let us start with the latter. In the past few years the study of coalescence of geodesics has gained focus. Methods for the study of geodesics of growth models can be traced back to Newman and co-authors in [13,14,17,19] for First Passage Percolation (FPP), another random growth model believed to be in the KPZ universality class. These methods were then used by Ferrari and Pimentel [9] and Coupier [4] to show that in LPP, for a fixed direction, from any point in the lattice there exists a.s. a unique infinite geodesic and that these geodesics coalesce. A first quantitive result on the coalescence of geodesics in LPP came from Pimentel [20], who showed that two infinite geodesics with the same direction, coming out of two points that are k away from each other will coalesce after about k 3 {2 steps. The tail of the decay was conjectured to be of exponent´2{3. The proof used the fact that the geodesic tree has the same distribution as its dual tree and existing bounds on the distribution of exit point of a geodesic of stationary LPP. The question of showing that the geodesics will not coalesce too far compared to k i.e. a matching upper bound, was left open. This question was then taken up by Basu, Sarkar and Sly [2] who proved the´2{3 exponent for the lower bound and a matching upper bound. In that paper, the authors also proved a polynomial upper bound for point to point coalescence. In [25] Seppäläinen and Shen, studied coalescence of infinite geodesics. Without relying on integrable probability methods, they proved the upper bound and a new exponential lower bound for fast coalescence of the geodesics. In [27] Zhang proved the optimal bounds of´2{3 for point to point coalescence of two geodesics leaving from two points of constant distance k. The proof relies on diffusive concentration of geodesics fluctuations coming from integrable probability.
In this work, we also study the coalescence of two geodesics starting from two points whose distance scales with the length of the geodesics. For a related work in Poissonian last passage percolation see [8]. More precisely, we are interested in the following question; if π 1 and π 2 are the geodesics starting from p0, 0q and p0, N 2 3 q respectively, terminating at pN, N q, what is the typical distance of the coalescence point from the three endpoints? We show that the coalescence point will not be too close, on a macroscopic scale, to any of the end points. We emphasize that the methods used in [20] and [25] cannot be used here, as they are predicated on a well understood duality principle for stationary LPP geodesics [24].
Let us now turn to stabilization. Let π be the geodesic going from p0, 0q to the point pN, N q. Since the work of Johansson in [15] it is known that the fluctuations of π around the diagonal at any macroscopic point should be of order N 2 {3 . If 1 ď l ăă N , as the geodesic is expected to have a self-similarity property, one would expect the fluctuation of π in a square of size l 2 around the origin to be of order l 2 {3 . A proof of this was given in [2, Theorem 3] with diffusive concentration bounds. In [12] Hammond considered the regularity of the spatial fluctuation around the point pl, lq for the Brownian LPP while for the Corner Growth Model with exponential weights this was proven in [1, Theorem 3] by Basu and Ganguly. The behaviour of infinite geodesics is somewhat better understood. This is due to the fact that the Busemann functions 'point out' the way in which the geodesic go, through the minimum gradient principle [10,11,24]. Consider a small square of side M around the origin. From each point in the square leaves a unique geodesic that terminate at the point pN, N q. Let us denote by T pp the tree consisting of all the geodesics starting from the square and ending at the point pN, N q. Similarly let T 8 be the tree that consists of all infinite geodesics of direction 45˝starting from the square. Our stabilization result shows that on a square of side M " δN 2 3 , the trees T pp and T 8 agree outside a set of probability of order power of δ.

Main results
Let ω " tω x u xPZ 2 be a set of random weights on the vertices of Z 2 . We assume that ω is i.i.d. of Expp1q distribution. For o P Z 2 , we define the last-passage time on o`Z 2 ě0 to be Π o,y is the set of paths x ‚ " px k q n k"0 that start at x 0 " o, end at x n " y with n " |y´o| 1 , and have increments x k`1´xk P te 1 , e 2 u. The a.s. unique path π o,y P Π o,y that attains the maximum in (2.1) is called the geodesic from o to y. Similarly we define the stationary LPP (see (3.7)) G 1 2 o,y associated with the direction p1, 1q. Let p R c " rN ξ´cN 2 3 ξ, N ξs be the rectangle whose lower left corner is pN´cN 2 3 , N´cN 2 3 q and whose upper right corner is pN, N q. Let Ep p R c q be the set of directed edges in the subgraph of Z 2 induced by the vertices in p R c . We define the following random variables indexed by Ep p Let d TV p¨,¨q denote the total variation distance between two distributions on a measurable set. If X " µ and Y " ν, we abuse notation and write d TV pX, Y q for d TV pµ, νq. The following result shows that on the scale of N  The set of possible asymptotic velocities or direction vectors for semi-infinite up-right paths is U " tpt, 1´tq : 0 ď t ď 1u, with relative interior ri U " tpt, 1´tq : 0 ă t ă 1u. For ξ P ri U, let R ξ,N " r0, N ξs be the rectangle whose lower left corner is p0, 0q and upper right corner is N ξ. Let π be an up-right path whose origin is p0, 0q. Let I π " ti : π i P R ξ,N u be the set of indices of π for which π is in R ξ,N . We define P ξ,N pπq to be the restriction of the path π on the rectangle R ξ,N , that is, P ξ,N pπq is a finite path defined by pP ξ,N pπqq i " π i @i P I π . Let π x,ξ8 be the infinite geodesic starting from x whose direction is ξ [24]. For M ă N , define the following event S ξ,M " tP ξ,M pπ x,ξ8 q " P ξ,M pπ x,ξN q for all x P R ξ,M u.    PpS ξ,f pN q q " 1. (2.7) In words, the sequence R ξ,f stablizes if with high probability the tree of all the geodesics starting at points in R ξ,f pN q and terminating at ξN agree on R ξ,f pN q with the tree of infinite geodesics in direction ξ starting from R ξ,f pN q . As f is a function of N we shall often write f instead of f pN q so that R ξ,f " R ξ,f pN q . As a corollary of Theorem 2.2 we have the following.   Stabilization can help relating results on infinite geodesics to results on point-to-point geodesics and vice versa. Consider the points v 1 " p0, 0q and v 2 " k 2 {3 e 2 for some k ě 1. Let π v 1 ,ξ8 and π v 2 ,ξ8 be the infinite geodesics in direction ξ starting from v 1 and v 2 respectively. Let v˚" pv1 , v2 q be the point in π v 1 ,ξ8 X π v 2 ,ξ8 that is closest to the origin. Similarly let u N be the closest point in π v 1 ,ξN X π v 2 ,ξN to the origin. In [2] Basu, Sarkar and Sly showed that there exist universal constants C 1 , C 2 , R 0 such that for every k ą 0 and R ą R 0 Moreover, they showed that there exist C, R 0 , c ą 0 such that for every k ą 0 The constant c in (2.10) was not identified but was conjectured to be 2 {3. This was recently settled by Zhang in [27] using input from integrable probability. We now show how this can be approached via our stabilization result.
Theorem 2.4. The sequence |u N | converges weakly to |v˚|. Moreover, there exist universal constants C 1 , C 2 , R 0 ą 0 such that for R ą R 0 , for any k ě 1 and N ą pRkq 5 Proof. If exactly one of the events t|v˚| ą Rku, t|u N | ą Rku occurs then paths must not have coalesced in R ξ,Rk , in other words, S ξ,Rk does not occur. Therefore, via the symmetric difference and using Theorem 2.2, which shows that |v N | converges weakly to |v˚|. Taking N " pRkq 5 and using Theorem 2.2 Let us now turn to our coalescence results. In R ξ,N , consider the points o " ξN , q 1 " p0, 0q and q 2 " aN 2 3 e 2 where a ą 0 and where we assume that N is large enough so that q 2 P R ξ,N . Let C a,ξ " π q 1 ,o X π q 2 ,o , (2.12) be the points shared by the geodesics starting from q i and terminating at o for i P 1, 2. We define the coalescence point p c to be the unique point such that p c P C a,ξ and p c ď x @x P C a,ξ , (2.13) as in Figure 2.2. Our next result shows that the point p c is not likely to be too close to the point o on a macroscopic scale.
Theorem 2.5. For every a ą 0 and ξ P ri U, there exists a constant Cpξ, aq ą 0, locally bounded in a, such that for every 0 ă α ă 1 and N ą N pαq Pp|o´p c | ď αN q ď Cα (2.14) The following result shows that the geodesics π q 1 ,o and π q 2 ,o do not coalesce too close to their origins on a macroscopic scale.
Theorem 2.6. For every a ą 0 and ξ P ri U, there exists a constants Cpξ, aq ą 0 such that for every 0 ă α ă 1 and N ą N pαq

Preliminaries
Some general notation and terminology Z ě0 " t0, 1, 2, 3, . . . u and Z ą0 " t1, 2, 3, . . . u. For n P Z ą0 we abbreviate rns " t1, 2, . . . , nu. A sequence of n points is denoted by x 0,n " px k q n k"0 " tx 0 , x 1 , . . . , x n u, and in case it is a path of length n also by x ‚ . a _ b " maxta, bu. C is a constant whose value can change from line to line.
The standard basis vectors of R 2 are e 1 " p1, 0q and e 2 " p0, 1q. For a point x " px 1 , x 2 q P R 2 the 1 -norm is |x| 1 " |x 1 |`|x 2 | and integer parts are taken coordinatewise: txu " ptx 1 u, tx 2 uq. We call the x-axis occasionally the e 1 -axis, and similarly the y-axis and the e 2 -axis are the same thing. Inequalities on R 2 are interpreted coordinatewise: for x " px 1 , x 2 q P R 2 and y " py 1 , y 2 q P R 2 , x ď y means x 1 ď y 1 and x 2 ď y 2 . Notation rx, ys represents both the line segment rx, ys " ttx`p1´tqy : 0 ď t ď 1u for x, y P R and the rectangle rx, ys " tpz 1 , z 2 q P R 2 : x i ď z i ď y i for i " 1, 2u for x " px 1 , x 2 q, y " py 1 , y 2 q P R 2 . The context will make clear which case is used. 0 denotes the origin of both R and R 2 .
Path segments are abbreviated by π rm,ns " pπ i q n i"m . X " Exp(λ) for 0 ă λ ă 8 means that random variable X has exponential distribution with rate λ, in other words P pX ą tq " e´λ t for t ě 0. The mean is EpXq " λ´1 and variance VarpXq " λ´2.
We write ω x and ωpxq interchangeably for the weight attached to lattice point x. In general, X " X´EX denotes a random variable X centered at its mean. If x ă y we write x, y for the set of integers rx, ys X Z. If x, y P R 2 such that x ď y we denote by x, y " rx, ys X Z 2 . If A Ă Z 2 is connected, we let EpAq denote the set of edges induced by A in Z 2 .

Order on Geodesics
We would like to construct a partial order on the set of non-intersecting paths in Z 2 . For x, y P Z 2 we write x ĺ y if y is below and to the right of x, i.e.
x 1 ď y 1 and x 2 ě y 2 . We also write xăy if A down-right path is a bi-infinite sequence Y " py k q kPZ in Z 2 such that y k´yk´1 P te 1 ,´e 2 u for all k P Z. Let DR be the set of infinite down-right paths in Z 2 . Let γ 1 , γ 2 be two up-right paths in Z 2 we write γ 1 ĺγ 2 if where we assume the inequality to be vacuously true if one of the intersections in (3.4) is empty(see Figure 3.1).

Stationary LPP
For each o " po 1 , o 2 q P Z 2 and a parameter value ρ P p0, 1q we introduce the stationary last-passage percolation process G ρ o,‚ on o`Z 2 ě0 . This process has boundary conditions given by two independent sequences Then in the bulk for x " px 1 , .
from the west and south boundaries of o`Z 2 ą0 . More precisely, (3.8) The value G ρ o,x can be determined by (3.6) and the following recursive relation Relation (3.9) implies that one can backtrack the geodesic π o,p in the box ro`e 1`e2 , ps in the following way; for each (directed) edge px, yq in ro`e 1`e2 , ps assign the weight w x,y " G ρ o,y´G ρ o,x . Let m " |p´o|, and denote p i " π o,p i . We have p m " p, (3.10) In other words, we trace the geodesic π o,p backwards up to the exit point from the boundaries, by following the edges on which the increments of the process G ρ o,p are minimal. Next we consider LPP maximizing down-left paths. For y ď o, define For each o P Z 2 and a parameter value ρ P p0, 1q define a stationary last-passage percolation processes , with boundary variables on the north and east, in the following way. Let tI ρ o´ie 1 u 8 i"1 and tJ ρ o´je 2 u 8 be mutually independent sequences of i.i.d. random variables with marginal distributions I ρ o´ie 1 " Expp1´ρq and J ρ o´je 2 " Exppρq. The boundary variables in (3.5) and those in (3.12) are taken independent of each other. Put p G ρ o, o " 0 and on the boundaries Then in the bulk for x " px 1 , .
from the north and east boundaries of o`Z 2 ă0 . Precisely, (3.15) Similar to (3.10), one can backtrack the geodesic π o,p in the box rp, o´e 1´e2 s in the following way; for each edge px, yq (where y ď x) in rp, o´e 1´e2 s assign the weight Since we see that (3.17) can be written as . The following is a construction we shall refer to often. For general weights tY x u xPZ 2 on the lattice and a point u P Z 2 , let G u,x be the LPP by Now let v P Z 2 be such that u ď v. One can construct a new LPP on Z 2 ąv as follows. Define the south-west boundary weight Let tG rus v,x u xPZąv be the LPP defined through relations (3.6)-(3.7) using the boundary conditions (3.21) and the bluk weights tY x u xPZąv . We call G rus the induced LPP at v by G u,x . The superscript rus indicates that G rus uses boundary weights determined by the process G u,‚ with base point u.
rus v,y . The restriction of any geodesic of G u,y to v`Z 2 ě0 is part of a geodesic of G rus v,y . The edges with one endpoint in v`Z 2 ą0 that belong to a geodesic of G rus v,y extend to a geodesic of G u,y .
In case the process inherited is associated to a stationary process G ρ we shall use the notation G ρ,rus to indicate the density ρ as well. Bpω, x`z, y`zq " Bpτ z ω, x, yq (stationarity) (4.1) Bpω, x, yq`Bpω, y, zq " Bpω, x, zq (additivity) (4.2) Given a down-right path Y P DR, the lattice decomposes into a disjoint union Z 2 " G´Y Y Y Gẁ here the two regions are G´" tx P Z 2 : Dj P Z ą0 such that x`jpe 1`e2 q P Yu and G`" tx P Z 2 : Dj P Z ą0 such that x´jpe 1`e2 q P Yu.
Definition 4.2. Let 0 ă α ă 1. Let us say that a process is an exponential-α last-passage percolation system if the following properties (a)-(b) hold.
(a) The process is stationary with marginal distributions For any down-right path Y " py k q kPZ in Z 2 , the random variables (4.5) tq η z : z P G´u, ttpty k´1 , y k uq : k P Zu, and tη x : x P G`u are all mutually independent, where the undirected edge variables tpeq are defined as The following equations are in force at all x P Z 2 : The following Theorem was proven in [24] Theorem 4.3. For each 0 ă α ă 1 there exist a stationary cocycle B α and a family of random weights tX α x u xPZ 2 on pΩ, S, Pq with the following properties.
(i) For each 0 ă α ă 1, process is an exponential-α last-passage system as described in Definition 4.2.
(ii) There exists a single event Ω 2 of full probability such that for all ω P Ω 2 , all x P Z 2 and all λ ă ρ in p0, 1q we have the inequalities x`e 2 pωq. Furthermore, for all ω P Ω 2 and x, y P Z 2 , the function λ Þ Ñ B λ x,y pωq is right-continuous with left limits. of full probability such that the following holds: for each ω P Ω pαq 2 and any sequence v n P Z 2 such that |v n | 1 Ñ 8 and The LPP process G x,y is now defined by (2.1). Furthermore, for all ω P Ω pαq 2 and x, y P Z 2 , To each direction ξ P ri U we associate a density ρ P p0, 1q through the relations ξpρq "ˆp 1´ρq 2 p1´ρq 2`ρ2 , ρ 2 p1´ρq 2`ρ2( 4. 16) In some literature ξpρq is called the characteristic direction associated with the parameter ρ. It is known that for ξ P ri U, with probability one, every point x P Z 2 has a unique geodesic π x,8 of direction ξ. Busemann functions can be used to construct infinite geodesic. Consider the family of random variables defined in (4.10). Let ξ :" ξpαq be the characteristic direction associated with ρ. Let us denote by π x,ξ8 the infinite geodesic with respect to the weights tω x u xPZ , starting from x in direction ξ. In [11], it was shown that one can trace the infinite geodesic π x,8 by following the gradient of the Busemann function B α . Let tp i u iPZ ą0 be an enumeration of the vertices in π x,ξ8 , i.e.
where p 0 " x. Then the vertices tp i u iPZ ě0 are given recursively through Note that in (4.18) p i is attained by taking an up\right step from the point p i´1 in the direction where the minimal increment of the Busemann function is attained.
Since ω x " B α x,x`e 1^B α x,x`e 2 , π x,ξ8 is the unique path that satisfies Lemma 4.4. Let x P Z 2 and ξ 1 , ξ 2 P ri U such that ξ 1 ĺξ 2 . For i P t1, 2u let π x,ξ i 8 be the infinite geodesic starting from x in direction ξ i . Then Proof. Suppose (4.21) does not hold. Then there exists y 1 P π x,ξ 1 8 and y 2 P π x,ξ 2 8 such that x ď y 1 , y 2 and y 2 ăy 1 . Since both geodesics have a direction and ξ 1 ĺξ 2 , there exists w 1 P π x,ξ 1 8 and w 2 P π x,ξ 2 8 such that y 1 , y 2 ď w 1 , w 2 and w 1 ĺw 2 . It follows that there exists a point x ď z such that z P π x,ξ 1 8 X π x,ξ 2 8 and that the geodesics start from x and terminate at z. As y 2 ăy 1 it follows that γ 1 ‰ γ 2 which violates the uniqueness of geodesics.

4.2.
Coupling Busemann functions. In [7] a coupling between Busemann functions of different densities was given which relies on the queueing mapping. Consider the queueing mapping D : R Z Ñ R Z from Appendix A. x,x`e 2 ě Bρ x,x`e 2 (4.24) ą0 . For every k, l P Z the following sets of random variables are independent tBρ x`ie 2 ,x`pi`1qe 2 u 0ďiďl´1 , tB ρ x`ie 2 ,x`pi`1qe 2 u´k ďiď´1 and so are (4.25) tB ρ x`ie 1 ,x`pi`1qe 1 u 0ďiďl´1 , tBρ x`ie 1 ,x`pi`1qe 1 u´k ďiď´1 .

Local stationarity and stabilization
In this section we prove Theorem 2.1 and Theorem 2.2. Let us define the event H ξ,M " t p G N ξ,y´p G N ξ,x " B ρpξq x,y for all px, yq P EpR ξ,M qu. H ξ,M is the event where the increments of G along the edges in EpR ξ,M q coincide with those of the Busemann function associated with the direction ξ. It will be clear from the proof that H ξ,M and S ξ,M defined in (2.4) and are equivalent events. 5.1. Bounds on PpS ξ,M q and PpH ξ,M q. Let ξ P ri U and let ρpξq " ρpξq´rN´1 3 andρpξq " ρpξq`rN´1 3 . We also let p o " ξN`e 1`e2 . Assign weights on the edges of the north-east boundary of R ξ,N by Use the boundary weights tIρ i u 0ďiďN ξ 1 , tJρ i u 0ďiďN ξ 2 and the bulk weights tω x u xPR ξ,N to construct the stationary LPP p Gρ as in (3.13)-(3.14). Similarly we construct p G ρ . As in (3.15) we let p Zρ Similarly we define tI ρ i u 0ďiďM ξ 1 and tJ ρ i u 0ďiďM ξ 2 . Define the event is determined uniquely by the bulk weights tωu xPR ξ,M and the boundary weights tI ρ i u 0ďiďM ξ 1 and tJ ρ i u 0ďiďM ξ 2 constructed as in (5.8). This follows from the recursive relation ((4.8)-(4.9)) B ρ x,x`e 1 " ω x`p B ρ x`e 2 ,x`e 1`e2´B ρ x`e 1 ,x`e 1`e2 qB ρ x,x`e 2 " ω x`p B ρ x`e 2 ,x`e 1`e2´B ρ x`e 1 ,x`e 1`e2 q´.
On the event C ξ,M the boundary conditions in (5.8) are equal for the processes p Gρ and p G ρ i.e. which, together with (5.14), implies (5.10). The geodesics P ξ,M pπ x,ξ8 q and P ξ,M pπ x,ξ8 q are determined by tBρ e u ePEpR ξ,M q and tB ρ e u ePEpR ξ,M q respectively using (4.18). (5.11) is now implied by (5.10).

Upper bound on Ppp p
A ξ,M q c q.
Proof. We only prove (5.21) as (5.20) is similar. Given ξ P ri U, abbreviate ρ " ρpξq andρ "ρpξq. Let x 0 " pM ξ 1 , 0q be the lower-right corner of R ξ,M . By the order on geodesics we have In order to upper bound (5.22) we must show that the characteristic line of direction ξpρq that leaves from o goes, on the scale of N 2 3 , well below the point x 0 . We have, via (4.16), For large enough N and r ď N 1 3 plogpN qq´1 plug into (5.24) to obtain such that for N large enough For r ą cξ 1 ρ 2 rξ 2 p1´ρq`ξ 1 ρ´p1´ρq 2 s _ 1, the right hand side of (5.25) is smaller than´N 2 3 . This in turn implies that there exists a constant C 1 pξ, cq ą 0 (locally bounded in c) such that It then follows by [23][Corollary 5.10] that there exists a constant C 1 pξ, cq ą 0 which proves the result.
Corollary 5.6. Fix ξ P p0, 1q and M ą 0 such that M ď ct 2 3 for some constant c ą 0. There exists Cpc, ξq ą 0, locally bounded in c, such that Proof. By the definition of p A ξ,M we see that Taking probability on both sides of (5.27) and using (5.20) and (5.21) we obtain the result.    where d " Dpa, sq(see Appendix A), and a " pa j q jPZ and s " ps j q jPZ are two independent i.i.d sequences of exponential random variables of intensity ρ andρ respectively, such that 0 ă ρ ăρ ă 1.
Using (A.9) It follows that Let us now try to explain the idea behind the proof. Let x j " s j´1´aj , from (A.6) we see that
The dynamics behind (5.33) is as follows. The waiting time w j increases when the service times are longer then usual and the interarrival times are shorter i.e. when the random walk S 0,j goes up. Similarly, the w j decreases when the service times are fast compared to the arrival of customers i.e. S 0,j goes down. This dynamics hold until the random walk goes below´w 0 where the waiting time at the queue vanishes. The r.v. ř ξ 1 M i"1 e i can be thought of as the local time of the queue at zero, i.e. the accumulated time of the queue being empty. The main idea behind the proof of Proposition 5.7 is the observation that whenρ´ρ " N´1 3 , that is when the queue is in the so-called heavy traffic regime, at stationarity, the waiting time w 0 of customer 0, is of order N 1 3 . As the difference between the average service time rate and the average inter-arrival time rate is of orderρ´ρ " N´1 3 , the simple random walk S 0,j has drift´N´1 3 . (5.33) implies that the queue's waiting time vanishes by time of order N 2 3 . Over time t " opN 2 3 q the random walk S t will not change the waiting time at the queue by much so that with high probability w t will be of order N 1 3 and the r.v. ř t i"1 e i will be zero (see Figure 5.1).
Lemma 5.8. Let ξ P ri U and let M ą 0. For 0 ă β ă α ă 1   The two cases of a queue at stationarity. S t is the random walk whose incremental step is x t " s t´1´at . As the rate of service at the queue is higher than the rate of interarrival Epx t q ă 0 and so S t is a simple random walk with a negative drift. The waiting time at the queue decreases by S t until it vanishes.
Proof. By (A.11) Next we bound from above the probability that the infimum of the path of tS 1,i x u 1ďiďξ 1 M drops too low. Let C ą 0, then As´S 1,i x is a submartingale and φ θ pxq " e θx is a strictly increasing convex function for θ ą 0 φ θ p´S 1,i x q is again a submartingale. By Doob's inequality Note that by the independence of a " pa j q jPZ and s " ps j q jPZ , for´α ă θ ă β Note that by the definition of w 0 ((A.5)), w 0 is independent of tS 1,i x u iPZ ą0 and so Lemma 5.9. Let ξ P ri U and let M ą 0. For ρpξq " ρpξq´rN´1 3 ,ρpξq " ρpξq`rN´1 3 and 0 ă θ ăρ, Proof. Set β " ρ and α "ρ so that Lemma 5.10. Let ξ P ri U and let M ą 0 such that M ď cN Note that by our assumption on M the numerator in (5.50) is dominated by 2cN 2 3 _ pξ 1 M q´1 and where Cpcq ą 0 is locally bounded in c. In particular, there exists C B 2 pρq ą 0 such that

Coalescence of geodesics
In this section we prove Theorem 2.5 and Theorem 2.6. For technical reasons, namely the direction in which we send v n to infinity in (4.12), we prove the results for a setup that is a bit different, yet equivalent, to the one in Figure 2.2 (see Figure 6.1).

Upper bound on Pp|o´p c | ď αN q. Let p
Gρ and p G ρ be the stationary LPP withρ " ρ`rN´1 3 and ρ " ρ´rN´1 3 constructed through (3.13)-(3.14) with the boundary weights on the north-east boundaries of R ξN as in (5.8) and the bulk weights tω x u xPR ξN . Similarly to (5.2) define Similarly to Corollary 5.6 we have Lemma 6.1. Fix ξ P ri U and a ą 0. There exist Cpξ, aq ą 0, locally bounded in a, and N 0 pξ, rq ą 0 such that Ppp p A r q c q ď Cr´3. Let u " pu 1 , u 2 q " ξN´aN 2 3 e 2 , and let p G ρ,rq 1 s u,x be the LPP induced by p G ρ q 1 ,x at u. By Lemma 3.1 we see that where c 1 " p1´ρq 2 2rξ 1 ρ`ξ 2 p1´ρqs . (6.4) It follows that there exists N 0 pξ, rq such that for N ą N 0 where c " p1´ρq 2 4rξ 1 ρ`ξ 2 p1´ρqs . (6.5) It then follows by [23][Corollary 5.10] that there exists a constant C 1 pξq ą 0 such that P`p Z rq 1 s u,x ą 0˘ď C 1 pr´caq´3, the proof is now complete.
Let 0 ă α ă 1 and o α " αξN . We define R α " ro, o α s to be the rectangle whose left bottom corner is o and whose upper right corner is o α . We shall need the following result. Proof. We prove (6.6) as (6.7) is similar. In fact we only prove here the upper bound for P`p Zρ oα,o ą tN 2 3˘a s the bound on P`p Zρ oα,o ă tN 2 3˘i s similar. Let p Gρ ,roαs u,x be the LPP induced by p Gρ oα,x at u where u " αξN´A 1 tN 2 3 e 1 , and where p Z roαs u,x and p Zρ oα,x are the exit points of p Gρ ,roαs u,x and p Gρ oα,x respectively. We would like to show that the characteristic ξpρq emanating from the point u " pu 1 , u 2 q goes well above the point o on the scale of N 2 3 . Computè By (6.8), for t ě rά u 2´ρ 2 p1´ρq 2 u 1¯´o2 ě tN 2 3`2`ξ 1 ρ`ξ 2 p1´ρq˘`α r 2 t N´1 3 pξ 2´ξ1 qp 1´ρq 2 .
Setting C " C 1 _ C 2 implies the result.

Define the sets
In words, B α is the north-east boundary of R α , B α,t c are all the points in B α whose l 1 distance from o α is less or equal to tN 2 3 while B α,t f are the set of points in B α whose l 1 distance from o α is larger or equal to tN 2 3 . Letπ q 1 ,o and π q 1 ,o be the geodesics that start from q 1 and terminate at o, associated to p Gρ and p G ρ respectively. Define The superscript r in B r,α,t appears implicitly inρ, ρ. The following result shows that with high probability the geodesicsπ q 1 ,o and π q 1 ,o will not wonder too far from the point o α . Corollary 6.3. Fix ξ P ri U, 0 ă α ă 1 and r ą 0. There exists Cpξq ą 0 such that for t ą αr and N ą N 0 pξ, rq P`pB r,α,t q c˘ď Cα 2 t´3.
Taking probabilities on both sides of (6.14) and (6.15), using Lemma 6.2 and union bound we obtain (6.13).
Lemma 6.4. For every ξ P ri U and 0 ă α ă 1, there exists Cpξq ą 0 so that for every r ě 1 and t ď r´2 there exists N 0 prq ą 0 such that for N ě N 0 P´pD r,α,t q c¯ď Ct  Sending N to 8, the right hand site of (6.18) converges to e ρ´2`2rt 1 2`1˘´1`2 t´1 2 r´1¯´1 ď e ρ´2`2rt Plugging (6.19) in (6.17), by our assumption on t, rt 1 2 ď 1, and so we see that there exists C 2 pξq ą 0 such that for every r ě 1, there exists N 0 prq ą 0 such that for N ě N 0 P´pD r,α,t 2 q c¯ď C 2 t 1 2 r. Similar bound can be obtained for D r,α,t 2 the result then follows by union bound.
Proof of Theorem 2.5. We first claim that on the event p A r X B r,α,t X D r,α,t the geodesics π q 1 ,o and π q 2 ,o must coalesce outside R α (see Figure 6.1). On the event p A r This means that coalescence of the geodesicsπ q 1 ,o and π q 1 ,o outside R α implies the coalescence of the geodesics π q 1 ,o and π q 2 ,o outside R α . It is therefore enough to show that on the set B r,α,t X D r,α,t P ξ,αN pπ q 1 ,o q " P ξ,αN pπ q 1 ,o q. (6.21) On the event B r,α,t the geodesicsπ q 1 ,o and π q 1 ,o do not cross B α,t f and therefore use only the weights Bρ e , B ρ e where e P EpB α,t c q and the bulk weights tω x u xPRα . It follows that on B r,α,t X D r,α,t (6.21) holds. Set r " α´2 27 , t " α 16 27 so that t " α 16 27 ě α 25 27 " αr holds (since 0 ă α ă 1). Use (6.1), (6.13) and (6.16)  The result now follows.

6.2.
Upper bound on Pp|q 2´p c | ď αN q. For every ξ P ri U and m P Z, define the set C ξm ,t " tmuˆty P Z : m ξ 2{ξ 1´y ď tN Let ξ 1 " ξ and ξ 2 " ξ´p0, aN´1 3 q be two vectors whose direction is that of the characteristics emanating from o associated with the point q 1 and q 2 respectively. Letρ " ρpξq`rN´1 3 , ρ "  .7). For x P o`Z 2 ą0 , letπ o,x and π o,x be the geodesics associated with the last passage time Gρ o,x and G ρ o,x respectively. We shall need the following auxiliary result.

It follows that
P´p c P r`p1´αqξ 1 N, 0˘, ξN s¯ď 2Cα 2 . (6.31) (6.31) implies the result. and the summation operator Summing e j we obtain the cumulative idle time [26][Chapter 9.2, Eq. 2.7] as in the following lemma. To see that the last equality holds, note that It follows that w l "´sup 1ďiďl S i,l¯`_`S1,l`w 0˘(
Summing on both sides of (A.14) and using (A.13) The next lemma is a deterministic property of the mappings.
Lemma A.2. The identity D`Dpb, aq, s˘" D`Dpb, Rpa, sqq, Dpa, sq˘holds whenever the sequences a, b, s are such that the operations are well-defined.
Proof. For the computation choose pA j q and pB j q so that A j´Aj´1 " a j and B j´Bj´1 " b j . Then the output of Dpb, aq is the increment sequence of .
Next, the output of DpDpb, aq, sq is the increment sequence of This can be proved with a case-by-case analysis. See Lemma 4.3 in [7].
Specialize to stationary M/M/1 queues. Let σ be a service rate and α 1 , α 2 arrival rates. Assume σ ą α 1 ą α 2 ą 0. Let b 1 , b 2 , s be mutually independent i.i.d. sequences with marginals b k j " Exppα k q for k P t1, 2u and s j " Exppσq. Define a jointly distributed pair of arrival sequences by pa 1 , a 2 q "`b 1 , Dpb 2 , b 1 q˘. From these and services s, define jointly distributed output variables: d k " Dpa k , sq, t k " Spa k , sq, and q s k " Rpa k , sq for k P t1, 2u.
Lemma A.3. We have the following properties.
(i) Marginally a 2 is a sequence of i.i.d. Exppα 2 q variables.
(ii) For a fixed k P t1, 2u and each m P Z, the random variables td k j u jďm , t k m , and tq s k j u jďm are mutually independent with marginal distributions d k j " Exppα k q, t k m " Exppσ´α k q, and q s k j " Exppσq.
(iii) For a fixed k P t1, 2u, sequences d k and q s k are mutually independent sequences of i.i.d. random variables with marginal distributions d k j " Exppα k q and q s k j " Exppσq. (iv) pd 1 , d 2 q d " pa 1 , a 2 q, in other words, we have found a distributional fixed point for this joint queueing operator.
(v) For any m P Z, the random variables ta 2 i u iďm and ta 1 j u jěm`1 are mutually independent.
Proof. Parts (i)-(iii) are well-known M/M/1 queueing theory. Proofs can be found for example in Lemma B.2 in Appendix B of [7]. For part (iv), the marginal distributions of d 1 and d 2 are the correct ones by Lemma A.3(iii). To establish the correct joint distribution, the definition of pa 1 , a 2 q points us to find an i.i.d. Exppα 2 q random sequence z that is independent of d 1 and satisfies d 2 " Dpz, d 1 q. From the definitions and Lemma A.2, d 2 " Dpa 2 , sq " D`Dpb 2 , a 1 q, s˘" D`Dpb 2 , Rpa 1 , sqq, Dpa 1 , sq˘" D`Dpb 2 , q s 1 q, d 1˘.
By assumption b 2 , a 1 , s are independent. Hence by Lemma A.3(iii) b 2 , q s 1 , d 1 are independent. So we take z " Dpb 2 , q s 1 q which is an i.i.d. Exppα 2 q sequence by Lemma A.3(iii). This proves part (iv).
We know that marginally a 1 and a 2 are i.i.d. sequences. In queueing language observation (v) becomes obvious. Namely, since a 2 " Dpb 2 , a 1 q, the statement is that past inter-departure times ta 2 i u iďm are independent of future inter-arrival times ta 1 j u jěm`1 . Rigorously, (A.2) and (A.3) show that variables ta 2 i u iďm are functions of ptb 2 i u iďm , ta 1 i u iďm q which are independent of ta 1 j u jěm`1 .
Proof. The proofs of all parts are similar. We prove the second inequality in (B.1), that is, The geodesics π o, x`e 2 and π o`e 2 , x must cross. Let u be the first point where they meet. Note that This inequality can be proved also from Lemma B.1, by writing G o`e 2 , x`e 2´G o`e 2 , x " r G o, x`e 2ŕ G o, x with environment r ω o`y " ω o`y when y 2 ą 0 and r ω o`ie 1 "´M for large enough M .
Fix base points u ď v on Z 2 . On the quadrant v`Z 2 ě0 , put a corner weight η v " 0 and define boundary weights (B.5) η v`ke i " G u, v`ke i´Gu, v`pk´1qe i for k P Z ą0 and i P t1, 2u.
In the bulk use η x " ω x for x P v`Z 2 ą0 . Denote the LPP process in v`Z 2 ě0 that uses weights Assume now that the weights are such that geodesics are unique. Define the exit point Z u, p as in (3.8). For k ě 1 let Z rus u`ke 1 , p be the exit point of the geodesic of G rus u`ke 1 , p . The lemma below follows from taking v " u`ke 1 in Lemma 3.1.