Coalescence estimates for the corner growth model with exponential weights

We establish estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model with i.i.d. exponential weights. There are four estimates: upper and lower bounds for both fast and slow coalescence on the correct scale with exponent $3/2$. The lower bound for fast coalescence is new and has optimal exponential order of magnitude. For the other three we provide proofs that do not rely on integrable probability or on the connection with the totally asymmetric simple exclusion process, in order to provide a template for extension to other models. We utilize a geodesic duality introduced by Pimentel and properties of the increment-stationary last-passage percolation process.


Introduction
Random growth models of the first-and last-passage type have been a central part of the mathematical theory of spatial stochastic processes since the seminal work of Eden [11] and Hammersley and Welsh [14]. In these models growth proceeds along optimal paths called geodesics, determined by a random environment. The interesting and challenging objects of study are the directed semiinfinite geodesics. These pose an immediate existence question because they are asymptotic objects and hence cannot be defined locally in a simple manner. Once the existence question is resolved, questions concerning their multiplicity and geometric behavior such as coalescence arise.
Techniques for establishing the existence, uniqueness, and coalescence of semi-infinite geodesics were first introduced by Newman and co-authors in the 1990s [15,16,18,19] in the context of planar undirected first-passage percolation (FPP) with i.i.d. weights. These methods were subsequently applied to the exactly solvable planar directed last-passage percolation (LPP) model with i.i.d. exponential weights by Ferrari and Pimentel [13] and Coupier [10]. This model is also known as the exponential corner growth model (CGM).
A key technical point here is that the strict curvature hypotheses of Newman's work can be verified in the exactly solvable LPP model. A second key feature is that the exponential LPP model can be coupled with the totally asymmetric simple exclusion process (TASEP). This connection provides another suite of powerful tools for analyzing exponential LPP.
The work of [13] and [10] established for the exponential LPP model that, almost surely for a fixed direction, directed semi-infinite geodesics from each lattice point are unique and they coalesce. An alternative approach to these results was recently developed by one of the authors [24], by utilizing properties of the increment-stationary LPP process.
Once coalescence is known, attention turns to quantifying it: how fast do semi-infinite geodesics started from two distinct points coalesce? The scaling properties of planar models in the Kardar-Parisi-Zhang (KPZ) class come into the picture here. This class consists of interacting particle systems, random growth models and directed polymer models in two dimensions (one of which can be time) that share universal fluctuation exponents and limit distributions from random matrix theory. For surveys of the field, see [9,21].
It is expected that, subject to mild moment assumptions on the weights, planar FPP and LPP are members of the KPZ class. It is conjectured in general and proved in exactly solvable cases that a geodesic of length N fluctuates on the scale N 2/3 . Thus if two semi-infinite geodesics start at distance k apart, we expect coalescence to happen on the scale k 3/2 .
The first step in this direction was taken by Pimentel [20], again in the context of the exponential LPP model. By relying on the TASEP connection, he proved that in a fixed direction, the so-called dual geodesic graph is equal in distribution (modulo a lattice reflection) to the original geodesic tree. Next, by appeal to fluctuation bounds derived by coupling techniques in [3], he derived an asymptotic lower bound on the coalescence time, of the expected order of magnitude. The next step taken by Basu, Sarkar, and Sly [6] utilized the considerably more powerful estimates from integrable probability. For the large tail of the coalescence time they established not only the correct order of magnitude k 3/2 but also upper and lower probability bounds of matching orders of magnitude. In the same paper the original estimate of Pimentel was also improved significantly.
Our goal in taking up the speed of coalescence is the development of proof techniques that rely on the stationary version of the model and avoid both the TASEP connection and integrable probability. The applicability of this approach covers all 1+1 dimensional KPZ models with a tractable stationary version. This includes not only various last-passage models in both discrete and continuous space, but also the four currently known solvable positive temperature polymer models [8].
Extension beyond solvable models may also be possible, as indicated by the exact KPZ fluctuation exponents derived in [4] for a class of zero-range processes outside currently known exactly solvable models. This is work left for the future. Another somewhat philosophical point is that capturing exponents should be possible without integrable probability. This has been demonstrated for fluctuation exponents by [3] for the exponential LPP and by [22] for a positive-temperature directed polymer model.
The results of this paper come from a unified approach based on controlling the exit point of the geodesic in a stationary LPP process and on Pimentel's duality of geodesics and dual geodesics. This involves coupling, random walk estimates, planar monotonicity, and distributional properties of the stationary LPP process. Here are the precise contributions of the present paper (details in Section 2.2): (i) The two bounds of Basu et al. [6] without integrable probability (Theorem 2.2), but with an upper bound short of the optimal order.
(iii) A new lower bound on fast coalescence with optimal exponential order (Theorem 2.3 lower bound).
(iv) A new quantified lower bound on the transversal fluctuations of a directed semi-infinite geodesic without integrable probability (Theorem 2.8).
(v) Strengthened exit time estimates for the stationary LPP process without integrable probability, some uniform over endpoints beyond a given distance (Theorems 4.1-4.4).
We mention two more general but related points about the exponential CGM.
(a) When all directions are considered simultaneously, the overall picture of semi-infinite geodesics is richer than the simple almost-sure-uniqueness-plus-coalescence valid for a fixed direction. Part of this was already explained by Coupier [10]. Recently the global picture of uniqueness and coalescence was captured in [17]. Coalescence bounds that go beyond the almost surely unique geodesics in a fixed direction are left as an open problem for the future.
(b) Various geometric features of the exponential LPP process can now be proved without appeal to properties of TASEP. An exception is a deep result of Coupier [10] on the absence of triple geodesics in any random direction. This fact currently has no proof except the original one that relies on the TASEP speed process introduced in [1].

Organization of the paper
Precise definition of the exponential LPP model and the main results appear in Section 2. Section 3 collects known facts about the CGM used in the proofs. This includes properties of the stationary growth process and the construction of the directed semi-infinite geodesics in terms of Busemann functions. Section 4 derives new exit time estimates for the geodesic of the stationary growth process, stated as Theorems 4.1 through 4.4. In the final Section 5 the exit time estimates and duality are combined to prove the main results of Section 2. The appendix contains a random walk estimate and a moment bound on the Radon-Nikodym derivative between two product-form exponential distributions.

Notation and conventions
Points x = (x 1 , x 2 ), y = (y 1 , y 2 ) ∈ R 2 are ordered coordinatewise: x ≤ y iff x 1 ≤ y 1 and x 2 ≤ y 2 . The ℓ 1 norm is |x| 1 = |x 1 |+|x 2 |. The origin of R 2 is denoted by both 0 and (0, 0). The two standard basis vectors are e 1 = (1, 0) and e 2 = (0, 1). For a ≤ b in Z 2 , a, b = {x ∈ Z 2 : a ≤ x ≤ b} is the rectangle in Z 2 with corners a and b. a, b is a segment if a and b are on the same horizontal or vertical line. We use a − e 1 , a , a − e 2 , a to denote unit edges when it is clear from the context. Subscripts indicate restricted subsets of the reals and integers: for example Z >0 = {1, 2, 3, . . . } and Z 2 >0 = (Z >0 ) 2 is the positive first quadrant of the planar integer lattice. For 0 < α < ∞, X ∼ Exp(α) means that the random variable X has exponential distribution with rate α, in other words P (X > t) = e −αt for t > 0 and E(X) = α −1 .

The corner growth model and semi-infinite geodesics
The standard exponential corner growth model (CGM) is defined on the planar integer lattice Z 2 through independent and identically distributed (i.i.d.) weights {ω z } z∈Z 2 , indexed by the vertices of Z 2 , with marginal distribution ω z ∼ Exp(1). The last-passage value G x,y between two coordinatewise-ordered vertices x ≤ y of Z 2 is the maximal total weight of an up-right nearestneighbor path from x to y: that satisfy z 0 = x, z |y−x| 1 = y, and z k+1 − z k ∈ {e 1 , e 2 }. The almost surely unique maximizing path is the point-to-point geodesic. G x,y is also called (directed) last-passage percolation (LPP). If x ≤ y fails our convention is G x,y = −∞.
A semi-infinite up-right path (z i ) ∞ i=0 is a semi-infinite geodesic if it is the maximizing path between any two points on this path, that is, In the exponential CGM it is natural to index spatial directions ξ by a real parameter ρ ∈ (0, 1) through the equation We call ξ[ρ] the characteristic direction associated to parameter ρ. This notion acquires meaning when we discuss the stationary LPP process in Section 3. Throughout, N will be a scaling parameter that goes to infinity. When ρ is understood, we write for the lattice point moving in direction ξ[ρ]. The theorem below summarizes the key facts about directed semi-infinite geodesics that set the stage for our paper. It goes back to the work of Ferrari and Pimentel [13] and Coupier [10] on the CGM, and the general geodesic techniques introduced by Newman and coworkers [15,16,18,19]. A different proof is given in [24]. Theorem 2.1. Fix ρ ∈ (0, 1). Then the following holds almost surely. For each For each pair x, y ∈ Z 2 , the geodesics coalesce: there is a coalescence point z ρ (x, y) such that b ρ,x ∩ b ρ,y = b ρ,z for z = z ρ (x, y).

Coalescence estimates for semi-infinite geodesics in a fixed direction
The two main results below give upper and lower bounds on the probability that two ξ[ρ]-directed semi-infinite geodesics initially separated by a distance of order Theorem 2.2. For each 0 < ρ < 1 there exist finite positive constants δ 0 , C 1 , C 2 and N 0 that depend only on ρ and for which the following holds: for all N ≥ N 0 and N −2/3 ≤ δ ≤ δ 0 , The requirement δ ≥ N −2/3 in Theorem 2.2 is needed only for the lower bound and only to ensure that ⌊δN 2/3 ⌋ = 0. Theorem 2.3. For each 0 < ρ < 1 there exist finite positive constants r 0 , C 1 , C 2 and N 0 that depend only on ρ and for which the following holds: for all N ≥ N 0 and r 0 ≤ r ≤ ((1 − ρ) 2 ∧ ρ 2 )N 1/3 , The requirement r ≤ ((1 − ρ) 2 ∧ ρ 2 )N 1/3 in Theorem 2.3 is needed only for the lower bound and only to ensure that both geodesics start inside the rectangle 0, v N .
If we replace one of the starting points with the origin 0, the upper bound of Theorem 2.2 and the lower bound of Theorem 2.3 hold automatically because b ρ,0 stays between b ρ,(⌊rN 2/3 ⌋,0) and b ρ,(0,⌊rN 2/3 ⌋) . The following corollary states that the other two tail estimates also hold with possibly different constants under this minor alteration in the geometry.
For direct comparison with [6], we state two corollaries for geodesics whose locations are not expressed in terms of the large parameter N . Corollary 2.6. For each 0 < ρ < 1 there exist finite positive constants R 0 , C 1 and C 2 that depend only on ρ and for which the following holds: for all k ≥ 1 and R ≥ R 0 , . Given k ≥ 1 and R ≥ R 0 , let N = Rk ≥ N 0 and δ = R −2/3 ≤ δ 0 . Now k 2/3 = δN 2/3 . The next Corollary 2.7 below is derived from Theorem 2.3 in a similar way.
Corollary 2.7. For each 0 < ρ < 1 there exist finite positive constants R 1 , C 1 and C 2 that depend only on ρ and for which the following holds: for all k ≥ 1 and R > 0 that satisfy Again, the lower bound R ≥ ((1−ρ) 2 ∧ρ 2 ) −1 k −1/3 is imposed only to ensure that both geodesics start inside the rectangle 0, v Rk , for otherwise the probability in Corollary 2.7 is zero.
The lower bounds in Theorem 2.2 and Corollary 2.6 are optimal, but the upper bounds are not. Optimal upper and lower bounds (both of order R −2/3 ) were proved for Corollary 2.6 by Basu, Sarkar, and Sly [6] with inputs from integrable probability. Thus in Theorem 2.2 and Corollary 2.6 our contribution is to provide bounds without relying on integrable probability.
In Theorem 2.3 the upper bound Cr −3 was proved by Pimentel [20] in the asymptotic sense, as N → ∞. This was strengthened to e −Cr 3/2 and without sending N to infinity in [6] with inputs from integrable probability, see [6,Remark 6.5]. (The parameter R in Remark 6.5 of [6] is the same as in our Corollary 2.7.) The expected optimal upper bound in Theorem 2.3 is e −Cr 3 . This is suggested by a combination of integrable probability, random matrix theory, and the duality approach. Remark 1.3 of [6] indicates that an optimal upper bound e −Cr 3 for transversal fluctuations of a point-to-point geodesic can be obtained through a tail estimate for the largest eigenvalue of the Laguerre Unitary Ensemble. This bound can be extended to semi-infinite geodesics as shown in Proposition 6.2 of [6]. By our Proposition 5.2, this replaces the bound Cr −3 in our Theorem 3.5 with e −Cr 3 . An application of duality, as in our proof of Theorem 2.3 in Section 5, converts this fluctuation bound into a bound on coalescence.
The lower bound e −C 1 r 3 in Theorem 2.3 is new and matches the expected optimal exponential order.
Among the results, the one obviously most in need of improvement is the upper bound of Theorem 2.3. As the reader sees below (5.5) in Section 5, after the application of duality this bound comes as a trivial weakening of the known exit time estimate Theorem 3.5.
It is by now well-known that over distances of order N , geodesics fluctuate on the scale N 2/3 . A by-product of our proof is the following lower bound on the size of the transversal fluctuation of a semi-infinite geodesic. It is an improvement over previous bounds obtained without integrable probability (see Theorem 5.3(b) in [23]).
Theorem 2.8. For each 0 < ρ < 1 there exist positive constants C, N 0 and δ 0 that depend only on ρ for which the following holds: for all N ≥ N 0 and 0 < δ ≤ δ 0 , The probability in (2.8) is essentially bounded above by the probability in (2.4). This is demonstrated through their proofs in Section 5. With inputs from integrable probability, the upper bound δ 3/8 in (2.8) can be improved to δ, the optimal upper bound for (2.4) obtained in [6].
We turn to develop the groundwork for the proofs. As in Pimentel [20], our proof takes advantage of the increment-stationary growth process and fluctuation bounds that go back to [3].

Preliminaries on the corner growth model
This section covers aspects of the CGM used in the proofs. We provide illustrations, some intuitive arguments, and references to precise proofs. The two main results are a fluctuation upper bound for the exit point of a stationary LPP process (Theorem 3.5) and the construction of semi-infinite geodesics with Busemann functions (Theorem 3.7). These are proved in the lecture notes [23] and article [24] without using anything beyond the stationary LPP process.
z,• uses boundary weights defined by the LPP process G x,• . Path x-a-y is the geodesic of G x,y and path z-a-y the geodesic of G (x) z,y . These geodesics share the segment a-y.

Nonrandom properties
We begin with two basic features of LPP that involve increments. We state them for our exponential case but in fact these properties do not need any probability. Let G x,• be defined by (2.1) and define increment variables for a ≥ x + e 1 and b ≥ x + e 2 by The first property is a monotonicity valid for planar LPP. Proof can be found for example in Lemma 4.6 of [23].
Fix distinct lattice points x ≤ z and define a second LPP process G (x) z,• with base point at z that uses boundary weights given by the increments of G x,• , as illustrated in Figure 3.1. Precisely, for y ≥ z, where the weights are given by Proof of the lemma below is elementary and can be found in Lemma A.1 of [23].
Lemma 3.2. Let x ≤ z and y ∈ z + Z 2 >0 . Then the unique geodesics of G x,y and G (x) z,y coincide in the quadrant z + Z 2 >0 .

Stationary last-passage percolation
The stationary LPP process G ρ is defined on a positive quadrant x + Z 2 ≥0 with a fixed base point x ∈ Z 2 . It is parametrized by ρ ∈ (0, 1). Start with mutually independent bulk weights {ω z : The probability distribution of these weights is denoted by P ρ . The LPP process G ρ x,• is defined on the boundary of the quadrant by G ρ x, In the bulk we perform LPP that uses both the boundary and the bulk weights: for The LPP value G a,b inside the braces is the standard one defined by (2.1) with the i.i.d. bulk weights ω. Call the almost surely unique maximizing path a ρ-geodesic. The exit time Z x → y is the Z \ {0}-valued random variable that records where the ρ-geodesic from x to y exits the boundary, relative to the base point x, with a sign that indicates choice between the axes: The definition above implies I x ke 1 = I ke 1 and J x le 2 = J le 2 for k, l ≥ 1. The term stationary LPP is justified by the next fact. Its proof is an induction argument and can be found for example in [23,Thm. 3.1].
(0, 0) is another stationary LPP process. Lemma 3.2 gives the statement below which will be used extensively in our proofs.
Since the boundary weights in (3.3) are stochastically larger than the bulk weights, the ρgeodesic prefers the boundaries. The characteristic direction ξ[ρ] = ((1 − ρ) 2 , ρ 2 ) defined earlier in (2.2) is the unique direction in which the attraction of the e 1 -and e 2 -axes balance each other out. A consequence of this is that the ρ-geodesic from x to x + v N spends order N 2/3 steps on the boundary. Here we encounter the 2/3 wandering exponent of KPZ universality. This is described in Theorems 3.5 and 4.1 below. The macroscopic picture is in Figure 3.3. This matter is discussed more thoroughly in Section 3.2 of [23].
Theorem 3.5. [23, Prop. 5.9] There exist positive constants N 0 , C that depend only on ρ such that for all r > 0, N ≥ N 0 , and |v − v N | 1 ≤ 10, In the next corollary the Θ(N 2/3 ) deviation is transferred from the basepoint 0 to the endpoint v N . Figure 3.4 illustrates how Lemma 3.4 reduces claim (3.9) to Theorem 3.5. (Corollary 3.6 is proved as Corollary 5.10 in the arXiv version of [23].) Corollary 3.6. There exist positive constants N 0 , C that depend only on ρ such that for N ≥ N 0 , and b > 0,

Busemann functions and semi-infinite geodesics
The key to our results is that the directed semi-infinite geodesics can be defined through Busemann functions, which themselves are instances of stationary LPP. Thus estimates proved for stationary LPP provide information about the behavior of directed semi-infinite geodesics. The next theorem summarizes the properties of Busemann functions needed. It is a combination of results from Section 4 of [23] and Lemma 4.1 of [24]. (ii) The unique ξ[ρ]-directed semi-infinite geodesic from x is defined by b ρ,x 0 = x and for k ≥ 0, Part (iii) above implies that for fixed x ∈ Z 2 , the process {B ρ x,y : y ≥ x} is exactly a stationary LPP process G ρ x,• as defined in (3.4), with boundary weights I x+ke 1 = B ρ x+(k−1)e 1 ,x+ke 1 and J x+le 2 = B ρ x+(l−1)e 2 ,x+le 2 and bulk weights ω z .

Exit time estimates
This section proves estimates on the exit time for stationary LPP processes defined in (3.4) and (3.5). These results are applied in Section 5 to prove the main theorems stated in Section 2. The key idea of these proofs is a perturbation of the parameter ρ of the stationary LPP process to another parameter λ = ρ + rN −1/3 . This allows us to control the exit point on the scale N 2/3 . This idea goes back to the seminal paper [7]. The first theorem below quantifies the lower bound on the exit point on the scale N 2/3 . This strengthens the estimates accessible without integrable probability, for previously no quantification was attained (Theorem 2.2(b) in [3]).
Theorem 4.1. For each 0 < ρ < 1, there exist positive constants q 0 , C, δ 0 and N 0 that depend only on ρ for which the following holds: for all q ∈ [0, q 0 ], N ≥ N 0 and 0 < δ < δ 0 , with Proof. We prove the case 1 ≤ Z 0 → w N ≤ δN 2/3 , the other case −δN 2/3 ≤ Z 0 → w N ≤ −1 being similar. First pick N 0 (ρ) large enough so that the two coordinates of w N are greater than 1. The probability in (4.1) is zero if δN 2/3 < 1. Thus we can always assume Also, it is enough to prove (4.1) for δ ∈ (0, δ 0 (ρ)] for any constant δ 0 (ρ) > 0 because, if necessary, we can increase the constant C(ρ) to δ 0 (ρ) −3/8 . Set r = δ −1/8 and introduce the perturbed parameter To guarantee that (4.4) λ ≤ ρ + (1 − ρ)/2 < 1, we must have N ≥ 2r 1−ρ 3 . The choice of ρ + (1 − ρ)/2 is only to bound λ by a constant less than one that depends only on ρ. This bound on N is automatically satisfied by (4.2) as long as δ −3/2 ≥ ( 2r 1−ρ ) 3 . With r = δ −1/8 , we can ensure bound (4.4) by considering δ > 0 subject to Fix a positive constant t 0 > δ 0 (ρ). (We can take t 0 = 1. It does not produce any additional dependencies in the constants of the theorem.) Then for all 0 < δ < δ 0 (ρ), The difference G (k,1),w N −G (l,1),w N above will be controlled by a random walk, through a coupling with another stationary LPP process whose boundary weights are on the north and east. For this, put independent λ-parametrized boundary variables on the north and east outer boundary of the rectangle 0, w N . Precisely, set w ′ N = w N + e 1 + e 2 . Then put i.i.d. Exp(1 − λ) weights on the vertices of the northern horizontal segment (0, w ′ N · e 2 ), w ′ N − e 1 and i.i.d. Exp(λ) weights on the vertices of the eastern vertical segment (w ′ N · e 1 , 0), w ′ N − e 2 . This is illustrated in Figure 4.1. For x ∈ (1, 1), w N + e 2 , G λ,N x,w N +e 2 denotes the last-passage time from x to w N + e 2 that uses the Exp(1 − λ) weights on the north boundary (superscript N for north). Similarly, passage time G λ,N E x,w ′ N uses boundary weights on both the north and east boundaries. G λ,N E x,w ′ N is the exact analogue of G ρ from (3.4) but with reversed axis directions. In particular, G λ,N E x,w ′ N has zero weight at the vertex w ′ N . The exit time Z N E, x → w ′ N records the distance from the vertex w ′ N to the point where the geodesic enters the north (as positive) and east (as negative) boundary. In particular, on the event In the derivation below, Lemma 3.1 gives the first inequality. The first equality below is valid on the event Z N E,(⌊t 0 N 2/3 ⌋,1) → w ′ N ≥ 1 which forces the geodesics of G λ,N E (k,1),w ′ N and G λ,N E (l,1),w ′ N to enter the north boundary.
The last quantity S N E,λ k+1,l above is the sum of i.i.d. Exp(1 − λ) increments of the LPP process G λ,N E Returning to (4.6), we have We bound separately the probabilities (4.7) and (4.8) by C(ρ)δ 3/8 .
• Estimating (4.7). This comes immediately from bound (3.8) of Corollary 3.6 applied to a stationary LPP process with parameter λ, when viewed in the right way. This is illustrated in Figure 4.2. As in the right diagram of Figure 4.2, M is chosen so that ⌊M λ 2 ⌋ = ⌊N ρ 2 ⌋. Ignoring the floor function, we bound R as follows.
By Lemma 1 on p. 417 of [12] (the notation used in this lemma is given in Theorem 4 on p. 416),  Theorem 4.2. For each 0 < ρ < 1 there exist finite positive constants δ 0 (ρ), C(ρ) and N 0 (ρ) such that for all 0 < δ ≤ δ 0 (ρ) and N ≥ N 0 (ρ), Proof. We prove this for 1 ≤ Z ≤ δN 2/3 . The proof for −δN 2/3 ≤ Z ≤ −1 is similar. It suffices to look at the north and east boundaries of 0, v N since any geodesic from 0 to outside of 0, v N crosses the boundary. Decompose these boundaries into two parts D and L as in Figure 4.3. Here r = δ −1/8 and q is chosen so that Theorem 4.1 is valid. First, we look at D. Following the same idea from previous proof, The right-hand side above is (4.6) from Theorem 4.1. This finishes the proof for D.
For ρ-geodesics that enter L we use monotonicity that comes from uniqueness of finite geodesics: The last inequality comes from bound (3.9) from Corollary 3.6. Theorem 4.3. For each 0 < ρ < 1 there exist finite positive constants δ 0 (ρ), C(ρ) and N 0 (ρ) such that for all N ≥ N 0 (ρ) and N −2/3 ≤ δ ≤ δ 0 (ρ), Proof. Utilizing Theorem 3.5, fix constants r 0 , C 0 and N 0 (depending on ρ) such that, for N ≥ N 0 , Set v ′ N = v N + e 1 + e 2 . Given small δ > N −2/3 , partition [−r 0 , r 0 ] as We cannot control the exact location of i ⋆ . We compensate by varying the endpoint around v ′ N . Let denote the set of lattice points on the boundary of the rectangle 0, v ′ N within distance r 0 N 2/3 of the upper right corner v ′ N . We claim that for any integer i ∈ [0, ⌊ 2r 0 δ ⌋],  We prove claim (4.19). If p i ≤ 0 ≤ p i+1 , (4.19) is immediate. We argue the case p i+1 > p i > 0, the other one being analogous. Set z = (⌊p i N 2/3 ⌋ − 1)e 1 and apply Lemma 3.4 to the LPP process G (0),ρ z, • . Then To prove this bound we tilt the probability measure to make the event likely and pay for this with a moment bound on the Radon-Nikodym derivative. This argument was introduced in [5] in the context of ASEP, and adapted to a lower bound proof of the longitudinal fluctuation exponent in the stationary LPP in Section 5.5 of the lectures [23]. Lemma 4.5 below is an auxiliary estimate for the proof of Theorem 4.4. It utilizes a perturbed parameter λ = ρ + rN −1/3 , assumed to satisfy (4.21) ρ < λ ≤ c(ρ) < 1 for some constant c(ρ) < 1, as r and N vary. Lemma 4.5 shows that, for small enough a > 0 and large enough b, r > 0, the λ-geodesic to a target point w N slightly perturbed from v N exits the e 1 -axis through the interval [[arN 2/3 e 1 , brN 2/3 e 1 ]] with high probability. This is illustrated in Figure 4.6. The constants 1 − ρ and 2/ρ 2 in the lemma come from the following observation: if u N is the lattice point closest to the ξ[λ]-directed ray such that u N · e 2 = v N · e 2 , then  There exist positive constants C, N 0 that depend only on ρ such that, for any r > 0 and N ≥ N 0 such that (4.21) holds, we have Before the proof of Lemma 4.5, we separate an observation about geodesics in the next lemma, illustrated by the left diagram of To prove (4.25), refer back to Figure 4.6. By (4.22), the distance between the origin and the black dot is bounded above by 1 10 brN 2/3 . So the distance between the black dot to brN 2/3 e 1 is at least brN 2/3 − 1 10 brN 2/3 = 9 10 brN 2/3 . Refer to Figure 4.6, applying Lemma 3.4, Theorem 3.5 to the LPP process G To prove (4.24), this is where Lemma 4.6 is used. As shown in Figure 4.7, define a new origin with integer coordinates (⌊arN 2/3 ⌋, −h) close to the stranght line going through w N and the white dot. Lemma 4.6 gives (4.27) Theorem 3.5 states that it is unlikely for the geodesic from (⌊arN 2/3 ⌋, −h) to w N to exit very late when going in the characteristic direction ξ[λ]. It suffices to show h is bounded below by some k(ρ)rN 2/3 .
For this lower bound, note the distance between the white dot and ⌊arN 2/3 ⌋e 1 is bounded below by 8arN 2/3 , and the slope of the line going through w N and white dot is λ 2 (1−λ) 2 . Thus, we have Since λ is bounded above and below by constants depend on ρ, we get (4.29) h ≥ k(ρ)rN 2/3 which finishes the proof.
On the e 1 -axis, define  On the e 2 -axis, we define Note that Lemma 4.5 continues to hold if a is decreased and b is increased. The constants a, b, N 0 can always be adjusted so that the situation in the left diagram of Figure 4.8 appears.
Denote the probability measure for the environment ω by P. The goal is the estimate where A denotes the event in braces. We check that this implies Theorem 4.4. The Cauchy-Schwartz inequality gives where f is the Radon-Nikodym derivative. Lemma A.2 gives the bound Note that the event A in (4.33) has the lower bound ≥ arN 2/3 . To replace this with ≥ rN 2/3 , as required for Theorem 4.4, modify the constant C.
To show (4.33) we bound its complement: We treat the case 1 ≤ Z 0 → z ≤ arN 2/3 of (4.36). The same arguments give the analogous bound for the case −arN 2/3 ≤ Z ≤ −1. Define w N = v N − ⌊ 1 10 (1 − ρ)rN 2/3 ⌋e 1 , and break up the northeast boundary of 0, v N into two regions L and D as in the diagram on the right of Figure  4.8.
First consider geodesics that hit D. Let σ 0 → x 1 denote the exit time of the optimal 0 → x path among those paths whose first step is e 1 . (4.37) The second inequality comes from the uniqueness of maximizing paths: the maximizing path to w N cannot go to the right of a maximizing path to D. The switch from P to P λ increases the boundary weights on the e 1 axis outside the interval ⌊arN 2/3 ⌋e 1 , ⌊brN 2/3 ⌋e 1 , hence the fourth inequality. The last inequality is from Lemma 4.5.

Dual geodesics and proofs of the main theorems
The main theorems from Section 2 are proved by applying the exit time bounds of Section 4 to dual geodesics that live on the dual lattice. First define south and west directed semi-infinite paths (superscript sw) in terms of the Busemann functions from Theorem 3.7: Recall the dual weights { ω ρ x = B ρ x−e 1 ,x ∧ B ρ x−e 2 ,x } x∈Z 2 introduced in part (iii) of Theorem 3.7. x x + e 1 + e 2 x + e * Let e * = 1 2 (e 1 + e 2 ) = ( 1 2 , 1 2 ) denote the shift between the lattice Z 2 and its dual Z 2 * = Z + e * . Shift the dual weights to the dual lattice by defining ω * z = ω ρ z+e * for z ∈ Z 2 * . By Theorem 3.7(iii) these weights are i.i.d. Exp (1). The LPP process for these weights is defined as in (2.1): Shift the southwest paths to the dual lattice by defining These definitions reproduce on the dual lattice the semi-infinite geodesic setting described in Section 3.3, with reflected lattice directions. This is captured in the next theorem that summarizes the development from Section 4.2 of [24].
Part (ii), the distributional equality of the tree of directed geodesics and the dual, was first proved in [20]. The non-crossing property of part (iii) can be seen from a simple picture. The additivity of the Busemann functions gives By (3.10) b ρ,x 1 = x+e 1 if and only if B ρ x,x+e 1 ≤ B ρ x,x+e 2 . By (5.3) this is equivalent to B ρ x+e 2 ,x+e 1 +e 2 ≤ B ρ x+e 1 ,x+e 1 +e 2 which is the same as b sw,ρ,x+e 1 +e 2 1 = x + e 2 , and this last is equivalent to b * ,ρ,x+e * k = x + e * − e 1 . An analogous argument works for the e 2 step. The conclusion is that the increments of b ρ,• out of x and b * ,ρ,• out of x + e * cannot cross. See The bulk weights are {ω * x : x ∈ Z * 2 , x ≥ e * }.
Proposition 5.2. For any w ∈ e * +Z 2 ≥0 the following holds. The edges of the semi-infinite geodesic b * ,ρ,w that have at least one endpoint in e * + Z 2 ≥0 are also edges of the geodesic of G * , ρ −e * ,w .
Proposition 5.2, illustrated in Figure 5.2, is another version of Lemma 3.2. It is proved as Prop. 5.1 in [24] but without the shift to the dual lattice, so in terms of the southwest geodesics in (5.1) for the weights ω ρ .
We are ready to prove the main results.
The bounds claimed in Theorem 2.2 follow from Theorems 4.2 and 4.3.

A Appendix
Below is the random walk estimate for the proof of Theorem 4.1. It is proved as Lemma C.1 in Appendix C of [2].
Next the moment bound on the Radon-Nikodym for the proof of Theorem 4.4.