Universality of the geodesic tree in last passage percolation
aa r X i v : . [ m a t h . P R ] A ug Universality of the geodesic tree in last passagepercolation
Ofer Busani ∗ Patrik L. Ferrari † August 19, 2020
Abstract
In this paper we consider the geodesic tree in exponential last passage perco-lation. We show that for a large class of initial conditions around the origin, theline-to-point geodesic that terminates in a cylinder of width o ( N / ) and length o ( N ) agrees in the cylinder, with the stationary geodesic sharing the same endpoint. In the case of the point-to-point model, we consider width δN / and lengthup to δ / N/ (log( δ − )) and provide lower and upper bound for the probabilitythat the geodesics agree in that cylinder. The last passage percolation model (LPP) is one of the most well studied models in theKardar-Parisi-Zhang (KPZ) universality class of stochastic growth models. In this model,to each site ( i, j ) ∈ Z , one associates an independent random variable ω i,j exponentiallydistributed with parameter one. In the simplest case, the point-to-point LPP model, fora given point ( m, n ) in the first quadrant, one defines the last passage time as G ( m, n ) = max π :(0 , → ( m,n ) X ( i,j ) ∈ π ω i,j , (1.1)where the maximum is taken over all up-right paths, that is, paths whose incrementalsteps are either (1 ,
0) or (0 , x to be (1 , −
1) and the timedirection t to be (1 , h ( x, t = N ) = G ( N + x, N − x ),see [32] and also [40,42] for a continuous analogue related to the Hammersley process [27].The height function has been studied extensively: at time N , it has fluctuations oforder N / and non-trivial correlation over distance N / , which are the KPZ scalingexponents [13, 35, 36]. Furthermore, both the one-point distributions [1, 4, 5, 18, 30, 40]as well as the limiting processes [2, 14, 15, 32, 42] are known for a few initial conditions(or geometries in LPP framework). Finally, the correlations in time of the interfaceare non-trivial over macroscopic distances [19, 23] and they have been recently partiallystudied [3, 8, 25, 33, 34]. ∗ University of Bristol, School of Mathematics, Fry Building, Woodland Rd., Bristol BS8 1UG, UK.E-mail: [email protected] † Institute for Applied Mathematics, Bonn University, Endenicher Allee 60, 53115 Bonn, Germany.E-mail: [email protected] m, n ), there is a uniquegeodesic. Geodesics follow characteristic directions and if the end-point is at distance O ( N ) from the origin, then it has spatial fluctuations of order O ( N / ) with respect tothe line joining the origin with the end-point [12, 31].Consider two or more end-points. To each of the end-points there is one geodesic from(0 ,
0) and thus the set of geodesics as seen from the end-points in the direction of theorigin, have a non-trivial coalescing structure. Some recent studies of this structure inLPP and related models can be found in [6, 10, 11, 28, 29, 38]. One might expect that on alarge scale the coalescing structure is universal and thus not depending on the details ofthe chosen random variables defining the LPP models (provided, of course, that the modelis still in the KPZ class, which rules out, for instance, heavy tailed random variable).The fact that the height function decorrelates over macroscopic times is reflectedin the geometrical behaviour of the geodesics. For instance, taking two end-points atdistance O ( N / ) of each other, the coalescence point of the two geodesics will be atdistance of order O ( N ) from the end-points [26] and have a non-trivial distribution overthe full macroscopic scale, as already noticed in some numerical studies in [22]. Morerefined recent results are also available [43, 45].In the study of the covariance of the time-time correlations [25] it was proven thattaking one end-point as ( N, N ) and the second ( τ N, τ N ), then as τ →
1, the first ordercorrection to the covariance of the LPP is O ((1 − τ ) / N / ) and is completely independentof the geometry of the LPP, i.e., it is the same whether one considers the point-to-pointLPP as in (1.1) or the line-to-point LPP, for which the geodesics start from a point onthe antidiagonal crossing the origin. This suggests that the coalescing structure of theend-points in { ( N + k, N − k ) , | k | ≤ δN / } , for a small δ >
0, should be independentof the LPP geometry over a time-span o ( N ) from the end-points. In particular, thecoalescing structure should be locally the same as the one from the stationary model,introduced in [7]. In [6] a result in this direction has been proven. Among other results,they showed that the tree of point-to-point geodesics starting from every vertex in a boxof side length δN / going to a point at distance N agree inside the box with the tree ofstationary geodesics.The goal of this work is to improve on previous results in the following points:1. In the case of point-to-point LPP, we extend previous results by showing (Theo-rem 2.2) that the coalescence to the stationary geodesics holds with high probabilityfor any geodesic starting in a large box around the origin and terminating in a cylin-der whose width is of order N / and its length is of order N (see Figure 2.1). Inother words, we obtain the correct dimensions of the cylinder around the point( N, N ).2. In the case of point-to-point LPP, we improve the lower bound of the coalescenceresult from exponent 3 / / N / and length N . The order of the lower bound depends on theconcentration of the exit point of the geodesics around the origin.Another problem that is closely related to the coalescence of the point-to-pointgeodesic with the stationary one is the question of coalescence of point-to-point geodesics.More precisely, consider the probability that two infinite geodesics starting k / away fromeach other will coalesce after Rk steps. A lower bound of the order CR − c was obtainedin [38]. Matching upper bound together with the identification of the constant c = − / N / away from each otherand terminate at or around ( N, N ) coalesce. In the setup of Brownian LPP, one takes k geodesics leaving from a small interval of order ǫN / and terminating at time N in aninterval of the same order. Then, the probability that they are disjoint is of order ǫ ( k − / with a subpolynomial correction, see [29, Theorem 1.1]. It was conjectured there thatthe lower bound should have the same exponent. For k = 2 this is proven in [9, Theorem2.4]. Furthermore, an upper bound of order τ / on the probability that two geodesicsstarting from the points (0 ,
0) and (0 , N / ) and terminating at ( N, N ) do not coalesceby time (1 − τ ) N is obtained in [6, Theorem 2.8].Our Theorem 2.7 gives the exact exponent 1 /
2, for the probability that any twogeodesics starting from a large box of dimensions of order N × N / and terminating at acommon point in a small box of size N × N / coalesce (see Figure 2.1 for more accuratedimensions). Theorem 2.7 can be compared with rarity of disjoint geodesics that wasconsidered in [29] although for geodesics starting from a big box rather than a small one.What is then the reason for the discrepancy in the different exponents (exponent3 / / δN / in [29]. Let us try to give a heuristic argument fora possible settlement of this discrepancy. Let us divide the interval I := { (0 , i ) } ≤ i ≤ N / into δ − sub-intervals of size δN / . If the event that two geodesics starting from I donot meet by the time they reach the small interval around the point ( N, N ) is dominatedby the event that the two geodesics leave from the same small sub-interval and if theseevents decorrelate on the scale of N / then by [29, Theorem 1.1] we have roughly δ − decorrelated events of probability (up to logarithmic correction) δ / which would implythat the probability of two geodesics starting from I to not meet by the time they reacha small interval around ( N, N ) is (up to logarithmic correction) δ − δ / = δ / .Concerning the methods used in this paper, one input we use is a control over thelateral fluctuations of the geodesics in the LPP. In Theorem 2.8 we show that the prob-ability that the geodesic of the point-to-point LPP is not localized around a distance M N / from the characteristic line decay like e − cM , which is the optimal power of thedecay. This is proven using the approach of [12], see Theorem 4.3, once the mid-point3nalogue estimate is derived, see Theorem 2.2. The novelty here is a simple and shortproof of this latter by using only comparison with stationary models. This probabilisticmethod is much simpler than previous ones.To prove Theorem 2.2, the first step is to prove that with high probability the spatialtrajectories of both the geodesic of the point-to-point LPP as well as the one of thestationary model with density 1 / ρ + > / ρ − < / Outline of the paper.
In Section 2 we define the model and state the main results.We recall in Section 3 some recurrent notations and basic results on stationary LPP. InSection 4 we prove first Theorem 2.8 on the localization of the point-to-point geodesics andthen show that the geodesics can be sandwiched between two version of the stationarymodel, see Lemma 4.6. This allows us to prove Theorems 2.2 and 2.5 in Section 5.Section 6 deals with the proof of Theorem 2.6 and Theorem 2.7.
Acknowledgments.
The authors are grateful to M´arton Bal´azs for initial discussionson the topic that led to our collaboration. O. Busani was supported by the EPSRCEP/R021449/1 Standard Grant of the UK. This study did not involve any underlyingdata. The work of P.L. Ferrari was partly funded by the Deutsche Forschungsgemein-schaft (DFG, German Research Foundation) under Germanys Excellence Strategy - GZ2047/1, projekt-id 390685813 and by the Deutsche Forschungsgemeinschaft (DFG, Ger-man Research Foundation) - Projektnummer 211504053 - SFB 1060.
Let ω = { ω x } x ∈ Z be i.i.d. Exp(1)-distributed random weights on the vertices of Z . For o ∈ Z , define the last-passage percolation (LPP) process on o + Z ≥ by G o,y = max x • ∈ Π o,y | y − o | X k =0 ω x k for y ∈ o + Z ≥ . (2.1)Π o,y is the set of paths x • = ( x k ) nk =0 that start at x = o , end at x n = y with n = | y − o | ,and have increments x k +1 − x k ∈ { e , e } . The a.s. unique path π o,y ∈ Π o,y that attainsthe maximum in (2.1) is the geodesic from o to y .Let L = { x ∈ Z | x + x = 0 } be the antidiagonal crossing through the origin. Givensome random variables (in general non independent) { h ( x ) } x ∈L on L , independent from ω , define the last passage time with initial condition h by G h L ,y = max x • ∈ Π L ,y (cid:18) h ( x ) + | y − o | X k =1 ω x k (cid:19) for y > L , (2.2)4here y > L is meant in the sense of the order on the lattice. We also denote by Z h L ,y the point x from where the geodesic from L to y leaves the line L , and refer to it as the exit point of the last passage percolation with initial condition h .One can define stationary models parameterized by a density ρ ∈ (0 , L , see Section 3.3for detailed explanations. In that case we denote the stationary LPP by G ρo,y or G ρ L ,y respectively.For σ ∈ R + and 0 < τ < σN / and length τ N C σ,τ = { i e + j e : (1 − τ ) N ≤ i ≤ N, − σ N / ≤ j ≤ σ N / } . (2.3)Similarly, for σ ∈ R + and 0 < τ < σN / and length τ N R σ,τ = { i e + j e : 0 ≤ i ≤ τ N, − σ N / ≤ j ≤ σ N / , | j | < i } . (2.4) Remark . Note that the shape of R σ,τ in (2.4) is somewhat different than the bluecylinder in Figure 2.1. The reason for that is that in this paper we use exit points withrespect to the vertical and horizontal axis so that the shape defined in (2.4) is easier towork with. We stress that similar results can be obtained for a box as in Figure 2.1 byusing exit points with respect to the antidiagonal L .Due to the correspondence to stochastic growth models in the KPZ universality class,we denote the time direction by (1 ,
1) and the spatial direction by (1 , − < τ < L τ = { τ N e + i e : −∞ < i < ∞} , (2.5)Let x, y, z ∈ Z be such that x, y ≤ z . For the geodesics π x,z and π y,z we define thecoalescence point C p ( π x,z , π y,z ) = inf { u ∈ Z : u ∈ π x,z ∩ π y,z } , (2.6)where the infimum is with respect to the order ≤ on the lattice. Upper and lower bounds on the coalescing point
The first main result of this paper is that, with probability going to 1 as δ →
0, the setof geodesics ending at any point in the cylinder C δ,τ of the stationary LPP with density1 / τ ≤ δ / / (log( δ − )) . Theorem 2.2.
Let o = (0 , . There exist C, δ > such that for any δ ∈ (0 , δ ) and τ ≤ δ / / (log( δ − )) , P (cid:16) C p ( π / o,x , π y,x ) ≤ L − τ ∀ x ∈ C δ,τ , y ∈ R log δ − , / (cid:17) ≥ − Cδ / log( δ − ) (2.7) for all N large enough. As a direct corollary we have that in a cylinder of spatial width o ( N / ) and timewidth o ( N ) around the end-point ( N, N ), the geodesics are indistinguishable from thestationary ones.
Corollary 2.3.
For any ε > , lim N →∞ P (cid:16) C p ( π / o,x , π o,x ) ≤ L − N − ε ∀ x ∈ C N − ε ,N − ε (cid:17) = 1 . (2.8)5heorem 2.2 generalizes to LPP with a large class of initial conditions, i.e., considerLPP from L with initial condition h , like the ones considered in [18, 25]. We make thefollowing assumption on h . Recall the exit point Z h L ,x mentioned right after (2.2). Assumption 2.4.
Let x = N e + δN / e and x = N e − δN / e . Assume that P ( Z h L ,x ≤ log( δ − ) N / ) ≥ − Q ( δ ) (2.9) and P ( Z h L ,x ≥ − log( δ − ) N / ) ≥ − Q ( δ ) (2.10) for all N large enough, with a function Q ( δ ) satisfying lim δ → Q ( δ ) = 0 . Under Assumption 2.4 the analogue of Theorem 2.2 (and thus of Corollary 2.3) holdstrue.
Theorem 2.5.
Under Assumption 2.4, there exist
C, δ > such that for any δ ∈ (0 , δ ) and τ ≤ δ / / (log( δ − )) , P (cid:16) C p ( π / L ,x , π h L ,x ) ≤ L − τ ∀ x ∈ C δ,τ (cid:17) ≥ − Cδ / log( δ − ) − Q ( δ ) (2.11) for all N large enough. The exponent 1 / Theorem 2.6.
Let o = (0 , . There exist C, δ > such that for any δ ∈ (0 , δ ) and τ ≤ δ / / (log( δ − )) , P (cid:16) C p ( π / o,x , π y,x ) ≤ L − τ ∀ x ∈ C δ,τ , y ∈ R ( log δ − ) ,τ (cid:17) ≤ − Cδ / (2.12) for all N large enough. The following result is closely related to Theorem 2.2 and Theorem 2.6, it considersthe question of coalescence of point-to-point geodesics.
Theorem 2.7.
There exist
C, δ > such that for any δ ∈ (0 , δ ) and τ ≤ δ / / (log( δ − )) , − Cδ / log( δ − ) ≤ P (cid:16) C p ( π w,x , π y,x ) ≤ L − τ ∀ x ∈ C δ,τ , w, y ∈ R ( log δ − ) ,τ (cid:17) ≤ − Cδ / (2.13) for all N large enough. Cubic decay of localization
In order to prove the main theorems we will need some control on the spatial fluctuationsof the geodesics for the point-to-point problem. As this estimate has its own interest, westate it below as Theorem 4.2. For a an up-right path γ we denoteΓ uk ( γ ) = max { l : ( k, l ) ∈ γ } , Γ lk ( γ ) = min { l : ( k, l ) ∈ γ } . (2.14)When γ is a geodesic associated with a direction, we denote the direction by ξ = ( ξ , ξ )with ξ + ξ = 1, and we setΓ k ( γ ) = max {| Γ uk ( γ ) − ξ ξ k | , | Γ lk ( γ ) − ξ ξ k |} . (2.15)6 heorem 2.8. Let ε ∈ (0 , . Then there exists N ( ε ) and c ( ε ) such that for ξ satisfying ε ≤ ξ /ξ ≤ /ε , P (cid:0) Γ k ( π o,ξN ) > M ( τ N ) / for all k ∈ [0 , τ ξ N ] (cid:1) ≤ e − c M (2.16) for all τ N ≥ N and all M ≤ ( τ N ) / / log( N ) . A statement similar to Theorem 2.8 with Gaussian bound is Proposition 2.1 of [8].The authors employed Theorems 10.1 and 10.5 of [12]. Potentially their argument couldbe improved to get a cubic decay using the bounds from random matrices of [37], but wedid not verify this. Instead, we provide a short and self-contained proof of the localizationresult using comparison with stationarity only.
Remark . Theorem 2.8 states the optimal localization scale for small τ . By symmetryof the point-to-point problem, the same statement holds with τ replaced by 1 − τ andand gives the optimal localization scale for τ close to 1. A family of random initial conditions.
Now let us consider the family of initial conditions, interpolating between flat initialcondition (i.e., point-to-line LPP) and the stationary initial condition, for which thetime-time covariance was studied in [25]. For σ ≥
0, let us define h ( k, − k ) = σ × P kℓ =1 ( X ℓ − Y ℓ ) , for k ≥ , , for k = 0 , − P ℓ = k +1 ( X ℓ − Y ℓ ) , for k ≤ − . (2.17)where { X k , Y k } k ∈ Z are independent random variables X k , Y k ∼ Exp(1 / σ = 0 itcorresponds to the point-to-line LPP, while for σ = 1 it is the stationary case with density1 / Proposition 2.10.
For LPP with initial condition (2.17) , Assumption 2.4 holds with Q ( δ ) = Ce − c (log( δ − )) (2.18) for some constants C, c > .Proof. The estimates leading to the proofs are all contained in the proof of Lemma 5.2of [25], see part (b) Random initial conditions . Replacing in that proof τ = 1, M = ˜ M =2 − / δ and α = 2 − / log( δ − ) we obtain Q ( δ ) ≤ min { Ce − c ( α − M ) , Ce − c ( α − M ) /α } . As α − M ≃ α for small δ , the result follows. We mention here some of the notations which will be used throughout the paper. Wedenote Z ≥ = { , , , , . . . } and Z > = { , , , . . . } . We use four standard vectors in R , namely e = (1 ,
0) and e = (0 , = (1 , −
1) and e = (1 , represents the spatial direction , while e represents the temporal direction .7 N / δ / l og ( δ − ) N l og ( δ − ) N / N / N Figure 2.1: Illustration of Theorems 2.2 and 2.6. With high probability, any geodesic tree(black curve) consisting of all geodesics starting from a fixed point in the blue cylinderand terminating at any point in the red cylinder, will agree in the red box with thestationary tree (red curve) of intensity 1 / x = ( x , x ) ∈ R the ℓ -norm is | x | = | x | + | x | . We alsouse the partial ordering on R : for x = ( x , x ) ∈ R and y = ( y , y ) ∈ R , we write x ≤ y if x ≤ y and x ≤ y . Given two points x, y ∈ Z with x ≤ y , we define the box[ x, y ] = { z ∈ Z | x ≤ z ≤ y } . For u ∈ Z we denote Z ≥ u = { x : x ≥ u } .Finally, for λ > X ∼ Exp( λ ) denotes a random variable X which has exponentialdistribution with rate λ , in other words P ( X > t ) = e − λt for t ≥
0, and thus the mean is E ( X ) = λ − and variance Var( X ) = λ − . To lighten notation, we do not write explicitlythe integer parts, as our results are insensitive to shifting points by order 1. For instance,for a ξ = ( ξ , ξ ) ∈ R , ξN means ( ⌊ ξ N ⌋ , ⌊ ξ N ⌋ ). We construct two partial orders on directed paths in Z . ≤ : For x, y ∈ Z we write x ≤ y if y is above and to the right of x , i.e. x ≤ y and x ≤ y . (3.1)We also write x < y if x ≤ y and x = y (3.2)An up-right path is a (finite or infinite) sequence Y = ( y k ) k in Z such that y k − y k − ∈ { e , e } for all k . Let U R be the set of up-right paths in Z . If A, B ⊂ Z ,we write A ≤ B if x ≤ y ∀ x ∈ A ∩ Y , y ∈ B ∩ Y ∀Y ∈ U R . (3.3)8 γ x γ y Figure 3.1: The two geodesics γ and γ are ordered i.e., γ ≺ γ . For any down-rightpath Y in Z the set of points x = Y ∩ γ and y = Y ∩ γ are ordered, i.e., x ≺ y .where we take the inequality to be vacuously true if one of the intersections in (3.3)is empty. (cid:22) : For x, y ∈ Z we write x (cid:22) y if y is below and to the right of x , i.e. x ≤ y and x ≥ y . (3.4)We also write x ≺ y if x (cid:22) y and x = y (3.5)A down-right path is sequence Y = ( y k ) k ∈ Z in Z such that y k − y k − ∈ { e , − e } for all k ∈ Z . Let DR be the set of infinite down-right paths in Z . If A, B ⊂ Z ,we write A (cid:22) B if x (cid:22) y ∀ x ∈ A ∩ Y , y ∈ B ∩ Y ∀Y ∈ DR . (3.6)where we take the inequality to be vacuously true if one of the intersections in (3.6)is empty (see Figure 3.1). Stationary LPP on Z ≥ has been introduced in [41] by adding boundary terms on the e and e axis. In [7] it was shown that it can be set up by using more general boundarydomains. In this paper we are going to use two of them. Boundary weights on the axis.
For a base point o = ( o , o ) ∈ Z and a parametervalue ρ ∈ (0 ,
1) we introduce the stationary last-passage percolation process G ρo, • on o + Z ≥ . This process has boundary conditions given by two independent sequences { I ρo + i e } ∞ i =1 and { J ρo + j e } ∞ j =1 (3.7)of i.i.d. random variables with I ρo +e ∼ Exp(1 − ρ ) and J ρo +e ∼ Exp( ρ ). Put G ρo,o = 0 andon the boundaries G ρo, o + k e = k X i =1 I o + i e and G ρo, o + l e = l X j =1 J o + j e . (3.8)Then in the bulk for x = ( x , x ) ∈ o + Z > , G ρo, x = max ≤ k ≤ x − o n k X i =1 I o + i e + G o + k e +e , x o _ max ≤ ℓ ≤ x − o n ℓ X j =1 J o + j e + G o + ℓ e +e , x o . (3.9)9 oundary weights on antidiagonal. The stationary model with density ρ can berealized by putting boundary weights on L as follows. Let { X k , k ∈ Z } and { Y k , k ∈ Z } be independent random variables with X k ∼ Exp(1 − ρ ) and Y k ∼ Exp( ρ ). Then, define h ( k, − k ) = P kℓ =1 ( X ℓ − Y ℓ ) , for k ≥ , , for k = 0 , − P ℓ = k +1 ( X ℓ − Y ℓ ) , for k ≤ − . (3.10)Then the LPP defined by (2.2) with initial condition h is stationary, that is, the incre-ments G h L ,x +e − G h L ,x ∼ Exp(1 − ρ ) as well as G h L ,x +e − G h L ,x ∼ Exp( ρ ) for all x > L .Next we define the exit points of geodesics, these will play an important role in ouranalysis. Definition 3.1 (Exit points) . (a) For a point p ∈ o + Z > , let Z o,p be the signed exit point of the geodesic π o,p of G o,p from the west and south boundaries of o + Z > . More precisely, Z ρo,p = argmax k (cid:8) P ki =1 I o + i e + G o + k e +e , x (cid:9) if π o,p ∩ o + e = ∅ , − argmax ℓ (cid:8) P ℓj =1 J o + j e + G o + ℓ e +e , x (cid:9) if π o,p ∩ o + e = ∅ . (3.11) (b) For a point p > L , we denote by Z h L ,p ∈ Z the exit point of the LPP from L with initialcondition h , if the starting point of the geodesic from L to p is given by ( Z h L ,p , − Z h L ,p ) .In the case of the stationary model with parameter ρ , the exit point is denoted by Z ρ L ,p . As a consequence, the value G ρo,x can be determined by (3.8) and the recursion G ρo,x = ω x + G ρo,x − e ∨ G ρo,x − e . (3.12) Next we consider LPP maximizing down-left paths. For y ≤ o , define b G o,y = G y,o , (3.13)and let the associated geodesic be denoted by ˆ π o,y . For each o = ( o , o ) ∈ Z and aparameter value ρ ∈ (0 ,
1) define a stationary last-passage percolation processes b G ρ on o + Z ≤ , with boundary variables on the north and east, in the following way. Let { ˆ I ρo − i e } ∞ i =1 and { ˆ J ρo − j e } ∞ j =1 (3.14)be two independent sequences of i.i.d. random variables with marginal distributionsˆ I ρo − i e ∼ Exp(1 − ρ ) and ˆ J ρo − j e ∼ Exp( ρ ). The boundary variables in (3.7) and thosein (3.14) are taken independent of each other. Put b G ρo, o = 0 and on the boundaries b G ρo, o − k e = k X i =1 ˆ I o − i e and b G ρo, o − l e = l X j =1 ˆ J o − j e . (3.15)Then in the bulk for x = ( x , x ) ∈ o + Z < , b G ρo, x = max ≤ k ≤ o − x n k X i =1 ˆ I o − i e + b G o − k e − e ,x o _ max ≤ ℓ ≤ o − x n ℓ X j =1 ˆ J o − j e + b G o − ℓ e − e ,x o . (3.16)10or a southwest endpoint p ∈ o + Z < , let b Z ρo,p be the signed exit point of the geodesicˆ π o,p of b G ρo,p from the north and east boundaries of o + Z < . Precisely, b Z ρo, x = argmax k (cid:8) P ki =1 ˆ I o − i e + b G o − k e − e ,x (cid:9) , if ˆ π o,x ∩ o − e = ∅ , − argmax ℓ (cid:8) P ℓj =1 ˆ J o − j e + b G o − ℓ e − e ,x (cid:9) , if ˆ π o,x ∩ o − e = ∅ . (3.17) We are going to use the comparison between point-to-point LPP and stationary LPPusing the lemma by Cator and Pimentel.
Lemma 3.2.
Let o = (0 , and consider two points p (cid:22) p .If Z ρo,p ≥ , then G o,p − G o,p ≤ G ρo,p − G ρo,p . (3.18) If Z ρo,p ≤ , then G o,p − G o,p ≥ G ρo,p − G ρo,p . (3.19)Clearly by reversion of the space we can use this comparison lemma also for backwardsLPP. Lemma 3.3.
Consider two points p (cid:22) p with p , p > L .If Z ρ L ,p ≥ Z h L ,p , then G h L ,p − G h L ,p ≤ G ρ L ,p − G ρ L ,p . (3.20) If Z ρ L ,p ≤ Z h L ,p , then G h L ,p − G h L ,p ≥ G ρ L ,p − G ρ L ,p . (3.21)For p = p , Lemma 3.2 is proven as Lemma 1 of [17], while Lemma 3.3 is Lemma 2.1of [39]. The generalization to the case of geodesics starting from L (or from any down-right paths) is straightforward, see e.g. Lemma 3.5 of [24]. τ N . In this section we are going to prove Theorem 2.8. We shall need the following estimateson the tail of the exit point of a stationary process. For a density ρ ∈ (0 ,
1) we associatea direction ξ ( ρ ) = (cid:18) (1 − ρ ) (1 − ρ ) + ρ , ρ (1 − ρ ) + ρ (cid:19) (4.1)and, vice versa, to each direction ξ = ( ξ , ξ ) corresponds a density ρ ( ξ ) = √ ξ √ ξ + √ ξ . (4.2)11 ξNp ζ p p ζ Figure 4.1: Illustration of the geometry of the points and characteristics that appear inthe proof of Theorem 4.2.
Lemma 4.1 (Theorem 2.5 and Proposition 2.7 of [20]) . Let ε ∈ (0 , . Then there exists N ( ε ) , c ( ε ) , r ( ε ) > such that for every direction ξ with ε ≤ ξ /ξ ≤ /ε , N ≥ N and r ≥ r : P ( | Z νo,ξN | > rN / ) ≤ e − c r , (4.3) P ( Z νo,ξN − rN / e > ≤ e − c r , (4.4) P ( Z νo,ξN + rN / e < ≤ e − c r , (4.5) for all densities ν satisfying | ν − ρ ( ξ ) | ≤ N − / . Using Lemma 4.1 one can control the path of a point-to-point geodesic.
Theorem 4.2.
Let ε ∈ (0 , . Then there exist N ( ε ) and c ( ε ) such that for ξ satisfying ε ≤ ξ /ξ ≤ /ε , P (cid:0) Γ τξ N ( π o,ξN ) > M ( τ N ) / (cid:1) ≤ e − c M (4.6) for all τ N ≥ N and all M ≤ ( τ N ) / / log( N ) .Proof. We will show in detail that P (cid:16) Γ uτξ N ( π o ,o ) > τ ξ N + M ( τ N ) / (cid:17) ≤ e − c M , (4.7)where o = o and o = ξN . Similarly one proves P (cid:16) Γ lτξ N ( π o ,o ) < τ ξ N − M ( τ N ) / (cid:17) ≤ e − c M . (4.8)Then Theorem 4.2 follows directly from the definition of Γ τξ N ( π o,ξN ). Set the points (seeFigure 4.1) p = τ N ξ + M ( τ N ) / e ,p = τ N ξ − M − ττ ( τ N ) / e ,p = τ N ξ + M ( τ N ) / e , (4.9)12nd the characteristics associated with ( o , p ) and ( o , p ) ζ = ( τ ξ N, τ ξ N + M ( τ N ) / ) ,ζ = ((1 − τ ) ξ N, (1 − τ ) ξ N + M − ττ ( τ N ) / ) . (4.10)The associated densities are ρ = q τ ξ N + M ( τ N ) / √ τ ξ N + q τ ξ N + M ( τ N ) / ,ρ = q (1 − τ ) ξ N + M − ττ ( τ N ) / p (1 − τ ) ξ N + q (1 − τ ) ξ N + M − ττ ( τ N ) / . (4.11)Note that by (4.4)–(4.5) there exists c > P (cid:0) Z ρ o ,p > (cid:1) ≤ e − c M , P (cid:0) ˆ Z ρ o ,p < (cid:1) ≤ e − c M . (4.12)Define, for i ≥ J i = G o ,p +( i +1)e − G o ,p + i e , b J i = G o ,p +e + i e − G o ,p +e +( i +1)e ,J ρ i = G ρ o ,p +( i +1)e − G ρ o ,p + i e , b J ρ i = G ρ o ,p +e + i e − G ρ o ,p +e +( i +1)e . (4.13)Then, by the Lemma 3.2, it follows from (4.12) that with probability 1 − e − c M J i ≤ J ρ i and b J ρ i ≤ b J i (4.14)for all i ≥
0, and therefore that J i − b J i ≤ J ρ i − b J ρ i (4.15)for all i ≥
0. Set ρ = ρ ( ξ ). Note that for M ≤ ( τ N ) / / log( N ) , using series expansionwe have ρ = ρ + κ ( ρ ) M τ N ) − / + o (( τ N ) − / ) ,ρ = ρ + κ ( ρ ) M
16 ( τ N ) − / + o (( τ N ) − / ) ,ρ − ρ = κ ( ρ ) M
16 ( τ N ) − / + o (( τ N ) − / ) > , (4.16)with κ ( ρ ) = (1 − ρ )(1 − ρ (1 − ρ )) /ρ > ρ ∈ (0 , S i = i X k =0 J k − b J k and W i = i X k =0 J ρ k − b J ρ k (4.17)so that by (4.15) S i ≤ W i for i ≥ . (4.18)13ote that { Γ uτξ N ( π o ,o ) > τ ξ N + M ( τ N ) / } ⊆ (cid:26) sup i ≥ M τN ) / S i > (cid:27) ⊆ (cid:26) sup i ≥ M τN ) / W i > (cid:27) . (4.19)It follows that it is enough to show that there exists c > P (cid:18) sup i ≥ M τN ) / W i > (cid:19) ≤ e − c M . (4.20)Note that P (cid:18) sup i ≥ M τN ) / W i > (cid:19) ≤ P (cid:16) W M τN ) / > − χ ( ρ ) M
64 ( τ N ) / (cid:17) + P (cid:18) sup i ≥ M τN ) / W i − W M τN ) / > χ ( ρ ) M
64 ( τ N ) / (cid:19) (4.21)for χ ( ρ ) = κ ( ρ ) /ρ .Plugging (4.16) in Lemma A.1 P (cid:18) sup i ≥ M τN ) / W i − W M τN ) / > χ ( ρ ) M
64 ( τ N ) / (cid:19) ≤ ρ ρ e − ( ρ − ρ ) χ ( ρ ) M ( τN ) / ≤ e − χ ( ρ ) κ ( ρ ) M / (4.22)for all τ N large enough.Next, using exponential Tchebishev inequality, we show that P (cid:16) W M τN ) / > − χ ( ρ ) M
64 ( τ N ) / (cid:17) ≤ e − M χ ( ρ ) κ ( ρ ) / (4.23)for all τ N large enough, which completes the proof. Indeed, using (cid:18) ρ − ρ (cid:19) M ( τ N ) / = − M χ ( ρ )32 ( τ N ) / + o (( τ N ) / ) , (4.24)we get, using also the independence of the J ’s and b J ’s, that(4.23) = P M τN ) / X k =0 ( J ρ k − ρ − − b J ρ k + ρ − ) > χ ( ρ ) M
64 ( τ N ) / + o (( τ N ) / ) ! ≤ inf λ> E (cid:16) e λ ( J ρ − ρ − − b J ρ + ρ − ) (cid:17) M τN ) / e λ [ χ ( ρ ) M ( τN ) / + o (( τN ) / )] = inf µ> e − M ( Mκ ( ρ ) − µ ) µ/ (64 ρ )+ o (1) ≤ e − M κ ( ρ ) χ ( ρ ) / , (4.25)for all τ N large enough, where in the third step we set λ = µ ( τ N ) − / and performedsimple computations. 14 heorem 4.3. Let o = (0 , and ε ∈ (0 , . There there exists N ( ε ) , c ( ε ) , C ( ε ) suchthat for every direction ξ with ε ≤ ξ /ξ ≤ /ε , and v ≤ N / / log( N ) , for N > N P (cid:16) max k ∈ [0 ,ξ N ] Γ k ( π o,ξN ) < vN / (cid:17) ≥ − Ce − cv . (4.26) Proof.
The proof follows the approach of [12], using the pointwise control of the fluctu-ations of the geodesic around the characteristic from Theorem 4.2. Let m = min { j :2 − j N ≤ N / } . Choose u < u < . . . with u = v/
10 and u j − u j − = u − ( j − / . Wedefine u ( k ) = Γ uk ( π o,ξN ) − ξ ξ k, k ∈ [0 , ξ N ] (4.27)and the following events A j = { u ( k − j N ) ≤ u j N / , ≤ k ≤ j − } ,B j,k = { u ( k − j N ) > u j N / } , k = 1 , . . . , j − ,L = { sup x ∈ [0 , | u (( k + x )2 − m N ) − u ( k − m N ) | ≤ vN / , ≤ k ≤ m − } ,G = { u ( k ) ≤ vN / for all 0 ≤ k ≤ ξ N } . (4.28)Notice that A cj = S j − k =1 B j,k . Also, since lim j →∞ u j ≤ v/
2, we have m [ j =1 2 j − [ k =1 ( B j,k ∩ A j − ) ⊇ { u ( k − m N ) ≥ vN / for some k = 1 , . . . , m − } . (4.29)This implies that G ⊇ m [ j =1 2 j − [ k =1 ( B j,k ∩ A j − ) ! c ∩ L. (4.30)Thus we have P ( G ) ≤ P ( L c ) + m X j =1 2 j − X k =1 P ( B j,k ∩ A j − ) . (4.31)Since the geodesics have discrete steps, in n time steps a geodesic can wonder off byat most n steps from its characteristic. For all N large enough, N / < vN / andtherefore P ( L ) = 1. Thus we need to bound P ( B j,k ∩ A j − ) only. As for even k the twoevents are incompatible, we consider odd k .If A j − holds, then the geodesic at t = ( k − − j N and t = ( k + 1)2 − j N satisfies u ( t ) ≤ u j − N / and u ( t ) ≤ u j − N / . (4.32)Consider the point-to-point LPP from ˆ o to ˆ o withˆ o = ( t , t ξ ξ + u j − ) and ˆ o = ( t , t ξ ξ + u j − ) . (4.33)Let ˆ u ( i ) = Γ ui ( π ˆ o , ˆ o ) for i ∈ [ t , t ]. Then, by the order of geodesics u ( i ) ≤ ˆ u ( i ) for i ∈ [ t , t ] , (4.34)so that { u ( i ) > u j N / } ⊆ { ˆ u ( i ) > u j N / } for i ∈ [ t , t ] . (4.35)15his gives P ( B j,k ∩ A j − ) ≤ P (cid:0) ˆ u ( k − j N ) > u j N / ) . (4.36)Since the law of ˆ u is the one of a point-to-point LPP over a time distance t − t = 2 − j +1 N ,we can apply Theorem 4.2 with τ = 1 / N = t − t M satisfying ( u j − u j − ) N / = M ( ( t j − t j − )) / . This gives P (cid:0) ˆ u ( k − j N ) > u j N / ) ≤ e − c ( u − ( j − / j/ ) ≤ e − c u j/ . (4.37)This bound applied to (4.31) leads to P ( G ) ≤ Ce − cv for some constants C, c >
Proof of Theorem 2.8.
Theorem 4.2 implies that with probability at least 1 − e − c M / , thegeodesic from o to ξN does not deviate more than M ( τ N ) / away from the point τ ξN .Given this event, by order of geodesics, the geodesic from o to ξN is sandwiched betweenthe geodesics from M ( τ N ) / e to τ ξN + M ( τ N ) / e and the one from − M ( τ N ) / e to τ ξN − M ( τ N ) / e . By Theorem 4.3, the latter two geodesics fluctuates no morethan M ( τ N ) / , with probability at least 1 − Ce − cM /τ , which implies the claim. In this subsection, we estimate the location of the exit point for densities slightly largeror smaller than 1 /
2. This will allow us to sandwich the point-to-point geodesics by thoseof the stationary. Notice that to apply Lemma 3.2, it would be enough to set in theevent A (resp. A ) below that the exit point is positive (resp. negative) and boundedby 15 rN / (resp. − rN / ) as the exit point for the LPP G o,x is 0. However, withthis slight modification (that the exit point is rN / from the origin), the proof is thenapplicable for more general initial conditions provided the exit points of the LPP withinitial conditions h on L is localized in a [ − rN / , rN / ] with high probability.Fix r > < s r , t r . s r and t r will be determined later and represent thedimensions of space and time respectively. Let 0 < a <
1. Define the points x = N e + as r N / e ,x = N e − as r N / e . (4.38)Define the densities ρ + = + rN − / , ρ − = − rN − / , (4.39)and, with o = (0 , A = (cid:8) Z ρ + o,x ≥ rN / , Z ρ + o,x ≤ rN / (cid:9) , A = (cid:8) Z ρ − o,x ≥ − rN / , Z ρ − o,x ≤ − rN / (cid:9) , A = A ∪ A . (4.40)The event A is highly probable for large r as shows the following lemma. Lemma 4.4.
Assume ≤ s r ≤ r and < a < . There exists c, N > such that for N > N and < r < N / / log( N ) , P ( A ) ≥ − e − cr . (4.41)16 x ξ + ξ + D D as r N / t r N s r N / rt r N / rt r N / M ( t r N ) / s r N / π ρ + o,x π ρ + o,x Figure 4.2: Illustration of the geometry around the end-point (
N, N ) magnified androtated by π/
4. Choosing t r , s r properly forces the geodesic π ρ + o,x to traverse to the leftof D and π ρ + o,x to the right of D . Proof.
We will show the claim for A , one can similarly prove the claim for A . Theresult would then follow from union bound. To prove the claim for A , by union boundit is enough to show that P (cid:0) Z ρ + o,x ≥ rN / (cid:1) ≥ − e − cr , (4.42) P (cid:0) Z ρ + o,x ≤ rN / (cid:1) ≥ − e − cr . (4.43)Let x = x − rN / e . Then by stationarity of the model, P (cid:0) Z ρ + o,x < rN / (cid:1) = P (cid:0) Z ρ + o,x < (cid:1) . (4.44)Now we want to use (4.5). For that denote ˜ N = N − r N / and write x = ξ ( ρ + )2 ˜ N + ˜ r ˜ N / e . (4.45)Solving with respect to ˜ r we obtain˜ r = 72 r − as r + O ( r /N / ) . (4.46)Applying (4.5) with ν → ρ + , N → ˜ N and r → ˜ r gives P (cid:0) Z ρ + o,x < (cid:1) ≤ e − c ˜ r . (4.47)Since s r ≤ r and r ≤ N / / log( N ), we have ˜ r ≥ r for all N large enough, which proves(4.42). 17et x = x − rN / e . Then by stationarity of the model, P (cid:0) Z ρ + o,x > rN / (cid:1) = P (cid:0) Z ρ + o,x > (cid:1) . (4.48)We will apply this time (4.4). Denote ˜ N = N − rN / and write x = ξ ( ρ + )2 ˜ N − ˆ r ˜ N / e . (4.49)Solving with respect to ˆ r we obtainˆ r = 72 r − as r + O ( r /N / ) ≥ r (4.50)for all N large enough. Applying (4.4) with ν → ρ + , N → ˜ N and r → ˆ r proves (4.43). C s r / ,t r . Consider the following assumption.
Assumption 4.5.
Let M > , a = 3 / , s r ≤ min { r, } and make the following assump-tions on the parameters: r ≤ N / / log( N ) , M ≤ M ≤ s r t − / r − rt / r . (4.51)We shall later discuss this assumption in Remark 4.9 below. Under Assumption 4.5,the geodesics π / o,x and π y,x , for y ∈ R r/ , / , are controlled by the ones with densities ρ + and ρ − for all x ∈ C s r / ,t r . This is the content of the following result whose proof we deferto the end of this section. Lemma 4.6.
Under Assumption 4.5, there exists
C, c > such that P (cid:16) π ρ − o,x (cid:22) π / o,x , π y,x (cid:22) π ρ + o,x ∀ x ∈ C s r / ,t r , y ∈ R r/ , / (cid:17) ≥ − Ce − cM − e − cr (4.52) for all N large enough. Define c = π ρ + o,x ∩ L − t r ,c = π ρ + o,x ∩ L − t r . (4.53)To ease the notation we also denote w = (1 − t r ) N e − as r N / e = x − t r N e ,w = (1 − t r ) N e + as r N / e = x − t r N e . (4.54) Lemma 4.7.
There exists c, N , M > such that for t r N > N , r ≤ N / / log( N ) , M ≥ M P (cid:16) w − M ( t r N ) / e (cid:22) c (cid:22) w + (8 rt r N / + M ( t r N ) / )e (cid:17) ≥ − e − cM , (4.55) P (cid:16) w − M ( t r N ) / e (cid:22) c (cid:22) w + (8 rt r N / + M ( t r N ) / )e (cid:17) ≥ − e − cM . (4.56)18 roof. Let p be the point of intersection of the characteristic ξ + starting from x withthe line L − t r . We have p = w + (4 rt r N / + O ( r t r ))e = w + 4 rt r N / (1 + o (1))e (4.57)for r ≤ N / / log( N ) and N large enough, implying w (cid:22) p (cid:22) w + 8 rt r N / e =: z . (4.58)By the order on geodesics c (cid:22) π ρ + o,x +4 rt r N / e and if Z ρ + z + M ( t r N ) / e ,x +4 rt r N / e < π ρ + o,x +4 rt r N / e (cid:22) z + M ( t r N ) / e . Thus P ( c (cid:22) z + M ( t r N ) / e ) ≥ P ( Z ρ + z + M ( t r N ) / e ,x +4 rt r N / e < . (4.59)Using (4.4), the latter is bounded from above by 1 − e − c M provided M ≥ M and t r N ≥ N .A similar bound can be obtained for P ( w − M ( t r N ) / e (cid:22) c ) (4.60)using (4.5). Thus we have shown that (4.55) holds. The proof of (4.56) is almost identicaland thus we do not repeat the details.Set q = (1 − t r ) N e + N / ( as r − M t / r )e ,q = (1 − t r ) N e + N / (8 rt r − as r + 2 M t / r )e (4.61)and define the lines (see Figure 4.2) D = { q + αt r N e : 0 < α < } ,D = { q + αt r N e : 0 < α < } . (4.62) Lemma 4.8.
There exist N , c, C > such that for every N ≥ N and M ≤ N / / log( N ) , r ≤ N / / log( N ) , P (cid:16) D (cid:22) π ρ + o,x (cid:17) ≥ − Ce − cM , (4.63) P (cid:16) π ρ + o,x (cid:22) D (cid:17) ≥ − Ce − cM . (4.64) Proof.
We will show (4.64) as (4.63) can be proven similarly. Let u = q − M ( t r N ) / e . (4.65)By Theorem 4.3 we have P (cid:0) π u ,x (cid:22) D (cid:1) ≥ − Ce − cM . (4.66)Recall the definition (4.53) of c . By Lemma 4.7 P (cid:0) c (cid:22) u (cid:1) ≥ − e − cM , (4.67)which implies that P (cid:0) π ρ + o,x (cid:22) π u ,x (cid:1) ≥ − Ce − cM . (4.68)(4.66) and (4.68) imply (4.64). 19 emark . Now we can discuss the origin of the conditions in Assumption 4.5. Thebound on r comes from Lemma 4.8. The condition on M is a consequence of the conditions q (cid:22) (1 − t r ) N e − s r N / e and also (1 − t r ) N e + s r N / e (cid:22) q . As we want M togrow to infinity we need to take t r ≪ s r /r .For 0 < τ < σ ∈ R + , define the anti-diagonal segment I σ,τ = { (1 − τ ) N e + i e , i ∈ [ − σ N / , σ N / ] } , (4.69)located right below the cylinder C σ,τ . Define the events O = (cid:8) Z ρ − o,x ∈ [ − rN / , − rN / ] , Z ρ + o,x ∈ [ rN / , rN / ] ∀ x ∈ C s r / ,t r (cid:9) , B = n { π ρ − o,x ∩ I s r ,t r = ∅} ∩ { π ρ + o,x ∩ I s r ,t r = ∅} ∀ x ∈ C s r / ,t r o . (4.70) Corollary 4.10.
Under Assumption 4.5 there exists
C, c, N > such that for N > N P ( O ) ≥ − e − cr − Ce − cM , (4.71) P ( B ) ≥ − e − cM . (4.72) Proof.
It might be helpful to take a look at Figure 4.2 while reading the proof. We provein details the statements for ρ + , since the proof for ρ − is almost identical.By our choice of parameters, D (cid:22) C s r / ,t r (cid:22) D . (4.73)By Lemma 4.8 and order of geodesics, with probability at least 1 − Ce − cM , π ρ + o,x (cid:22) π ρ + o,x (cid:22) π ρ + o,x (4.74)for all x ∈ C s r / ,t r . By Lemma 4.4 the exit point of the geodesics to x and x for thestationary model with density ρ + lies between rN / and 15 rN / with probability atleast 1 − e − cr , which leads to (4.71).To prove (4.72), first notice that Assumption 4.5 implies w + (8 rt r N / + M ( t r N ) / )e (cid:22) w + s r e (4.75)and w − s r e (cid:22) w − M ( t r N ) / e . (4.76)Thus by Lemma 4.7 we know that the crossing of π ρ + o,x and π ρ + o,x with I s r ,t r occurs withprobability at least 1 − e − cM . This, together with (4.74) implies (4.72). Proof of Lemma 4.6.
Consider the straight line going from (0 , rN / ) to x parameterizedby ( u, l ( u )) with l ( u ) = rN / + u N − ( s r + r ) N / N + s r N / , (4.77)and the straight line ( u, l ( u )), which overlaps in its first part with the boundary of R r/ , / , defined through l ( u ) = r N / + u. (4.78)20y our assumption s r ≤ r ,inf ≤ u ≤ N ( l ( u ) − l ( u )) ≥ r N / − rN / N + rN / N ≥ rN / . (4.79)It follows from (4.79) and Theorem 4.3 that for some C, c > P (Γ lu ( π (0 ,rN / ) ,x ) < l ( u ) for some 0 ≤ u ≤ N ) ≤ Ce − cr . (4.80)By the analogue of Lemma 4.8 for ρ − , with probability at least 1 − C e − c M , π ρ − o,x (cid:22) π ρ − o,x for all x ∈ C s r / ,t r . Furthermore, on the event A , ρ ρ − o,x starts from a point above(0 , rN / ). Combining these two facts with (4.80) we get P (cid:16) π ρ − o,x (cid:22) π (0 ,rN / ) ,x (cid:22) R r/ , / ∀ x ∈ C s r / ,t r (cid:17) ≥ − C e − c M − Ce − cr . (4.81)A similar result can be obtained for π ρ + o,x , which combined with (4.81) gives P (cid:16) π ρ − o,x (cid:22) R r/ , / (cid:22) π ρ + o,x ∀ x ∈ C s r / ,t r (cid:17) ≥ − C e − c M − Ce − cr . (4.82)By the order of geodesics, on the event of (4.82), every geodesic starting in R r/ , / andending at x , is sandwiched between π ρ − o,x and π ρ + o,x .Next, for each point x ∈ C s r / ,t r , its associated density ρ ( x ) satisfies | ρ ( x ) − | ≤ N − / for all s r ≤ − t r ) and N large enough. By (4.3) of Lemma 4.1, with probability at least1 − e − c r the exit point of π / o,x is also between − rN / and rN / . Thus by appropriatechoice of constants C, c , the sandwitching of π / o,x in (4.52) holds. Using the results of Section 4 we first relate the bound for the coalescing point of π / o,x and π o,x to that of the coalescing point of π ρ + o,x and π ρ − o,x . Lemma 5.1.
Under Assumption 4.5, there exists
C, c > such that P (cid:16) C p ( π / o,x , π y,x ) ≤ L − t r ∀ x ∈ C s r / ,t r , y ∈ R r/ , / (cid:17) ≥ − Ce − cM − e − cr − P (cid:16) ∃ x ∈ C s r ,t r : C p ( π ρ + o,x , π ρ − o,x ) ≥ I s r / ,t r , B (cid:17) (5.1) for all N large enough.Proof. We bound the probability of the complement event. Define the event G = { π ρ − y,x (cid:22) π / o,x , π y,x (cid:22) π ρ + y,x ∀ x ∈ C s r / ,t r , y ∈ R r/ , / } . (5.2)Then P (cid:16) ∃ x ∈ C s r / ,t r , y ∈ R r/ , / : C p ( π / o,x , π y,x ) > L − t r (cid:17) ≤ P ( B c ) + P ( G c )+ P (cid:16) {∃ x ∈ C s r / ,t r , y ∈ R r/ , / : C p ( π / o,x , π y,x ) > L − t r } ∩ B ∩ G (cid:17) . (5.3)21 H C s r / ,t r I s r ,t r C p ( π / o,x , π o,x ) xv c Figure 5.1: On the event E ∩ E , the geodesics π ρ + o,x and π ρ − o,x coalesce before crossing I s r ,t r .Note that if B and G hold, then both geodesics π / o,x and π y,x are sandwiched between π ρ − o,x and π ρ + o,x and their crossings with the line L − t r occurs in the segment I s r ,t r . Furthermore,if C p ( π / o,x , π y,x ) > L − t r and G holds, then also C p ( π ρ + o,x , π ρ − o,x ) > L − t r . This, together withCorollary 4.10 and Lemma 4.6 proves the claim.Thus, to prove Theorem 2.2 it remains to get an upper bound for the last probabilityin (5.1).For x, y, z ∈ Z such that x ≤ y ≤ z , let γ x,z be an up-right path going from x to z .Define the exit point of γ x,z with respect to the point y Z y ( γ x,z ) = sup { u ∈ γ x,z : u = y or u = y } . (5.4)Define the sets H = { (1 − t r ) N e + s r N / e − i e , ≤ i ≤ s r N / } , V = { (1 − t r ) N e − s r N / e − i e , ≤ i ≤ s r N / } , (5.5)and the point v c = [(1 − t r ) N − s r N / ]e . (5.6)Define the event E = {Z v c ( π ρ + o,x ) ∈ H ∪ V , Z v c ( π ρ − o,x ) ∈ H ∪ V ∀ x ∈ C s r / ,t r } . (5.7)Note that (see Figure 5.1) E = B , (5.8)since to cross the set I s r ,t r the geodesic must cross either H or V and, viceversa, if thegeodesic crosses H ∪ V , then it crosses also I s r ,t r .22n the edges of Z ≥ define the random field B through B x,x +e k = G ( π o,x +e k ) − G ( π o,x ) , k = 1 , x > o . Similarly we define B ρ + x,x +e k = G ( π ρ + o,x +e k ) − G ( π ρ + o,x ) , k = 1 , ,B ρ − x,x +e k = G ( π ρ − o,x +e k ) − G ( π ρ − o,x ) , k = 1 , . (5.10)One can couple the random fields B, B ρ − and B ρ + (see [21, Theorem 2.1]) such that B ρ − x − e ,x ≤ B x − e ,x ≤ B ρ + x − e ,x ,B ρ + x − e ,x ≤ B x − e ,x ≤ B ρ − x − e ,x . (5.11)Let o ≤ v ≤ x . Then, since each geodesic has to pass either by one site on the rightor above v , we have G o,x = G o,v + max n sup ≤ l ≤ x − v l X i =0 B v + i e ,v +( i +1)e + G v +( l +1)e +e ,x , sup ≤ l ≤ x − v l X i =0 B v + i e ,v +( i +1)e + G v +( l +1)e +e ,x o . (5.12)Thus setting v = v c , on the event E ∩ G , for every x ∈ C s r / ,t r G o,x = G o,v + max n sup u ∈H u − v c X i =0 B v c + i e ,v c +( i +1)e + G u +e ,x , sup u ∈V u − v c X i =0 B v c + i e ,v c +( i +1)e + G u +e ,x o . (5.13) G ρ + o,x and G ρ − o,x can be decomposed in the same way.This shows that on the event E ∩ G the restriction of the geodesics π o,x , π ρ + o,x , π ρ − o,x to Z >v c is a function of the weights (5.9)–(5.10) and the bulk weights east-north to v c . Moreprecisely, define E Bv c to be the set of edges in the south-west boundary of Z >v c and thatare incident to H ∪ V i.e. E Hv c = { ( x − e , x ) : x ∈ H} , E Vv c = { ( x − e , x ) : x ∈ V} , E Bv c = E Hv c ∪ E Vv c . (5.14)The representation (5.13) show that on the event E ∩ G , for every x ∈ C s r / ,t r , therestrictions of the geodesics π o,x , π ρ + o,x , π ρ − o,x to Z >v c are functions of the bulk weights { ω x } x ∈ Z >vc (5.15)and the boundary weights { B e } e ∈E Bvc , { B ρ + e } e ∈E Bvc and { B ρ − e } e ∈E Bvc (5.16)respectively. The stationary geodesics with densities ρ + and ρ − will coalesce beforereaching I s r ,t r if the following event holds true: E = {∃ e ∈ E Bv c : B ρ + e = B ρ − e } . (5.17)23 emma 5.2. We have {∃ x ∈ C s r / ,t r : C p ( π ρ + o,x , π ρ − o,x ) ≥ I s r ,t r , B} ⊆ E ∩ E . (5.18) Proof.
The representation (5.13) for the stationary models imply that G ρ − o,x and G ρ + o,x arefunctions of the weights in (5.15) and the stationary weights in (5.16). This implies thaton ( E ∩ E ) c C p ( π ρ + o,x , π ρ − o,x ) ≤ H ∪ V ≤ I s r ,t r ∀ x ∈ C s r / ,t r , (5.19)which implies the result.Thus it remains to find an upper bound for P ( E ). For m >
0, let us define A m = (cid:8) B ρ + v c + i e ,v c +( i +1)e = B ρ − v c + i e ,v c +( i +1)e , ≤ i ≤ m (cid:9) . (5.20)Recall that ρ + = 1 / rN − / . In [6] the following bound is proven. Lemma 5.3 (Lemma 5.9 of [6]) . Let m ≥ . For < θ < ρ + , it holds P ( A m ) ≥ − rN − / + rN − / + − rN − / + rN − / " rθN − / + θ − ( r N − / + 2 rN − / θ + θ ) m
11 + 2 θr − N / . (5.21) Corollary 5.4.
There exists
C > , such that for every r > and < η < / , P ( A ηr − N / ) ≥ − Cη / . (5.22) for all N large enough.Proof. We set θ = η − / rN − / and plug this into (5.21). Taking N → ∞ we obtainlim N →∞ P ( A ηr − N / ) ≥ − e √ η √ η √ η . (5.23)Taking for instance C = 62, then for all 0 < η < / ≥ C √ η ≥ e √ η √ η √ η ,which implies the result.We are now able to bound the probability of E . Corollary 5.5.
There exists
C > , such that for every r > and < s r < / (2000 r ) P ( E ) ≤ Cs / r r (5.24) for all N large enough.Proof. Define E H = {∃ e ∈ E Hv c : B ρ + e = B ρ − e } ,E V = {∃ e ∈ E Vv c : B ρ + e = B ρ − e } , (5.25)and note that E = E H ∪ E V . (5.26)By the symmetry of the problem, it is enough to show P ( E H ) ≤ Cs / r r. (5.27)Apply Corollary 5.4 with m = s r N / i.e. with η = s r r . Then P (cid:0) ( E H ) c (cid:1) = P ( A ηr − N / )and (5.22) gives the claimed result. 24 orollary 5.6. Consider the parameters satisfying Assumption 4.5 and s r r < / .Then, there exist constants c, C > such that P (cid:16) C p ( π / o,x , π y,x ) ≤ L − t r ∀ x ∈ C s r / ,t r , y ∈ R r/ , / (cid:17) ≥ − Ce − cM − e − cr − Cs / r r (5.28) for all N large enough.Proof. By Lemma 5.2 P (cid:16) ∃ x ∈ C s r / ,t r : C p ( π ρ + o,x , π ρ − o,x ) ≥ I s r ,t r , B (cid:17) ≤ P ( E ) . (5.29)The result follows from Corollary 5.5 and Lemma 5.1. Proof of Theorem 2.2.
To end the proof of Theorem 2.2, we just need to express theparameters r, s r , t r , M in terms of δ so that Assumption 4.5 is in force. For a small δ > s r = 2 δ,t r = δ / / (log(1 /δ )) ,r = log(1 /δ ) ,M = log(1 /δ ) . (5.30)Let us verify the assumptions. For all δ ≤ .
05, the last inequality in (4.51) holds trueand also s r < r . Finally, for δ ≤ exp( − M ), we have M ≥ M as well. For small δ , thelargest error term in (5.28) is Cs / r r , which however goes to 0 as δ goes to 0. Remark . Of course, (5.30) is not the only possible choice of parameters. For instance,one can take t r = δ / / log(1 /δ ), but then we have to take smaller values of r and M ,e.g., r = M = p log(1 /δ ), for which the last inequality in (4.51) holds for δ ≤ . e − cr and e − cM are much slower. In this section we prove Theorem 2.5. The strategy of the proof is identical to the one ofTheorem 2.2 and thus we will focus only on the differences.
Proof of Theorem 2.5.
Here we think of the stationary LPP as leaving from the line L = { x ∈ Z | x + x = 0 } rather than the point o = (0 , Z ρ ± L ,x k rather than Z ρ ± o,x k . Next we use line-to-point versions of (4.44) and (4.48)which indeed can be obtained as { Z ρ + o,x < } = { Z ρ + L ,x < } and therefore the boundsfrom Lemma 4.1 can still be applied.Lastly, for general initial conditions, the exit point of Z h L ,x is not 0 anymore. ByAssumption 2.4 it is between − rN / and rN / with r = log( δ − ) with probability atleast 1 − Q ( δ ). Thus we replace the bound 1 − e − cr for r = log( δ − ) with 1 − Q ( δ ). Therest of the proof is unchanged. Since the geodesics have slope very close to 1, one might expect that rN / and 15 rN / should bereplaced by their half. However the statement don’t need to be changed since we did not choose theseboundary to be sharp. Upper bound for the probability of no coalescence
In this section we will prove Theorem 2.6, but for this we need some preparations.
Let a = ( a j ) j ∈ Z and s = ( s j ) j ∈ Z be two independent sequences of i.i.d. exponential randomvariables of intensity β and α respectively, where 0 < β < α <
1. We think of a j as theinter-arrival time between customer j and customer j −
1, and of s j as the service timeof customer j . The waiting time of the j ’th customer is given by w j = sup i ≤ j (cid:16) j X k = i s k − − a k (cid:17) + . (6.1)The distribution of w (and by stationarity the distribution of any w j for j ∈ Z ) is givenby P ( w ∈ dw ) := f ( dw ) = (cid:0) − βα (cid:1) δ ( dw ) + ( α − β ) βα e − ( α − β ) w dw. (6.2)The queueing map D : R Z + × R Z + → R Z + takes the sequence of interarrival times and theservice times and maps them to the inter-departure times d = D ( a , s ) ,d j = ( w j − + s j − − a j ) − + s j . (6.3)We denote by ν β,α the distribution of ( D ( a , s ) , s ) on R Z + × R Z + , that is ν β,α ∼ ( D ( a , s ) , s ) . (6.4)By Burke’s Theorem [16] D ( a , s ) is a sequence of i.i.d. exponential random variables ofintensity β , consequently, the measure ν β,α is referred to as a stationary measure of thequeue. One can write d j = e j + s j , (6.5)where e j is called the j ’th idle time and is given by e j = ( w j − + s j − − a j ) − . (6.6) e j is the time between the departure of customer j − j inwhich the sever is idle. Define x j = s j − − a j , (6.7)and the summation operator S kl = l X i = k x i . (6.8)Summing e j we obtain the cumulative idle time (see Chapter 9.2, Eq. 2.7 of [44]). Lemma 6.1 (Lemma A1 of [6]) . For any k ≤ l l X i = k e i = (cid:16) inf k ≤ i ≤ l w k − + S ki (cid:17) − . (6.9)26t has long been known that the LPP on the lattice can be seen as queues in tandem.In particular, the stationary distribution for LPP can be seen as a stationary distributionof queues in tandem. In [21] Fan an Sepp¨al¨ainen found the multi-species stationarydistribution for LPP. For 0 < β < α <
1, let I β = { I βi } i ∈ Z and I α = { I αi } i ∈ Z be two i.i.d.random sequences such that I β ∼ Exp(1 − β ) and I α ∼ Exp(1 − α ) . (6.10)Let x ∈ Z such that o = (0 , ≤ N e + x e . Let G α , G β be stationary LPP as inSection 3.3 with the weights in (6.10). Define the sequences I β,x and I α,x by I β,xi = G β (1 − t r ) N e + i e − G β (1 − t r ) N e +( i − for i > x,I α,xi = G α (1 − t r ) N e + i e − G α (1 − t r ) N e +( i − for i > x. (6.11)The multi-species results in [21], in particular Theorem 2.3 of [21], show that if we take( I α , I β ) ∼ ν − α, − β then ( I α,x , I β,x ) ∼ ν − α, − β | x + R Z + . (6.12)where ν − α, − β | x + R Z + is the restriction of ν β,α to x + R Z + . (1 − t r ) N . As main ingredient in the proof, in Proposition 6.2 below, we show that with positiveprobability the geodesics ending at N e for the stationary models with density ρ + and ρ − do not coalesce before time (1 − t r ) N .Let r > z = − r t / r N / , z = r t / r N / , and define H ρ = sup { i ∈ Z | (1 − t r ) N e + ( i, ∈ π ρo,N e } (6.13)be the exit point of the geodesic π ρo,N e with respect to the horizontal line ( Z , (1 − t r ) N ),which geometrically is at position e H ρ = (1 − t r ) N e + ( H ρ , I = I − ∪ I + where I − = { z , . . . , } , I + = { , . . . , z } . (6.14) Proposition 6.2.
Under the choice of parameters in (5.30) , there exist
C, δ > suchthat for δ < δ and N large enough P ( H ρ − ∈ I − , H ρ + > ≥ Cδ / . (6.15)Before we turn to the proof of Proposition 6.2, we need some preliminary results. Thefollowing lemma shows that with probability close to 1 / π ρo,N e will cross the interval I from its left half. Lemma 6.3.
Under (5.30) , there exists δ , c > such that for δ < δ and for largeenough N P ( H ρ − ∈ I − ) ≥ − e − cr . (6.16)27 roof. Let v = (1 − t r ) N e − r t / r N / e . Note that { Z ρ − v,Ne ∈ [0 , − z ] } = {H ρ − ∈ I − } . (6.17)Moreover P ( Z ρ − v,Ne ∈ [0 , − z ]) = P ( Z ρ − v,Ne ≤ − z ) − P ( Z ρ − v,Ne < . (6.18)The exit point Z ρv,Ne is stochastically monotone in ρ , that is, P ( Z ρv,Ne ≤ x ) ≥ P ( Z λv,Ne ≤ x ) for ρ ≤ λ. (6.19)It follows that P ( Z ρ − v,Ne ≤ − z ) ≥ P ( Z / v,Ne ≤ − z ) = P ( Z / v − z ,Ne ≤ P ( Z / − t r ) Ne ,Ne ≤
0) = 1 / , (6.20)where the last equality follows from symmetry. Consider the characteristic ρ − emanatingfrom N e and its intersection point c with the set { (1 − t r ) N e + ie } i ∈ Z . A simpleapproximation of the characteristic ( ρ − ) (1 − ρ − ) shows that c = (1 − t r ) N − rt r N / + O ( N / ) . (6.21)It follows that c − v ≥ r t / r N / − rt r N / (1 + o (1))= (cid:16) r δ (log( δ − )) − − δ / (log( δ − )) − (1 + o (1)) (cid:17) N / . (6.22)For δ small enough c − v ≥ r δ (log( δ − )) − N / (6.23)and c − v ( t r N ) / ≥ r . (6.24)It follows from Lemma 4.4 that P ( Z ρ − v,Ne < ≤ e − cr . (6.25)Plugging (6.20) and (6.25) in (6.18) and using (6.17) implies the result.Let o = N e and defineˆ I i = b G o , (1 − t r ) N e − ( i +1)e − b G o , (1 − t r ) N e − i e for i ∈ Z (6.26)and let ( a j ) j ∈ Z and ( s j ) j ∈ Z be two independent sequences of i.i.d. random variables,independent of ˆ I , such that a ∼ Exp(1 − ρ + ) , s ∼ Exp(1 − ρ − ) . (6.27)For i ∈ Z define the shifted random varaibles X i = ˆ I i − , X i = s i − , X i = a i − . (6.28)28inally define the random walks S a,xi = i X k = x X ak , for a ∈ { , , } ,S a,b,xi = S a,xi − S b,xi for a, b ∈ { , , } . (6.29)In particular, S , ,xi = i X k = x ( s k − a k ) . (6.30)We also define unbiased versions of S a,x for a ∈ { , } ¯ S ,xi = i X k = x ( s k − (1 − ρ − ) − ) , ¯ S ,xi = i X k = x ( a k − (1 − ρ + ) − ) . (6.31)A simple computation gives, for i > x , S ,xi ≤ ¯ S ,xi ≤ S ,xi + ( i − x ) rN − / + rN − / (6.32)¯ S ,xi ≤ S ,xi ≤ ¯ S ,xi + ( i − x ) rN − / − rN − / . (6.33)Our next result controls the maximum of S ,z on I . Lemma 6.4.
There exists c > such that for any fixed y > r / P (cid:16) sup z ≤ i ≤ z | S ,z i | ≤ y ( z − z ) / (cid:17) ≥ − Ce − c ( y − r / ) − e − cr . (6.34) for all N large enough.Proof. Let λ ± = 12 ± r ( t r N ) − / . (6.35)We first show that P ( b Z λ + N e ,x > , b Z λ − N e ,x < ∀ x ∈ I ) ≥ − e − cr . (6.36)We will show that P ( b Z λ + N e ,x > ∀ x ∈ I ) ≥ − e − cr , (6.37)a similar result can be shown for ˆ Z λ − N e ,x and the result follows by union bound. Note that P ( b Z λ + N e ,x > ∀ x ∈ I ) = P ( b Z λ + N e , (1 − t r ) Ne + z e > . (6.38)Consider the characteristic λ + emanating from N e and its intersection point d with theset { (1 − t r ) N e + ie } i ∈ Z . A simple approximation of the characteristic ( λ + ) (1 − λ + ) showsthat d = (1 − t r ) N + 8 r ( t r N ) / + O ( N / ) ≥ (1 − t r ) N + 2 r ( t r N ) / (6.39)29or all N large enough. It follows that d − [(1 − t r ) N + z ] ≥ r ( t r N ) / . (6.40)As in previous proofs, applying Lemma 4.4 we conclude (6.37) and therefore (6.36).Define ˆ I λ − i = G λ − Ne , (1 − t r ) Ne + ie − G λ − Ne , (1 − t r ) Ne +( i +1) e , ˆ I λ + i = G λ + Ne , (1 − t r ) Ne + ie − G λ + Ne , (1 − t r ) Ne +( i +1) e . (6.41)Using the Comparison Lemma, see Section 3.5, we obtain P ( ˆ I λ − i ≤ ˆ I i ≤ ˆ I λ + i i ∈ I ) ≥ − e − cr . (6.42)Therefore, we also have P (cid:16) i X k = z ( ˆ I λ − i − ≤ S ,z i ≤ i X k = z ( ˆ I λ + i − (cid:17) ≥ − e − cr . (6.43)Denote b S i := P ik = z ( ˆ I λ + k − (1 − λ + ) − ), which is a martingale starting at time i = z .Then P (cid:16) sup z ≤ i ≤ z i X k = z ( ˆ I λ + k − ≤ yr / ( t r N ) / (cid:17) ≥ P (cid:16) sup z ≤ i ≤ z b S i + ( z − z )((1 − λ + ) − − ≤ yr / ( t r N ) / (cid:17) ≥ P (cid:16) sup z ≤ i ≤ z r − / ( t r N ) − / b S i ≤ y − r / (cid:17) , (6.44)where in the third line we used( z − z )((1 − λ + ) − −
2) = 8 r ( t r N ) / + O (1) ≤ r ( t r N ) / (6.45)for all N large enough. The scaling is chosen such that, setting i = z + 2 τ r ( t r N ) / , thescaled random walk r − / ( t r N ) − / b S i converges as N → ∞ weakly to a Brownian motionon the interval τ ∈ [0 ,
1] for some (finite) diffusion constant. Thus there exists constants
C, c > y > r / P (cid:16) sup z ≤ i ≤ z i X k = z ( ˆ I λ + k − ≤ yr / ( t r N ) / (cid:17) ≥ − Ce − c ( y − r / ) (6.46)for all N large enough. Similarly we show that for any given y > r / P (cid:16) inf z ≤ i ≤ z i X k = z ( ˆ I λ − k − ≥ − yr / ( t r N ) / (cid:17) ≥ − Ce − c ( y − r / ) (6.47)for all N large enough. From (6.43), (6.46) and (6.47) it follows that P (cid:16) sup z ≤ i ≤ z | S ,z i | ≤ y ( z − z ) / (cid:17) = P (cid:16) sup z ≤ i ≤ z | S ,z i | ≤ yr / ( t r N ) / (cid:17) ≥ P (cid:16) inf z ≤ i ≤ z i X k = z ( ˆ I λ − k − ≥ − yr / ( t r N ) / , sup z ≤ i ≤ z i X k = z ( ˆ I λ + k − ≤ yr / ( t r N ) / (cid:17) ≥ − e − c ( y − r / ) − e − cr . (6.48)30ext we control the fluctuations of the random walk S ,z . Lemma 6.5.
Let C = { sup z ≤ i ≤ z | S ,z i | ≤ y ( z − z ) / } . Under the choice of parametersin (5.30) , there exists c, δ > such that for δ < δ , and any fixed y > r / P ( C ) > − Ce − c ( y − r / ) (6.49) for all N large enough.Proof. By (6.32) we have P ( C ) > P (cid:16) sup z ≤ i ≤ z | ( z − z ) − / ¯ S ,z i | ≤ y − ( z − z ) / rN − / + rN − / (cid:17) . (6.50)Note that( z − z ) / rN − / + rN − / = (2 r ) / ( t r N ) / rN − / + rN − / = (2 r ) / t / r r + rN − / = (2 r ) / δ / + rN − / . (6.51)Thus for all N large enough and δ small enough P ( C ) > P (cid:16) sup z ≤ i ≤ z | ( z − z ) − / ¯ S ,z i | ≤ y − r / (cid:17) . (6.52)Also notice that ( z − z ) − / ¯ S ,z i converges weakly to a Brownian motion as N → ∞ .Using Doob maximum inequality one deduces that for N large enough (6.49) indeedholds.For M > E = n sup i ∈ I | S ,z i | ≤ M ( z − z ) / o . (6.53)For x >
0, define the sets E = n H ρ − ∈ I − , sup i ∈ I | S , ,z i | ≤ M ( z − z ) / o , E ,x = n inf i ∈ I − ( x + S , ,z i ) > − M ( z − z ) / , inf i ∈ I + ( x + S , ,z i ) < − M ( z − z ) / o . (6.54)Note that on the event E S , ,z i > − M ( z − z ) / − S ,z i and S , ,z i < M ( z − z ) / − S ,z i , (6.55)which implies E ∩ E ⊇ {H ρ − ∈ I − , sup i ∈ I | S ,z i | ≤ M ( z − z ) / } ∩ E . (6.56)Similarly, on the event E S , ,z i > − M ( z − z ) / − S ,z i and S , ,z i < M ( z − z ) / − S ,z i , (6.57)31o that, for x ∈ [0 , ( z − z ) / ] and M ≥ E ,x ∩ E ⊇ { inf i ∈ I − ( x − S ,z i ) > − M ( z − z ) / , inf i ∈ I + ( x − S ,z i ) < − M ( z − z ) / } ∩ E ⊇ { sup i ∈ I − S ,z i < M ( z − z ) / , sup i ∈ I + S ,z i > M ( z − z ) / } ∩ E ⊇ { sup i ∈ I − S ,z i < M ( z − z ) / , sup i ∈ I + S ,z i − S ,z > M ( z − z ) / − S ,z } ∩ E ⊇ { sup i ∈ I − | S ,z i | < M ( z − z ) / , sup i ∈ I + S , i > M ( z − z ) / } ∩ E ⊇ { sup i ∈ I − | S ,z i | < M ( z − z ) / , S , z > M ( z − z ) / } ∩ E , (6.58)as in the first line we used (6.57), in the second x = 0 for the first term and x = ( z − z ) / for the second one.Next we are going to prove that, conditioned on E , E and E ,x occurs with positiveprobability. Lemma 6.6.
There exists c , r > and M ≥ such that for x ∈ [0 , ( z − z ) / ] P ( E , E ,x |E ) > c . (6.59) for all N large enough.Proof. Define F = {H ρ − ∈ I − , sup i ∈ I | S ,z i | ≤ M ( z − z ) / } , F = { sup i ∈ I − | S ,z i | < M ( z − z ) / , S , z > M ( z − z ) / } . (6.60)By (6.56) and (6.58) and the independence of S ,z and S ,z P ( E , E ,x |E ) ≥ P ( F , F |E ) = P ( F ) P ( F ) . (6.61)Next we want to derive lower bounds for P ( F ) and P ( F ).Note that P ( F ) ≥ P ( H ρ − ∈ I − ) − P (sup i ∈ I | S ,z i | ≥ M ( z − z ) / ) . (6.62)By Lemma 6.3 and Lemma 6.4 there exists r > M > r / for which P ( H ρ − ∈ I − ) ≥ / P (sup i ∈ I | S ,z i | ≥ M ( z − z ) / ) ≤ / , (6.63)so that P ( F ) ≥ / . (6.64)Let us now try to find a lower bound for P ( F ). P ( F ) = P (cid:16) sup i ∈ I − | S ,z i | < M ( z − z ) / , S , z > M ( z − z ) / (cid:17) (6.65)= P (cid:16) sup i ∈ I − | S ,z i | < M ( z − z ) / (cid:17) P (cid:16) S , z > M ( z − z ) / (cid:17) (6.66)32ince the processes { S ,z i } i ∈ I − and { S , i } i ∈ I + are independent. Note that the centeredrandom walks { ( z − z ) − / ¯ S ,z i } i ∈ I − and { ( z − z ) − / ¯ S , i } i ∈ I + converge weakly to aBrownian motion. Furthermore, the difference coming from the non-zero drift of S ,z i is,by (6.33), bounded bysup i ∈ I | ( z − z ) − / S ,z i − ( z − z ) − / ¯ S ,z i | ≤ ( z − z ) / rN − / − rN − / (6.67)which is o (cid:16) ( z − z ) / (cid:17) as δ → M large enough,there exists c > N large enough P (cid:16) sup i ∈ I − | S ,z i | < M ( z − z ) / (cid:17) ≥ / , P (cid:16) S , z > M ( z − z ) / (cid:17) ≥ c . (6.68)I follows that P ( F ) ≥ c (6.69)Plugging (6.64) and (6.69) in (6.61) we obtain the result.Now we can prove the main statement of this section. Proof of Proposition 6.2.
Note that n H ρ − ∈ I − , sup i ∈ I − i X k = z ( ˆ I ρ + k − ˆ I k ) < sup i ∈ I + i X k = z ( ˆ I ρ + k − ˆ I k ) o ⊆ n H ρ − ∈ I − , H ρ + > o . (6.70)Indeed, if H ρ − ∈ I − then also H ρ + ≥ z , and the second condition implies that H ρ + I − .Using a decomposition as in (6.5), we can write n H ρ − ∈ I − , sup i ∈ I − i X k = z ( ˆ I ρ + k − ˆ I k ) < sup i ∈ I + i X k = z ( ˆ I ρ + k − ˆ I k ) o = n H ρ − ∈ I − , sup i ∈ I − i X k = z e k + i X k = z ( ˆ I ρ − k − ˆ I k ) < sup i ∈ I + i X k = z e k + i X k = z ( ˆ I ρ − k − ˆ I k ) o (6.71)where e j = ˆ I ρ + j − ˆ I ρ − j . By (6.9) the following event has the same probability as (6.71) E = n H ρ − ∈ I − , sup i ∈ I − (cid:16) inf z ≤ l ≤ i w z − + S , ,z l (cid:17) − + S , ,z i < sup i ∈ I + (cid:16) inf z ≤ l ≤ i w z − + S , ,z l (cid:17) − + S , ,z i o . (6.72)It follows that P ( H ρ − ∈ I − , H ρ + > ≥ P ( E ) . (6.73)For x >
0, define E ,x = n sup i ∈ I − S , ,z i > sup i ∈ I + S , ,z i , sup i ∈ I − (cid:16) inf z ≤ l ≤ i x + S , ,z l (cid:17) − + S , ,z i < sup i ∈ I + (cid:16) inf z ≤ l ≤ i x + S , ,z l (cid:17) − + S , ,z i o . (6.74)33ote that E ∩ E ∩ E ,x ⊆ E ,x . (6.75)In our case, the value of x in E ,x is random and distributed according to (6.2).Therefore we have P ( E ) = Z ∞ P ( E | w z − = w ) f ( dw ) = Z ∞ P ( E ,w ) f ( dw ) ≥ Z ∞ P ( E , E , E ,w ) f ( dw ) ≥ Z ( z − z ) / P ( E , E , E ,w ) f ( dw ) , (6.76)where in the second equality we used the fact that the processes { S i,z j } j ≥ z ,i ∈{ , , } areindependent of w z − .Taking M large enough, Lemma 6.5 gives P ( E ) ≥ / N large enough. (6.77) and Lemma 6.6 imply that there exists c > w ∈ [0 , ( z − z ) / ], δ < δ , and large enough N P ( E , E , E ,w ) ≥ c . (6.78)Plugging (6.78) in (6.76) P ( E ) ≥ c (cid:16) − − rN − / + rN − / e − rN − / ( z − z ) / (cid:17) . (6.79)Note that 2 rN − / ( z − z ) / = 2 r (2 r ) / t / r = 2 / r / δ / → r δ → δ →
0. Then by first order approximation of the exponential function, thereexists
C > δ < δ , r ≤ δ − (log δ − ) − , and large enough N P ( E ) ≥ Cδ / . (6.81)Using (6.81) in (6.73) we obtain the result. Finally we prove the second main result of this paper.
Proof of Theorem 2.6.
Let ρ ′ + = 12 + 1120 rN − / , ρ ′− = 12 − rN − / , (6.82)and define A ′ = n − r N / ≤ Z ρ ′− o,N e ≤ Z ρ ′ + o,N e ≤ r N / o . (6.83)By Lemma 4.4 P ( A ′ ) ≥ − e − cr (6.84)Define y = 14 rN / e , y = 14 rN / e . (6.85)34et o = N e . Similar to (6.13) we define H j = sup { i : ( i, (1 − t r ) N ) ∈ π y j ,o } for i ∈ { , } . (6.86)Note that on the event A ′ , the geodesics π ρ ′− o,o , π ρ ′ + o,o are sandwiched between the geodesics π y ,o , π y ,o , which implies that if the geodesics π ρ ′− o,o , π ρ ′ + o,o did not coalesce then neitherdid π y ,o , π y ,o i.e. A ′ ∩ {H ρ ′− ∈ I − , H ρ ′ + > } ⊆ { C p ( π y ,o , π y ,o ) > L − t r } . (6.87)Indeed, on the event A ′ − y < Z ρ − o,o ≤ Z ρ + o,o ≤ y (6.88)so that π y ,o (cid:22) π ρ ′− o,o (cid:22) π ρ ′ + o,o (cid:22) π y ,o (6.89)which implies that under A ′ ∩ {H ρ ′− ∈ I − , H ρ ′ + > } L − t r ≤ C p ( π ρ ′ + o,o , π ρ ′− o,o ) ≤ C p ( π y ,o , π y ,o ) (6.90)Note that as y , y ∈ R r/ , / and o ∈ C s r / ,t r { C p ( π y ,o , π y ,o ) > L − t r } ⊆ {∃ x ∈ C s r / ,t r , y ∈ R r/ , / : C p ( π / o,x , π y,x ) > L − t r } . (6.91)Indeed, if the geodesics π y ,o and π y ,o do not meet before the time horizon L − t r , atleast one of the them did not coalesce with the geodesic π / o,o before L − t r . It follows from(6.87), (6.91) and (6.84) that P ( ∃ x ∈ C s r / ,t r , y ∈ R r/ , / : C p ( π / o,x , π y,x ) > L − t r ) ≥ Cδ / . (6.92) Proof of Theorem 2.7.
Note that { C p ( π / o,x , π y,x ) ≤ L − τ ∀ x ∈ C δ,τ , y ∈ R log δ − , / }⊆ { C p ( π w,x , π y,x ) ≤ L − τ ∀ x ∈ C δ,τ , w, y ∈ R log δ − , / } . (6.93)Indeed, on the event that any geodesic starting from R log δ − , / and terminating in C δ,τ coalesces with the stationary geodesic before the time horizon L − τ any two geodesicsstarting from R log δ − , / and terminating in C δ,τ must coalesce as well. Theorem 2.2and (6.93) imply the lower bound in Theorem 2.7.Next note that {∃ x ∈ C δ,τ , y ∈ R log δ − , / : C p ( π / o,x , π y,x ) > L − τ , | Z / o,x | ≤
116 log( δ − ) N / }⊆ { C p ( π w,x , π y,x ) ≤ L − τ ∀ x ∈ C δ,τ , w, y ∈ R log δ − , / } c . (6.94)To illustrate the validity of (6.94), assume w.l.o.g. that x = N e , y = o such that C p ( π / o,e N , π o,e N ) > L − τ and that log( δ − ) N / ≥ Z / o,Ne = a >
0. It follows that C p ( π ae ,Ne , π o,Ne ) > L − τ (6.95)35olds. The event in (6.95) is contained in the event in the last line of (6.94) which implies(6.94).(6.94) implies that P (cid:16) C p ( π / o,x , π y,x ) ≤ L − τ ∀ x ∈ C δ,τ , y ∈ R log δ − , / (cid:17) + P (cid:16) | Z / o,x | >
116 log( δ − ) N / for some x ∈ C δ,τ (cid:17) ≥ P (cid:16) C p ( π w,x , π y,x ) ≤ L − τ ∀ x ∈ C δ,τ , w, y ∈ R log δ − , / (cid:17) . (6.96)Next we claim that for some c > P (cid:16) | Z / o,x | >
116 log( δ − ) N / for some x ∈ C δ,τ (cid:17) ≤ e − c log( δ − ) . (6.97)Indeed, it follows by (4.71) with r =
115 116 log( δ − ) that P (cid:16) Z / o,x >
116 log( δ − ) N / for some x ∈ C δ,τ (cid:17) ≤ P (cid:16) Z ρ + o,x >
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