Geodesics, bigeodesics, and coalescence in first passage percolation in general dimension

We consider geodesics for first passage percolation (FPP) on $\mathbb{Z}^d$ with iid passage times. As has been common in the literature, we assume that the FPP system satisfies certain basic properties conjectured to be true, and derive consequences from these properties. The assumptions are roughly as follows: (i) the fluctuation scale $\sigma(r)$ of the passage time on scale $r$ grows approximately as a positive power $r^\chi$, in the sense that two natural definitions of $\sigma(r)$ and $\chi$ yield the same value $\chi$, and (ii) the limit shape boundary has curvature uniformly bounded away from 0 and $\infty$ (a requirement we can sometimes limit to a neighborhood of some fixed direction.) The main a.s. consequences derived are the following, with $\nu$ denoting a subpolynomial function and $\xi=(1+\chi)/2$ the transverse wandering exponent: (a) for one-ended geodesic rays with a given asymptotic direction $\theta$, starting in a natural halfspace $H$, for the hyperplane at distance $r$ from $H$, the density of"entry points"where some geodesic ray first crosses the hyperplane is at most $\nu(r)/r^{(d-1)\xi}$, (b) the system has no bigeodesics, i.e. two-ended infinite geodesics, (c) given two sites $x,y$, and a third site $z$ at distance at least $\ell$ from $x$ and $y$, the probability that the geodesic from $x$ to $y$ passes through $z$ is at most $\nu(\ell)/\ell^{(d-1)\xi}$, and (d) in $d=2$, the probability that the geodesic rays in a given direction from two sites have not coalesced after distance $r$ decays like $r^{-\xi}$ to within a subpolynomial factor. Our entry-point density bound compares to a natural conjecture of $c/r^{(d-1)\xi}$, corresponding to a spacing of order $r^\xi$ between entry points, which is the conjectured scale of the transverse wandering.

for which every finite subpath is a geodesic. A doubly infinite path with the same property is called a bigeodesic.
In the spirit of various past works on FPP ( [11], [24], [25], [30]), we take as assumptions a few basic properties believed to hold generally, but unproven for any specific FPP model. One such property states that fluctuations of passage times have an exponential bound on the scale of their standard deviation; a second says that the boundary of the limit shape has uniform curvature in a certain sense. We also assume for technical purposes that this standard deviation behaves mildly regularly as a function of distance and direction. See A2 below for details. Our purpose is to show how certain relatively strong conclusions about geodesic behavior, including aspects of coalescence, flow from essentially just the basic properties.
We will use percolation "site/bond" terminology in the lattice Z d , rather than "vertex/edge." For d = 2 it is known for lattice FPP that under (arguably) mild hypotheses, (i) every geodesic ray has an asymptotic direction [29]; (ii) for a fixed direction θ, with probability 1, there is exactly one geodesic ray with asymptotic direction θ starting from a given site ( [1], [24], [29]); (iii) for a fixed direction θ, with probability 1, for all sites x, y, the geodesic rays with asymptotic direction θ from x and y eventually coalesce ( [13], [24]); (iv) for a fixed direction θ, with probability 1 there is no bigeodesic for which either asymptotic direction is θ ([1], [14]; a weaker form is in [24]). Here by asymptotic direction we mean the value where v 0 , v 1 , . . . are the sites, in order, of the geodesic ray and S d−1 is the (d − 1)-sphere, and | · | denotes Euclidean length. Note that (iv) does not rule out the existence of all bigeodesics, as it allows a random null set of θ values for which such bigeodesics exist. We will call a geodesic ray with asymptotic direction θ a θ-ray. Given a halfspace H we call a θ-ray a halfspace θ-ray from H if its first site, and no other, is contained in H; here r ∈ R and α ∈ S d−1 . We may omit the "from H" if the appropriate halfspace is either apparent or not relevant. For d ≥ 3, under our hypotheses we will (among other things) prove (i), and prove (ii) with "at least one" in place of "exactly one," but it's not clear whether (iii) and the "exactly one" in (ii) should be true. Assuming continuous distributions of passage times to prevent ties between paths, a priori, for any two geodesic rays Γ,Γ, any of 3 things may happen: (1) Γ,Γ are disjoint, i.e. they have no bonds in common; (2) Γ,Γ coalesce, that is, there is a site v ∈ Γ ∩Γ such that the segments of Γ,Γ up to v are disjoint, and the two rays from v onwards coincide; (3) Γ ∩Γ is a single segment consisting of finitely many bonds. We refer to the phenomenon in (3) as temporary touching; when it occurs, the last site in the segment is called a branching site; see Figure 1. Despite the complications temporary touching and branching create, we can still quantify some aspects of the coalescence of θ-rays in the following way. Below we will associate to each direction θ a vector z θ , chosen so the hyperplane {x ∈ R d : x·z θ = 1} is tangent to the unit ball of the norm associated to the FPP process at the boundary point of the ball in direction θ. We define the hyperplanes and halfspaces H θ,s = {x ∈ R d : x · z θ = s}, H + θ,s = {x ∈ R d : x · z θ ≥ s}, H − θ,s = {x ∈ R d : x · z θ ≤ s}. For θ, θ 0 ∈ S d−1 (close together), consider the H + θ 0 ,s -entry points of θ-rays, meaning those lattice sites in H + θ 0 ,s where some θ-ray from H − θ 0 ,0 first intersects H + θ 0 ,s . We may ask, what is the density of such H + θ 0 ,s -entry points per unit volume near H θ 0 ,s , and how does it decrease as s → ∞? Does it approach 0, and if so how fast? We call this density the H θ 0 ,s -crossing density, postponing a precise definition for later.
Note that θ-rays passing through different H θ 0 ,s -entry points cannot be assumed disjoint up to those entry points, due to the possibilities of temporary touching and branching.
If all halfspace θ-rays from H − θ 0 ,0 coalesce a.s., their H θ 0 ,s -crossing density must approach 0 as s → ∞. The converse is true for d = 2, but for d ≥ 3 the H θ 0 ,s -crossing density approaching 0 does not in itself guarantee coalescence of all the θ-rays. We can make equivalence classes of halfspace θ-rays from H − θ 0 ,0 by writing Γ ∼Γ if Γ,Γ eventually coalesce; the H θ 0 ,s -crossing density will approach 0 if the number of equivalence classes is finite, but it is not clear that the converse holds. One could also ask about the existence of finite equivalence classes; if they exist with positive probability then their starting points must have a positive density near H θ 0 ,0 , but again the possibility of temporary touching means such a positive density is not immediately ruled out by the H θ 0 ,s -crossing density approaching 0.
For d = 2, we can predict the H θ 0 ,s -crossing density heuristically from the transverse wandering exponent of geodesics, that is, the value ξ such that for a geodesic of length s, the maximum distance of the geodesic from the straight line connecting its endpoints is of order s ξ . There must exist halfspace θ-rays, one passing through each H + θ 0 ,s -entry point, and any such rays must remain disjoint at least until they cross H θ 0 ,s , since there is a.s. no branching or temporary touching for a fixed θ in two dimensions. Heuristically, to remain disjoint until H + θ 0 ,s the θ-rays should be spaced apart by order s ξ , so the H θ 0 ,s -crossing density should be s −ξ . For d ≥ 3 this predicts an H θ 0 ,scrossing density of at least s −(d−1)ξ , but it is not clear a priori that the crossing density shouldn't be greater, since the θ-rays can weave around one another without meeting, and we cannot rule out the branching of some θ-rays each into multiple θ-rays.
We will show that in fact the H θ 0 ,s -crossing density approaches 0 faster than s −(d−1)ξ+ for all > 0; one can in fact replace the factor s here with a large power of log s. Along the way we will obtain results about the regularity of geodesics, and their transverse fluctuations. We will follow a heuristic of Newman (presented at the AIM 2015 workshop "First-passage percolation and related models") in using the convergence to 0 of the crossing density to show nonexistence of bigeodesics. Reforumlated to our context, it goes roughly as follows: suppose the crossing-point density bound holds not just for each single geodesic ray direction, say θ 0 -rays, but for the union of all θ-rays over |θ − θ 0 | ≤ for some , for each fixed θ 0 ; we'll call these near-θ 0 -rays. It is known under (again arguably) mild hypotheses that every geodesic ray has an asymptotic direction, so any bigeodesic must have an asymptotic direction "each way"; the two directions should always be ±θ for some θ. Let G θ 0 , be the set of all bigeodesics which have (by some directional labeling) forward asymptotic direction θ and backward direction −θ, with |θ − θ 0 | ≤ . Then a geodesic in G θ 0 , has a well-defined H + θ 0 ,s -entry point for all s ∈ R. For R > 0 consider the set P θ 0 ,R = x ∈ H + θ 0 ,R : x is the H + θ 0 ,R -entry point of some bigeodesic in G θ 0 , .
The density of P θ 0 ,R is bounded above by the density near H θ 0 ,R of entry points of halfspace nearθ 0 -rays, so by our assumption it approaches 0 as R → ∞. But by translation invariance, P θ 0 ,R has the same density as P θ 0 ,0 for all R, so the density of P θ 0 ,0 must be 0. By stationarity this means G θ 0 , = ∅ a.s. If this holds for all θ 0 , then by compactness, there are a.s. no bigeodesics.
The preceding heuristic assumes we know that the H θ 0 ,s -crossing density approaches 0 as s → ∞; a second heuristic for why that should be true is as follows. Suppose the transverse fluctuations of a typical geodesic of length s are of some order ∆ s , behaving roughly like s ξ . We can divide H θ 0 ,0 and H θ 0 ,s each into blocks of size ∆ s . If the H θ 0 ,s -crossing density of near-θ 0 rays is much more than ∆ −(d−1) s (i.e. "one entry point per block") then a typical block in H θ 0 ,s has many H + θ 0 ,s -entry points near it. This should mean we can find many pairs of blocks, say B 0 in H θ 0 ,0 and B s in H θ 0 ,s , for which there are a large number of near-θ 0 rays from B 0 passing through B s with distinct H + θ 0 ,s -entry points. The distinct entry points means none of these near-θ 0 rays have coalesced when reaching H θ 0 ,s despite starting and ending their paths from H θ 0 ,0 to H θ 0 ,s within ∆ s of each other. (Note such coalescence is "temporary" if two near-θ 0 rays have different asymptotic directions.) Thus a primary ingredient is to show there is a low probability of such overly-densely-packed nearly-parallel geodesics.
In addition to ξ, the other exponent of central interest is the χ for which the standard deviation of the passage time over distance r "grows like r χ ." Our precise assumptions related to this standard deviation are given in A2 below. It is known [11] that under "reasonable" hypotheses and definitions, χ, ξ are related by χ = 2ξ − 1.
As mentioned above, our assumptions of basic model properties cannot be verified for any specific model of FPP; the best known exponential bounds are on scale r 1/2 ( [20], [34], [15]) whereas for d = 2 the conjectured value of χ is 1/3. An exponential bound on scale r 1/3 is known for certain integrable models of last passage percolation (LPP) for d = 2, however-in [8] (extracted from [4]) and [23] for LPP on Z 2 with exponential passage times, in [26], [27] for LPP based on a Poisson process in the unit square, and in [12] for LPP on Z 2 with geometric passage times. For d ≥ 3 there is no generally-agreed-upon value of χ in the physics literature. Heuristics and simulations suggest that χ should decrease with dimension; simulations in [32] for a model believed to be in the same (KPZ) universality class as FPP show a decrease from χ = .33 to χ = .054 as d increases from 2 to 7. Some have predicted the existence of a finite upper critical dimension, possibly as low as 3.5, above which χ = 0 ( [16], [22]); others predict that χ is positive for all d ( [3], [28]), with simulations in [21] showing χ > 0 all the way to d = 12, decaying approximately as 1/(d + 1). Our results here require χ > 0 so they only have content below the upper critical dimension, should it be finite.
In [6], Basu, Hoffman, and Sly show that there are no bigeodesics for last-passage percolation (LPP) on Z 2 when passage times are exponential, essentially by following Newman's heuristic of bounding the density of entry points, which in turn involves bounding the probability of overlydensely-packed parallel geodesics. (See also the earlier papers [31], [19], and see [5] for a proof avoiding results from integrable probability.) The paper [6] exploits key ingredients not available in our general context-the restriction to d = 2, the exponential bound on the scale of the standard deviation in [8] (extracted from [4]), and the fact that the rescaled passage time distributions converge to a limit (Tracy-Widom) which has negative mean. We need here a completely different heuristic and proof to control overly-densely-packed geodesics, and this is the core of our main proof; see Remark 4.1.
In two dimensions one can use bounds on the probability of overly-densely-packed geodesics to bound the probability of non-coalescence before traveling distance cr, for θ-rays which start at separation r ξ . For the integrable case of LPP in d = 2 with exponential passage times, such a bound on non-coalescence probabilities was proved in [7]. But again, strong use is made of d = 2 and bounds obtained through integrable probability, so the methods do not apply in our context. The results are at the optimal rate, analogous to removing the powers of log in our Theorems 1.5 and 1.7. The reliance on integrable probability was removed in [33], but the results are still restricted to LPP in d = 2 with exponential passage times.
Let us now formalize our definitions. Let E d denote the set of all bonds (i.e. nearest-neighbor pairs) of Z d . The passage times of bonds are a collection of nonnegative iid variables τ = {τ e : e ∈ E d }. For x, y ∈ V , a (self-avoiding) path Γ from x to y is a finite sequence of alternating sites and bonds of G, of the form Γ = (x = x 0 , x 0 , x 1 , x 1 , . . . , x n−1 , x n−1 , x n , x n = y), with all x i distinct; we may identify a path by just the sequence of sites, or view it as a string of line segments, as convenient, when clear from the context. T (x, y) := inf{T (Γ) : Γ is a path from x to y in Z d }.
A path Γ achieving the infimum in (1.1) is called a geodesic from x to y. For technical reasons we extend (1.1) to x, y ∈ R d as follows. Define Z : R d → Z d by Z(x) = z for all z ∈ Z d and x ∈ z + [−1/2, 1/2) d , where + denotes translation, and set T (x, y) = T (Z(x), Z(y)), x, y ∈ R d .
We assume the following.
(i) τ e is a continuous random variable.
(ii) There exists λ > 0 such that Ee λτe < ∞. A1 guarantees that there is exactly one geodesic from x to y a.s., for each x, y; we denote it Γ xy . As is standard, since passage times T (x, y) are subadditive, assumption (ii) guarantees the a.s. existence (positive and finite for x = 0) of the limit for x ∈ Z d ; g extends to x with rational coordinates by considering only n with nx ∈ Z d , and then to a norm on R d by uniform continuity. We let B g denote the unit ball of g, and write y θ for the positive multiple of θ which lies in ∂B g (so g(y θ ) = 1.) The tangent hyperplane to ∂B g at y θ will be unique under our hypotheses, and there is a vector z θ such that this tangent hyperplane is is a geodesic ray if every finite segment of Γ is a geodesic. Given θ in the sphere S d−1 , we say Γ is a θ-ray if lim n x n /|x n | = θ.
Throughout the paper, c 1 , c 2 , . . . and C 1 , C 2 , . . . , and 0 , 1 , . . . represent unspecified constants which depend only on d and the distribution of the passage times τ e . We use C i for constants which occur outside of proofs and may be referenced later; any given C i has the same value at all occurrences. We use c i for those which do not recur and are only needed inside one proof. For the c i 's we restart the numbering with c 0 in each proof, and the values are different in different proofs.
To avoid technical clutter, at various times we will assume (sometimes tacitly) that certain points of R d are lattice sites, and certain (large) real numbers are integers; the modifications to be made when this fails are trivial.
As mentioned above, we cannot formally establish simple hypotheses on the distribution of τ e under which our conclusions hold, due to the lack of any results establishing an exponential bound on the scale of the standard deviation. Instead we will assume certain "macroscopic" properties which one expects to be consequences of such hypotheses, as follows. To that end, we call a nonnegative function {σ(r) = σ r , r > 0} powerlike (with exponent χ) if there exist 0 < γ 1 < χ < γ 2 and constants C i such that If (1.2) holds with γ 2 < 1 we say σ(·) is sublinearly powerlike. Note that (1.2) implies that for all r, s ≥ C 1 , A2. System properties.
When A2(ii) holds, for y ∈ H θ,1 with |y − y θ | ≥ 0 we have by (1.6) and convexity that so for all y ∈ H θ,1 , is powerlike, then so is the nondecreasing functionσ(r) = sup s≤r σ(s), and if A2(ii) holds for σ(·) then it also holds forσ(·). By (1.2) we have C 2 ≤ σ(r)/σ(r) ≤ 1 for all r. By further increasingσ (though by at most a constant factor) we may make it strictly increasing and continuous while still preserving these properties. Therefore without loss of generality we always assume σ(·) is strictly increasing and continuous. In addition, under (1.4), (1.5) is equivalent to the assumption that there exist constants η < 1/2, C 10 > 0 such that for all x, y ∈ R d with |x − y| ≥ C 4 the following both hold: Remark 1.
2. An equivalent way to state the local curvature condition A2(ii) is as follows. Let B r (x) denote the closed Euclidean ball of radius r centered at x, and recall z θ is perpendicular to the tangent plane H θ,1 to B g at y θ . Define the cones There exist contants C 11 < C 12 as follows. For all θ ∈ J(θ 0 , 0 ) ∩ S d−1 we have Equation (1.11) says that near y θ , B g is sandwiched between two balls which also have tangent plane H θ,1 at y θ . From this we see that for some 1 determined by 0 , the angle ψ z θ ,zα between z θ and z α (equivalently between hyperplanes H θ,0 and H α,0 ) grows at most linearly in |θ − α|: and also It follows from A2(i) that var(T (x, y)) is of order σ 2 (|x − y|). When A2(i) holds for some σ(·), the corresponding transverse wandering function is given by Since σ(·) is continuous and increasing, the inverse function ∆ −1 is well-defined. For ξ = (1+χ)/2 ∈ ( 1 2 , 1) we have ∆(r) r ξ and ∆ −1 (a) a 1/ξ in the sense that To motivate the definition of ∆(r), consider two sites x, y separated by distance r, and a third site z at some distance ∆ r from the line segment connecting x to y, somewhere near the middle of this line. The Euclidean distance via z, that is, |y − z| + |z − x|, exceeds the "straight" distance |y − x| by order ∆ 2 /r, and under the local curvature assumption A2(ii), the same will be true for the distance in the norm g. Heuristically, the geodesic may nonetheless pass through z, meaning T (x, z) + T (z, y) = T (x, y), if the fluctuation scale σ r of the random "distance" T (·, ·) is larger than the deterministic excess distance, that is, σ r ≥ ∆ 2 /r or equivalently ∆ ≤ ∆(r).
When H θ,0 is rationally oriented, we can apply the multidimensional ergodic theorem (see [18], Appendix 14.A) to obtain the existence of a (nonrandom) crossing density for θ-rays. For other θ an additional argument would be required for this; since we are primarily interested in upper bounds, we avoid the matter by defining the crossing density as a lim sup.
Let L θ (u) denote the line though a point u in direction θ, and write L θ for L θ (0). The θ-projection of a point x into a hyperplane H θ,s is the projection along L θ .
For A ⊂ H θ,s let C θ,s (A) be the set of all H + θ,s -entry points x of halfspace θ-rays from H − θ,0 , for which the θ-projection of x into H θ,s lies in A. Similarly let C J(θ, ),s (A) be the set of all sites x which are H + θ,s -entry points of halfspace α-rays from H − θ,0 for some α ∈ J(θ, ), for which the θ-projection of x into H θ,s lies in A. Formally, the mean H θ,s -crossing density ρ θ (s) for θ-rays, and the mean combined H θ,s -crossing density ρ J(θ, ) (s), are given by (1.15) ρ θ (s) = lim sup Existence of a limit in (1.15) and (1.16) follows here from periodicity in x of P (x ∈ C θ,s (R d )) and P (x ∈ C J(θ, ),s (R d ), and the equality with almost sure limits follows from the multidimensional ergodic theorem (see [18], Appendix 14.A.) The following is our main result.
(i) With probability 1, for all θ ∈ J(θ 0 , 2 ) and x ∈ Z d , there is at least one θ-ray from x.
(ii) There exist constants C i such that and for all θ ∈ J(θ 0 , 2 ), (iii) With probability 1, there exists no bigeodesic containing a subsequential θ-ray with θ ∈ J(θ 0 , 2 ). (iv) Suppose also A2(ii') holds. Then with probability 1, (a) every geodesic ray has an asymptotic direction, (b) for every θ ∈ S d−1 and every x ∈ Z d there is at least one θ-ray from x, and (c) there are no bigeodesics.
We will prove (i), (iv)(a), and (iv)(b) in Section 3, (ii) in Section 5, and (iii) and (iv)(c) in Section 6. Theorem 1.5(iv) improves on existing results even for d = 2 (though under somewhat stronger hypotheses), as it rules out bigeodesics in all directions simultaneously, instead of almost surely for a fixed direction as in [1], [14], [24].
As we have noted, one expects the spacing of entry points at distance R to be of the same order as the transverse fluctuation of geodesics at the same distance R; in other words, two geodesics which are close enough that their transverse fluctuations allow them to coalesce should generally do so. This means the bound (1.21) should be sharp up to the power of log in the numerator. The bound (1.20) is likely not sharp, though, as we expect the combining of a small sector of directions should not significantly increase the number of entry points; a bound like (1.21) should apply to the mean combined crossing density as well. But for the purpose of banning bigeodesics, the fact that the bound in (1.20) approaches 0 as s → ∞ is sufficient. Remark 1.6. Suppose we fix θ and consider the collection of all halfspace θ-rays from H − θ,0 . By the time these reach H θ,R , based on (1.21) enough coalescence or temporary touching must occur so that on average at least ∆ d−1 R /(log R) C 16 θ-rays pass through any given H + θ,R -entry point x; this number is roughly of order R (d−1)ξ . To the extent this is due to temporary touching rather than coalescence, these θ-rays will later separate again. But this separating can occur at most only slowly: it can be shown using Proposition 4.10 that the number of H + θ,2R -entry points of θ-rays passing through such an x is with high probability at most of order (log R) K for some K. All the while there should be significant additional coalescence and/or temporary touching initiated between H θ,R and H θ,2R , since the crossing point density is lower for H θ,2R . All this brings up the question, do the conclusions of Theorem 1.5 already force coalescence of all θ-rays? We do not have an answer.
When θ is fixed it will be convenient to express a general u ∈ R d in terms of a basis B θ in which the first vector is y θ , and the other d − 1 form an orthonormal basis for H θ,0 . (The particular choice of orthonormal basis does not matter.) In a mild abuse of notation we will simply write u = (u θ 1 , u θ 2 ) θ for the representation in this basis, with u θ 1 = u · z θ ∈ R and u θ 2 ∈ R d−1 ; we call these θ-coordinates. The corresponding decomposition of u is Figure 2), and we refer to u θ 1 y θ and u − u θ 1 y θ as the first and second θ-components of u. For d = 2 it is known ( [13], [24]) that, under hypotheses weaker than the combination of A1 and A2(i), (ii), for each fixed θ ∈ J(θ 0 , 0 ), with probability one there is a unique θ-ray, which we denote Γ θ x , starting from each x ∈ Z 2 , and any two such θ-rays eventually coalesce; there is no temporary touching or branching. (We note again, however, that there must be a random countable set of directions θ for which branching does occur, producing multiple θ-rays from a single site.) So we may ask, how far do two θ-rays go before they coalesce? To formulate the question more precisely we first make some definitions. Fix θ,θ ∈ J(θ 0 , 0 ). Aθ-start site is a site in H − θ,0 which is adjacent to a site in the interior of H + θ 0 ,0 . Aθ-start site x is a θ-source (in a configuration τ ) if Γ θ x is a halfspace θ-ray from H − θ,0 . With probability one, for any twoθ-start sites x, y, there exists a unique coalescence site U θ xy such that Γ θ x and Γ θ y are disjoint up to U θ xy and coincide from U θ xy onward. Thẽ θ-coalescence time of Γ θ x and Γ θ y is theθ-coordinate (U θ xy )θ 1 . Throughout the preceding, it would be convenient to take θ =θ, but we will needθ to be rationally oriented.
We can bound the tail of the coalescence time as a consequence of Theorem 1.5(ii), as follows.
Theorem 1.7. Suppose for some FPP process on Z 2 , A1 and A2(i),(ii) hold for some θ 0 , 0 . There exist constants 3 , C i as follows. Fix θ ∈ J(θ 0 , 3 ), and suppose x, y are θ-start sites with C 18 ≤ |x − y| ≤ ∆ r . Then for all r ≥ ∆ −1 (|x − y|), A natural scaling in Theorem 1.7 is r = t∆ −1 (|x − y|) with t ≥ 1. If we strengthen the assumption in A2 that σ(·) is powerlike to instead require that σ(s) ∼ Cs χ for some C (as is known to be true with χ = 1/3 for integrable models of LPP [8]), then (1.22) says that for t ≥ 1, One expects that the probability in (1.23) is actually of order t −(1+χ)/2 , uniformly in |x−y|, without any logs involved; such a result is proved for 2d LPP in [7], with related bounds in [31] and [33]. Let e j denote the jth unit coordinate vector. Under assumptions much milder than ours, it is proved in [1] that for all sequences v k in In particular, taking v k = ke 1 solves a conjecture made in [10]. Here, under stronger hypotheses, in general dimension we can establish a rate at which this probability converges to 0; as with (1.21) we expect this rate to be optimal up to the power of log in the numerator. The statement is as follows.
Theorem 1.8. Suppose for some FPP process on Z d , A1 and A2(i),(ii) hold for some θ 0 , 0 . There exist constants 4 , C i as follows.

The cost of bad geodesic behavior
Let is nonnegative by subadditivity of h. A variant of the following bound was proved in [2], with error term C 25 |x| 1/2 log |x|; in [35] this was improved to C 25 |x| 1/2 (log |x|) 1/2 . The proof in [2] adapts readily to the present situation and we don't need improvement to (log |x|) 1/2 , so we will work here without that improvement.
Recall that Γ xy denotes the geodesic between x and y. In general we view Γ xy as an undirected path, but at times we will refer to, for example, the first point of Γ xy with some property. Hence when appropriate, and clear from the context, we view Γ xy as a path from x to y.
A number of our arguments involve the following theme: for points x, y ∈ R 2 , if y is far enough from the straight line from 0 to x, then Γ 0x is unlikely to pass through y, because g(y) + g(x − y) significantly exceeds g(x). To express this in more than a crude way, since (1.4) involves centering at the expectation, we also need to quantify how increases in g(x) relate to increases in h(x), but for the moment we consider just g. For Euclidean distance the following is useful: Under the local curvature assumption A1(ii), the following analog for g-distance is straightforward, under A1(ii), proved similarly to Lemma 2.3 below (see also Remark 1.2.) There exist constants This is really the significance of the local curvature assumption for the boundary of B g : it says that in dealing with vectors with direction near θ 0 , the norm g is "Euclidean-like" in that (2.4) holds, and, most importantly, there is consequently a discrepancy as in (2.26) below in the triangle inequality. In (2.4) we can view the "min" term as a lower bound for the cost, in extra distance, of deviating by |u θ 2 | from a target point at g-distance u θ 1 in direction θ. To obtain a probability cost of such deviation by a geodesic, in view of (1.4) roughly we can divide the extra distance by σ(u θ 1 ); we will incorporate an extra log factor to handle the entropy that arises when handling many scales of |u| with a single bound. Keeping this in mind, we define first σ * (s) and Φ(s) by .
Here factoring out a power of s ensures that Φ is strictly increasing, and C 3 , γ 2 are from (1.2). Note that by (1.2) we have (2.5) C −1 3 σ s ≤ σ * (s) ≤ σ s , the first inequality being valid for s ≥ C 28 for some C 28 . We then define (2.6) Ξ(s) = (sσ(s) log(2 + s)) 1/2 , D θ (u) = min Roughly speaking, if we ignore the above-mentioned entropy-controlling log factors in Φ and Ξ, then for a geodesic or geodesic ray with ultimate direction θ, D θ (u) represents the cost of that geodesic deviating from direction θ to pass through u. When the direction of u is far from θ, this cost has form Φ(|u| θ,∞ ), and when it is close to θ the cost has form |u θ 2 | 2 /Ξ(u θ 1 ) 2 . Note that by (1.2) and (2.5), for some C 29 , for all s ≥ C 29 , This tells us in part which term in the "min" in (2.6) is smaller: we have for |u| ≥ 2C 29 that Here we used the fact that 0 ≤ u θ 1 ≤ |u θ 2 | implies |u| θ,∞ = |u θ 2 |, and |u θ 2 | ≤ u θ 1 implies |u| θ,∞ = u θ 1 . Let Π xy denote the line segment from x to y. To deal with paths from 0 to some ry θ it is useful to have the following symmetric version of D θ : This makes the right half of the region E θ,r,c := {u : D θ,r (u) ≤ c} symmetric with the left half; this region is a "tube" surrounding Π 0,ry θ bounded by the shell {u : |u θ 2 | = c 1/2 Ξ(u θ 1 )}, augmented by a "tilted cylinder" around each endpoint; see Figure 3. (In a mild abuse of terminology, we will simply call it a cylinder.) The cylinder around 0 is We call this the 0-cylinder of E θ,r,c ; it has one inside end in the hyperplane H θ,Φ −1 (c) , and an outside end in H θ,−Φ −1 (c) . We thus call E θ,r,c a tube-and-cylinders region. By monotonicity of Φ and Ξ, {u : D θ,r (u) = c} is the boundary of the tube-and-cylinders region.
By (2.8), on {u : 0 < u θ 1 = |u θ 2 |} (which is a cone boundary), Φ(|u| θ,∞ ) is the "min" in (2.6).; this uses the fact that u θ 1 = |u| θ,∞ on that cone boundary. This means that the boundary of the 0cylinder meets the shell in the inside end of the 0-cylinder, the intersection being the (d − 2)-sphere of radius Let ψ ab denote the angle, taken in [0, π], between nonzero vectors a and b. The vector θ (or its multiple y θ ) and the vector z θ ⊥ H θ,0 need not be parallel, but we can bound the angle between them as follows. Let a > 0 be such that az θ ∈ H θ,1 ; then 1 = az θ · z θ so a = 1/|z θ | 2 . Also, g(az θ ) ≥ 1, and az θ is the orthogonal projection of y θ onto the line through 0 and z θ so From lattice symmetry, there exists an othonormal basis for R d containing y θ and consisting of vectors in ∂B g having the same Euclidean length; by inverting basis vectors we may assume z θ has all nonnegative coefficients in this basis. Then the convex hull of the basis vectors includes a Figure 2. Top: Illlustration of the relationships in (2.10). u θ 1 y θ and u − u θ 1 y θ are the θ-components of u. u θ 1 y θ may instead lie on the opposite side of (u · θ)θ. Bottom: Illustration of (2.11) and (2.12). The angle between L θ (v) and H θ,0 is bounded below by arcsin 1/ √ d.
multiple bz θ with b > 0; the convex hull is contained in B g so we must have b ≤ a. The minimum Euclidean length of vectors in this convex hull is |y θ |/ √ d, so Therefore ψ y θ ,z θ = ψ θ,z θ ≤ arccos 1/ √ d; alternatively we can say the angle between θ and H θ,0 is at least arcsin 1/ √ d. This has several consequences. First, for all u ∈ R d , (2.10) (see Figure 2.) Second, there exists 5 , 6 > 0 and C 30 as follows. Suppose α, θ ∈ S d−1 and the angle ψ zα,z θ between H α,0 and H θ,0 is at most 5 . Suppose also that for some v, L θ (v) intersects H α,0 and H θ,0 in points x α and x θ respectively. Then which by (1.12) means that In addition, for α ∈ S d−1 , letting Here and in what follows, for a line L, a hyperplane H, and a point w, we write w = L ∩ H as a shorthand for {w} = L ∩ H.
Note that in both (2.11) and (2.12), we can view the context as starting with a line L θ (v) through v that intersects a hyperplane H θ,0 at an angle of at least arcsin 1/ √ d. Equation (2.11) bounds the change in the intersection point if we rotate the hyperplane around 0 from H θ,0 to H α,0 , keeping the line fixed. Equation (2.12) bounds the change in the intersection point if we instead rotate the line through v from direction α to α (both near θ), keeping the hyperplane fixed.
Another consequence of (2.11) and (2.12) is the following, relating change in θ-coordinate values to change in the angle θ. When we change from θ to α, in comparing x θ 2 to x α 2 it is not appropriate to simply consider |x θ 2 − x α 2 |, as these are coordinate vectors under different bases, used in different spaces (H θ,· vs H α,· ). Instead we compare them in R d by considering |(0, x θ 2 ) θ − (0, x α 2 ) α |. Lemma 2.2. Suppose A2(ii) holds for some θ 0 and 0 > 0, and let 6 be as in (2.11), (2.12). There exist C i as follows. Suppose α, θ ∈ S d−1 with ψ αθ ≤ 6 , and 0 = x ∈ R d . Then ) Then from (2.10), (2.11), and (2.12), (2.15) and using the last two inequalities in (2.15), We can use (2.10) to relate the θ-ratio of u to the tangent of ψ uθ : In the other direction, by (2.9), the tangent of angle between θ and u − u θ 1 y θ has magnitude at least 1/ √ d − 1 so letting w be the closest point to u in L θ , satisfying |w| = u · θ, we have The bound on the angle between y θ and z θ also gives information about the 0-cylinder of E θ,r,c . If u lies in either end of C θ,c then |u| ≥ |y Let µ = g(e 1 ). Convexity and lattice symmetry yield that µB g contains the 1 -unit ball in R d and is contained in the ∞ -unit ball, so In addition, from the triangle inequality, Hence suppose c is large and x lies in the tube portion of E θ,r,c , that is, |x θ , which we call the fattened and backwards-fattened H θ,s , respectively; generically we call any such slab of thickness µ √ d a fattened hyperplane. This thickness is chosen so that, by (2.20), any lattice path crossing a fattened hyperplane must have at least one site in it. If x ∈ H rfat θ,s ∪ H fat θ,s then by (2.20) the θ-projectionx of x into H θ,s satisfies More generally, for a set B contained in some H θ,s we write The values θ, s will be uniquely determined by B in all instances here.
Given δ > 0, a geodesic Γ xy , and a site u preceding a site v in Γ xy , we say that that is, the extra distance associated with this triangle is at least (δ 2 ∧ δ)|v − x|. An analog for g is the following variant of (2.4).
Proof. Let u, v be as in the lemma statement. Consider first the case of g(u) ≥ g(v); we then have from (2.20) that Consider next u / ∈ Ω θ (0, g(v)), noting that this slab has 0, v in its boundary. From symmetry we ]. We letŵ = u θ 1 y θ be the first θ-component of u. Thenŵ ∈ Π 0v so from symmetry we may assume |ŵ| ≥ |v|/2, so that using (2.20), Note that H θ,g(ŵ) contains u,ŵ and is tangent to ∂(g(ŵ)B g ) atŵ. It therefore follows from (2.28) and (1.8) that and then using (2.20), since g(ŵ) ≥ g(v)/2, Now H θ,g(ŵ) contains u and is tangent to the boundary of the translate v + g(ŵ − v)B g atŵ. It follows that g(u − v) ≥ g(ŵ − v). Therefore using (2.29), The proof of the next proposition is based on the fact that if a path γ from 0 to some site ry θ contains a site u with D θ,r (u) ≥ t, then there are necessarily 3 sites in γ (one of which is an endpoint, 0 or ry θ ) which form a δ-fat triangle for some appropriate δ, and Lemma 2.3 can be used to help show that the probablity of this is small. Analogous results based on the same general principle appear in [6] for an integrable last passage percolation model in d = 2, and in [17] for FPP in d = 2 under hypotheses similar to ours here. Proposition 2.4. Suppose A1 and A2(i),(ii) hold for some θ 0 and 0 > 0. There exist constants C i as follows. For all r, θ with ψ θθ 0 < 0 and 0 = ry θ ∈ Z d , and all t > 0, Proof. It is enough to consider t sufficiently large (not depending on r, θ.) Let U = (U θ 1 , U θ 2 ) be the site which maximizes D θ,r (·) over Γ 0x , with ties broken arbitrarily, and let C = D θ,r (U ) ≥ t be the corresponding maximum value. By monotonicity of Φ, U must lie on the boundary of {u : D θ,r (u) ≤ C}. From symmetry we may assume U θ 1 ≤ r/2. We show there exists W ∈ Γ 0x such that 0, U, W form a δ-fat triangle for appropriate δ. Fix κ large enough so Note that from (2.5), (1.2), and (2.31), given δ > 0, provided we take κ sufficiently large, There are four cases; see Figure 3.

Consider now the events
We have by (2.38) that Here the factor of 2 accounts for the fact we assumed U θ 1 ≤ r/2. Let A k (u, w) denote the event that one of the following holds: For u, w as in the event A k , one of these inequalities must hold, so A k ⊂ ∪ u,w A k (u, w), where the union is over u, w as in the definition of A k . For each such u, w we have from (1.4) that Summing the last bound over u, w, k and using (2.39), (2.40) yields (The latter means "the cylinder is not small relative to the tube.") Then U again lies on the boundary of the 0-cylinder in E θ,r,C , but this time we take W = ry θ . We consider two subcases.
Case 2a: Suppose Case 2 holds with Φ −1 (C) ≤ r/2. This means the tube is not completely contained inside the two cylinders; this can only occur when Φ −1 (t) ≤ r/2. Using (2.19) we then have and as in Case 1, using (2.7), Thus 0, U, W form a c 7 -fat triangle. Define the event Similarly to Case 1 we have using Φ −1 (t) ≤ r/2 that Case 2b: Suppose Case 2 holds with Φ −1 (C) > r/2. This means the two cylinders contain the entire tube, and here we need only consider Φ −1 (t) ≥ r/2. This is generally similar to Case 1, except that we do not know |U | ≤ |W | = r|y θ |. Analogously We also know from and we define K t,2 by so that similarly to (2.42), , meaning that U lies on the tube boundary {u : |u θ 2 | = C 1/2 Ξ(u θ 1 )} near the 0 end (but outside the 0-cylinder.) Similarly to Case 1, let W be the first point of Γ 0,ry θ after U with W θ 1 = κU θ 1 , and let W be the first site in Γ 0,ry θ after W . Then using (2.7) and (2.32), (2.45) and the θ-ratio of U satisfies As in Case 1 let V be the closest point to U in Π 0W , so |V | ≤ |U |. From (2.31) and (2.46), the θ-ratio of W satisfies so by (2.17), (2.32), and (2.47), On the other hand, we have from (2.10) that Thus 0, U, W form a δ-fat triangle with From (2.48) and (2.21) we have Hence from Lemma 2.3, (2.48), and (2.7) we have and since |W − W | ≤ 1, the same holds for W in place of W , possibly with a smaller c 19 . Defining the events and defining K t,3 by in view of (2.45) and (2.47) we have similarly to (2.42) and (2.44), provided t is large: , meaning that U lies on the tube boundary but not near an end. This can only occur when Φ −1 (t) ≤ r/2, and this time we take W = ry θ . We have using (2.10) that From the definition (2.6) we have Similarly to Case 3 we have δ < 1. From (2.21) and (2.7) we obtain The following is a purely deterministic result about norms on Z d when the local curvature assumption is satisfied.
Lemma 2.5. Suppose the norm g satisfies the local curvature assumption A2(ii) for some θ 0 , 0 . There exist constants 7 and C i as follows. Suppose > C 37 , ψ θθ 0 < 7 , and u, v ∈ H θ, with Then Proof. We bound g(v) − g(u) as the opposite bound is nearly symmetric. Let α = u/|u|. By A2(ii) and (1.12), provided we take 7 small, the first inequality in (2.53) guarantees that the angle between H θ,0 and H α,0 is at most so by A2(ii), (2.20), and (2.56), Combining this with (2.20) and (2.55) yields We next consider the transverse increments of T (0, u), that is, we bound |T (0, u) − T (0, v)| when |g(u) − g(v)| is small. (We can't require it be exactly 0 since u, v are lattice points.) Heuristically, assuming |u − v| ∆(|u|), ∆ −1 (|u − v|) may be viewed as the typical distance traveled by Γ u0 and Γ v0 before they can get close enough to coalesce. Then σ(∆ −1 (|u − v|)) becomes the scale of "fluctuations before coalescing," and we show that |T (0, u) − T (0, v)| is unlikely to be much larger than this scale. A variant of the following proposition, for d = 2, appears in [17]. Figure 4. Illustration for the proof of Lemma 2.5.
There exist constants C i as follows. For all u, v ∈ Z d with and all λ ≥ C 41 , we have with c 1 , c 2 to be specified; note log and log |u − v| are of the same order. Provided C 40 is taken small (depending on c 1 ), we have ≤ |u|/2|y θ | = u θ 1 /2. We consider first the case of "moderately large λ": We intersect a tube-and-cyliners region with a fattened hyperplane to get It follows using Proposition 2.4 that Illustration for the proof of Proposition 2.6. T (0, u) and T (0, v) are likely to be close because Γ 0u has the option to pass through the same site x ∈ Υ that Γ 0v passes through.
To prove this, let w = (u θ 1 − )y θ ∈ Π 0u . We first approximate x by its θ-projectionx = (w θ 1 , x θ 2 ) into H θ,u θ 1 − . Then from the definition of Υ , Letv be the closest point to v in H θ,u θ 1 . Since |g(v) − g(u)| ≤ 4µd, provided |u| is large (so by (2.57) ψ uv is small) there exists a point v on the line through v andv satisfying g(v) = g(u) and g(v − v) ≤ 5µd. In order to bound g(v −v), we first observe that and the last fraction can be made small by taking c 1 large in the definition of (so itself is large), and it then follows from A2(ii) and (2.20) that In view of (2.62) and (2.64), to prove (2.61) it remains to bound g(v −x) − g(u −x), for which we use Lemma 2.5 with the origin shifted tox. First observe that provided c 1 (and hence ) is large, using (2.59) we have Therefore > Φ −1 (t), meaning that w, x lie in the tube part of the tube-and-cylinders region E θ,u θ 1 ,t . Therefore , and hence from the second inequality in (2.65), provided c 1 (and therefore ) is large, so Lemma 2.5 applies, giving Using again |v − u| ≤ ∆( /c 1 ) along with (2.66) and (2.67) lets us conclude It follows using (1.4) that the probability in (2.60) satisfies and similarly for |u − x|. Hence the right side of (2.69) is bounded above by 2C 5 e −c 13 λ log |u−v| , which with (2.60) yields that for all λ ≥ c 14 /2 satisfying (2.59), It remains to consider "large λ," meaning (2.59) does not hold: Here we have using (1.4) that With (2.60) and (2.71) this proves (2.58).

Existence and transverse fluctuations of θ-rays
For a geodesic ray Γ = (v 0 , v 1 , . . . ) (as a sequence of sites), we say Γ is a subsequential θ-ray if there exists a subsequence {v n k } for which v n k /|v n k | → θ. We say a sequence {Γ n } of geodesics or geodesic rays from v 0 converges to Γ if for each j ≥ 1, for all sufficiently large n, Γ[v 0 , v j ] is an initial segment of Γ n . If {Γ n } is a sequence of geodesics or geodesic rays from a fixed v 0 with length |Γ n | → ∞, then {Γ n } has a converging subsequence.
In Proposition 2.4, the bound on the probability is uniform in r. This enables us to turn that lemma (or more precisely, its proof) into a result about θ-rays, which is part (ii) of the next proposition. The proposition also includes parts (i), (iv)(a), and (iv)(b) of Theorem 1.5.
If also A2(ii') holds then this is true without the condition ψ θθ 0 < 0 .
(ii) There exist constants C i as follows. For t > 1, If also A2(ii') holds then this is true without the condition ψ θθ 0 < 0 .
If also A2(ii') holds then P there exists a geodesic ray with no asymptotic direction = 0. (3.4) Proof. Observe that event in (3.2) is contained in the event This can be seen by fixing Γ = (v 0 , v 1 , . . . ) (as a sequence of sites) and θ as in (3.2), and a site is a subsequential θ-ray which is not a θ-ray, then there are subsequences v n(k) /|v n(k) | → θ and ζ(j) = v n (j) /|v n (j) | → ζ for some θ = ζ. But this means that given t > 0, fixing k sufficiently large we have D ζ(j) (v n(k) ) > t for all sufficiently large j (depending on k), which in turn means that A t occurs with z j = v n (j) . It follows that the complement of the Therefore to prove both (3.2) and (3.3) it is enough to show Note that for fixed z n we can use Proposition 2.4, but we cannot sum over possible z n as the entropy is too large. However, in the proof of that lemma, all that we use is (after converting the notation for our present context) the existence of a δ-fat triangle of sites 0, u, w in Γ x,zn for sufficiently large δ (depending on |u|, |w|.) The bounds in that proof do not involve z n , so in the 4 cases there it is only necessary to sum over ranges of possible values of |u| or |w|. Thus the proof of Proposition 2.4 also proves (3.5).
The last sentence in (ii) follows from (3.2) and the compactness of S d−1 , and similarly in (iii). Turning to (i), it is enough to consider v = 0. Given θ ∈ S d−1 with ψ θθ 0 < 0 , let z n ∈ Z d with |z n | → ∞ and θ n = z n /|z n | → θ. Then some subsequence of {Γ 0zn } converges to a geodesic ray which means A t occurs for all t > 0. Thus Γ ∞ is a θ-ray a.s. Equation (3.1) then follows from (3.5), and the last sentence of (i) again follows from compactness of S d−1 .

Crowded geodesics
We call a path or geodesic from a set A ⊂ R d to B ⊂ R d nonreturning if only the first bond x 0 x 1 (viewed as a line segment) intersects A and only the last bond x m−1 x m intersects B. A nonreturning path from H − θ,s 1 to H + θ,s 2 for some s 1 < s 2 is called a θ-slab path, and similarly for a θ-slab geodesic.
In Theorem 1.5(ii), one can view the density of H + θ 0 ,R -entry points as bounding the maximum possible longitudinal density of any set of halfspace θ 0 -rays with distinct H + θ 0 ,R -entry points. Given an upper bound for this density larger than its heuristically-suggested order of ∆ for some β 0 > 0 and n ≥ 1, which we may read as n θ-rays crossing per volume (n −β 0 ∆ R ) d−1 in H θ,0 . Thus to bound the mean H θ 0 ,R -crossing density our main task is roughly to bound P there exist n halfspace θ 0 -rays with distinct H + θ 0 ,R -entry points Obtaining such a bound with n = (log R) K for some (large) K corresponds-modulo a few technicalities-to bounding the mean H θ 0 ,R -crossing density by (log R) K /∆ d−1 R , with K = (1 + (d − 1)β 0 )K. Similarly, we can bound the mean combined H θ 0 ,R -crossing density of θ-rays over θ ∈ J(θ 0 , ) for some , mainly by bounding where by the initial orientation of a geodesic we mean its direction from its starting point to its H + θ 0 ,R -entry point. Remark 4.1. The idea of the bound on (4.2) is as follows, for some values β i ∈ (0, 1), with some details and definitions altered to reduce technicalities. Consider first the probability there exist n slab geodesics as in (4.2) which are disjoint. Divide the initial portions (i.e. up to H θ 0 ,R ) of these slab geodesics into n β 2 segments of equal length = n −β 2 R, cutting at the H + θ 0 ,i -entry points for i ≤ n β 2 . The "natural" transverse spacing of these segments is ∆( ) = ∆(n −β 2 R); for given i let us call the ith segments of two slab geodesics Γ (1) , Γ (2) neighbors if the respective H + θ 0 ,i -entry points u We show that with very high probability, no pair of neighbors (for any i) have passage times that differ by even a small multiple of σ . The neighbors have close passage times because, except near their endpoints, they are "geodesics chosen from the same set of possible paths." The close passage times mean that, modulo a small-probability event, if we fix any one slab geodesic Γ and the passage times of its segments, those segment passage times in Γ effectively nearly determine the passage times of all neighbor segments of other slab geodesics. In particular, in view of (1.9) and (1.10), we say a segment is fast if its passage time is at most ET (0, y θ 0 ) + η 8 σ ; up to a small error, for every fast segment of our fixed θ-ray, all neighbor segments in other θ-rays must be fast. We then show that a fixed one of our n slab geodesics (let's call it "special") likely has at least of order n γ 2 β 3 fast segments before crossing H θ 0 ,R , and every one of those fast segments has one or more neighbors, all of which are also fast. However, we can use the FKG inequality to say that the presence of the special slab geodesic (because it's a geodesic) stochastically increases the passage times of bonds not in the geodesic; hence the probability is small that all the neighbors will be fast. This provides the desired bound on (4.2), in the disjoint case.
In summary, the fast segments of length in the special geodesic force all their neighbor segments to be fast (with high probability), while simultaneously reducing the neighbor segments' probability to be fast, in the FKG sense. These contradictory aspects can only coexist if the probability of crowded geodesics as in (4.2) is very small.
For the general (non-disjoint) case, for segments to qualify as neighbors, we add the requirement that length of their intersection, if any, is not more than n −β 4 . We consider two possibilities: (i) popular site case: there exists a site in H − θ 0 ,R in the intersection of at least n β 3 −β 4 of the n slab geodesics.
(ii) no popular site: there is no site as in (i). In the no-popular-site case, the arguments from the disjoint case still apply with some modifications. In the popular-site case, we take a subset of n β 5 out of the n β 3 −β 4 slab geodesics sharing a common point, with all in the subset having H + θ 0 ,R -entry points within distance of order n −β 0 R. Because these slab geodesics share a common point before H θ 0 ,R and have distinct H + θ 0 ,R -entry points, they must be disjoint after H θ 0 ,R . Hence the disjoint case can be applied to these n β 5 geodesics from H θ 0 ,R to H θ 0 ,2R , instead of n geodesics from H θ 0 ,0 to H θ 0 ,R .
For sites u, v in a path γ, let γ[u, v] denote the segment of γ between u and v. Given a geodesic Γ from H − θ,0 to H + θ,R for some R and given 0 ≤ s ≤ R, let x θ,s (Γ) denote the H + θ,s -entry point of Γ, and let x θ,s (Γ) be the last site in Γ before x θ,s (Γ). An -segment of Γ is the segment To start formalizing the argument outlined in Remark 4.1, we show that with high probability, a geodesic contains at least a certain minimum number of fast -segments. There exist constants C i as follows. Let with γ 2 from (1.2) and k an integer. For ψ θθ 0 < 0 and Γ a geodesic from H − θ,0 to H + θ,R , let Then for all v ∈ H rfat θ,0 and all w ∈ H fat θ,R with we have Note that (4.4) allows the angle ψ θ,w−v to be nonzero but not too large. Such a bound is necessary for (4.10) in the proof.
Proof of Lemma 4.2. We need a slightly modified definition to take into account that Γ vw might not be a slab geodesic, i.e. we might not have x θ,0 (Γ vw ) = v and x θ,R (Γ vw ) = w: σ .
For technical convenience we will assume y θ is a lattice point; the adjustments when this is false are minor. It follows from (1.2) and (4.3) that provided C 47 is large enough in (4.3), we have with c 1 chosen so that h(Ry θ ) ≤ g(Ry θ ) + c 1 σ R log R, from Proposition 2.1. With this we obtain that provided C 48 is small (so is large), for some c 1 , for all 1 ≤ i ≤ k, where the second inequality follows from k(x θ,i (Γ vw ) −x θ,(i−1) (Γ vw )) ∈ H + θ,R . Therefore The key point is that Equation (1.2) gives σ /σ r ≥ C −1 3 k −γ 2 , so using also (1.4) and (4.8), it follows that To control the next-to-last probability we need to bound h(w − v) − h(Ry θ ). Letŵ,v be the θprojections of w, v into H θ,R and H θ,0 , respectively, so by (2.24) we have |w −ŵ| ≤ d and |v −v| ≤ d. From (1.2) and (4.4) we have and hence provided C 48 is small (so is large), Therefore A2(ii) applies and, provided we take C 49 small in (4.4), we get From (4.10), With Proposition 2.1, (4.6), and (4.11) this gives the desired bound Therefore in (4.9), using (1.2), (1.4), and (4.12) we have It remains to bound the last probability in (4.9). Write Λ i for H fat θ,i , which must contain x θ,i (Γ vw ). For t = k 1−γ 2 we have using Proposition 2.4 and (4.8) that Note that the lower bound of 3 in the third line here means the ith -segment of Γ vw has direction quite far from θ. For the last probability we have from (1.4) For the next-to-last probability in (4.14) we have from subadditivity and Proposition 2.1 that for all a, b in the double sum, so from (1.4) again, Here the last inequality follows from the fact that for large , by (1.2) and (4.3), The number of a or b in the sums in (4.14) is at most of order R d , so provided C 47 is large enough in (4.3), the right side of (4.14) is bounded above by With (4.9) and (4.13) this proves (4.5).
Remark 4.3. The results to follow involve several different length scales, and other quantities, expressed using R and small (less than 1) powers of the number n of geodesics we are dealing with. We summarize them here for ready reference, with precise definitions to follow: (i) a length scale n −β 0 ∆ R for cubic blocks in each hyperplane H θ,s , with geodesics considered close when they start and end with separation n −β 0 ∆ R or less; (ii) a width n β 1 ∆ R for the "target box" to which length-R geodesics are confined, with high probability; (iii) a length scale = n −β 2 R for segments of length-R geodesics; (iv) for a transition (i.e. choice of starting and ending blocks) made by a length-segment of a geodesic, a maximum number n β 3 of other geodesics which can make the same transition, for that transition to be considered "sparse"; (v) a maximum overlap length n −β 4 for two length-geodesic segments to be considered "low overlap"; this length also serves as the minimum significant backtrack in a geodesic. Other quantities are defined in terms of these powers, such as a minimum number n β 3 −β 4 of geodesics passing through a site for it to qualify as a popular site in the sense of Remark 4.1.
We start with some formal definitions related to the items in Remark 4.3. The following definitions are for fixed R, n, θ, which don't necessarily appear in the notation.
by an integer multiple of 2n −β 0 ∆ R (so such θ-blocks tile H θ,s .) The center point of any θ-block is called a θ-block center. An enlarged θ-block has the same center as a block but larger linear dimensions by a factor 2 √ d; more precisely it has the form (in θ-coordinates) with y a θ-block center in H θ,s , and the + denoting translation. The factor 2 √ d ensures that if u lies in a θ-block with center a, and v lies outside an enlarged θ-block with center b, then |v − b| ≥ 2|u − a|.
For Γ a geodesic which starts in H − θ,0 and crosses H θ,R , the pre-H θ,R segment of Γ is the segment of Γ from its starting point to x R (Γ). In what is to follow, as in Lemma 4.2, we will divide the pre-H θ,R segment of a θ-slab geodesic into sub-segments of some length . In this context, a geodesic Φ is an ( , θ)-interval geodesic if for some i, the initial site of Φ is in H fat θ,(i−1) , and the last bond of Φ is the first bond of Φ to cross H θ,i (so the last site of Φ must lie in H fat θ,i .) An -segment of a θ-ray (defined before Lemma 4.2) is thus one example of an ( , θ)-interval geodesic.
The target θ-box is a tube around L θ : in θ-coordinates. A geodesic is θ-target-directed if it is contained in the target θ-box. A geodesic from H − θ,0 to H + θ,s for some s > 0 is θ-target-directed up to s if its pre-H θ,s segment is target-directed. A target θ-block is a θ-block which intersects Q R,n,θ .
For θ ∈ S d−1 , Γ a geodesic with a designated direction, and sites x preceding y in Γ, we say Γ has a θ-backtrack of size r from x to y if x θ 1 − y θ 1 ≥ r.
We may omit the parameters in the preceding terminology when it is clearly understood, e.g. referring to a block rather than a θ-block.
Before proceeding to the proof of Theorem 1.5, in the next few lemmas we consider various forms of "bad geodesic behavior" specialized to our context, and apply our previous results to show that these have small probability. These lemmas involve the quantities n ±β i as in Remark 4.3, so we now specify the relations that we assume to hold among these exponents.
We now consider four cases; see Figure 6.
Here we find a lower bound for d(u, Π xy ).

It follows from Proposition 2.1 and (4.20) that
For the event G 2 , the idea is that B rfat s,θ,home,+ lies in front of, and nearly parallel to, the much smaller block B rfat θ 0 ,home , so any geodesic from B rfat θ 0 ,home to B θ 0 ,cross will very likely cross H rfat θ,s by passing through B rfat s,θ,home,+ , whereas G 2 says to the contrary. The picture with the larger B fat r,θ,home,+ just behind the smaller B rfat θ 0 ,cross is analogous.
Proof of Lemma 4.6. Suppose τ ∈ G 2 and Γ = Γ[x, y] is a geodesic as described in G 2 , from some x ∈ B rfat θ 0 ,home to y ∈ B fat θ 0 ,cross , containing a site u ∈ H rfat θ,s \B rfat s,θ,home,+ . Letû = (s, u θ 2 ) θ be the θ-projection of u into H θ,s andx = (0, x θ 0 2 ) θ 0 the θ 0 -projection of x into H θ 0 ,0 , so |u −û| ≤ d and Hence using (2.10) and (2.2), so using (4.23) and the last inequality in (4.22), Having (4.45), by (2.8), to obtain a lower bound for D α,|(y−x) α 1 | (u − x) we need only obtain a lower bound for From the definition of s, there must exist a point v ∈ H θ 0 ,0 ∩ H θ,s ∩ Q R,n,θ , so using (2.11) we have Now in view of (4.28) the θ-ratio of y − x satisfies From Lemma 2.2, (2.21), and (4.46) we obtain that, after reducing min if necessary, With (4.45) this shows that (4.50) D α,|(y−x) α 1 | (u − x) ≥ n 2β 1 , and then summing (2.30) over x ∈ B rfat θ 0 ,home and y ∈ B fat θ 0 ,cross shows Lemma 4.7. There exist constants C i > 0 as follows. Suppose A1 and A2(i),(ii) hold for some θ 0 , 0 , and A3 holds with θ = θ 0 for some R, n, , min . For the events we have Proof. Since θ = θ 0 we'll simply call it θ. We can handle G 3 ∪G 4 as one, as follows. Suppose τ ∈ G 3 . This means there exists an ( , θ)-interval geodesic Γ xy ⊂ Q R,n,θ containing sites u preceding v with This shows that for the event G 5 : (bad-direction segment in a geodesic) There exists an ( , θ)-interval geodesic Γ ⊂ Q R,n,θ for some i ≤ R/ , containing sites u preceding v as in G 5 , that is, Let α = (y − u)/|y − u|; we want a lower bound for D α,|(y−u) α 1 | (v − u) so we can use Proposition 2.4. We may assume u θ 1 ≤ (i − 1/2) , as the other case is symmetric. Then u, y ∈ Q R,n,θ with (y −u) θ 1 ≥ /2, so from (4.24), provided R is large the θ-ratio of y −u is at most 8 Define the intersection points (2.20) and (4.53), provided R (and hence and |u − v|) is large we have As a shorthand we say a point y is strictly behind a hyperplane H ϕ,t if y lies in the interior of H − ϕ,t , and y is ϕ-behind a point z if z ϕ 1 − y ϕ 1 is nonnegative. We now consider 3 cases; see Figure 9.
Suppose not; we will show that the second inequality in (4.53) is contradicted. From (4.26) and the fact that t ≥ t 0 , the θ-ratio of v − u satisfies Since v lies in the tangent plane H θ,u θ 1 +t to u + tB g at u + ty θ = q, it follows from (1.6), using the first inequality in (4.66), that d(v, u + tB g ) ≤ C 9 n 2β 1 σ t .
We then have using Lemma 2.2, (4.54), (4.65), and (4.68) that It follows from (1.2), the first inequality in (4.17), and (4.63) that which with (4.70) shows that From this and (4.64), where the first inequality uses |y θ | ≤ 2|y α |, from (1.13). Then in view of (4.58) we conclude Therefore as with (4.60),  There exist constants C i > 0 as follows. Suppose A1 and A2(i),(ii) hold for some θ 0 , 0 , and A3 holds with θ = θ 0 for some R, n, , min . For the event G 6 : (there are close ( , θ)-interval geodesics with dissimilar passage times) There exist i ≤ n/ we have Proof. Let i ≤ n/ and suppose u, We have and we want to use Proposition 2.6 to bound the probability that either difference on the right exceeds ησ /8. Letû,v,ŵ,x be the θ-projections of u, w, v, x, withû,ŵ into H θ,(i−1) andv,x into H θ,i . Sinceû,v,x ∈ Q R,n,θ we have using (4.24) and using (2.24) We may assume the points are labeled so that g(x −û) ≤ g(v −û); there then exists a pointẑ ∈ Πûv (close tov) with g(ẑ −û) = g(x −û), and z ∈ Z d with |z −ẑ| ≤ √ d. We then have and the latter implies Checking the conditions of Proposition 2.6 for the first probability on the right of (4.79), we note first that L θ (û) Figure 9. Illustration for Lemma 4.8. The arc containingx andẑ is the boundary of a g-ball centered atû. The dashed lines are the geodesics Γ uv and Γ wx , which are unlikely to have highly dissimilar passage times because they can follow the same path, once away from their endpoints. so using (4.25), (4.75), and the second inequality in (4.17), for C 40 from Proposition 2.6, In applying Proposition 2.6 to the first probability on the right of (4.79) we take λ = n 2β 1 , so we need to verify that for this λ, If we can show that which is equivalent to (4.80). In fact we have using the second inequality in (4.17), along with (4.25), (4.75), and log |v − x| ≤ log R ≤ n 2β 1 , that proving (4.81) and hence also (4.80). Now (4.80) and Proposition 2.6 give Turning now to the last probability in (4.79), we have using (1.2), the second inequality in (4.17), and (4.78) that and similarly |v − z| ≤ n β 1 +β 2 (1+γ 2 )−β 0 σ , so from (1.2), (1.4), and (4.26), Combining this with (4.76), (4.79), and (4.82), along with a similar computation for the second term on the right in (4.76) in place of the first term, we get Summing this over the O(R 4(d−1 ) possible values of (u, v, w, x) we get as in (4.42) that (4.83) P (G 6 ) ≤ c 9 exp − 1 2 c 10 n 2β 1 .
Lemma 4.9. There exist constants C i > 0 as follows. Suppose A1 and A2(i),(ii) hold for some θ 0 , 0 , and A3 holds with θ = θ 0 for some R, n, , min . For the event G 7 : (unusual-speed short segment) There exist u, v ∈ Q R,n,θ ∩ Z d with we have Proof. When |u − v| ≤ 3n −β 4 we have using (1.2) and the last inequality in (4.16) that As in (4.42), summing over the O(R 2d ) possible values of (u, v) gives We are now ready for the core of our main proof, given by the next proposition.
Proposition 4.10. Suppose A1 and A2(i),(ii) hold for some θ 0 , 0 . There exist constants β j ∈ (0, 1) and C i as follows. Let R, n, min be as in A3, and let B θ 0 ,cross be a θ 0 -block in H θ 0 ,R with center point y. Let θ = y/|y|, and suppose ψ θθ 0 < min /2. Then P there exist n θ 0 -slab geodesics from B rfat θ 0 ,home to H + The upper bound on n in (4.20) can be written which is not really a restriction at all, since the density of H + θ 0 ,R -entry points is bounded. It is only there for technical use in Lemmas 4.5-4.9.
As a shorthand, a θ 0 -slab geodesic from B rfat θ 0 ,home to H + θ 0 ,2R will be called a 2R-geodesic. Given a 2R-geodesic Γ, we can decompose the pre-H θ 0 ,R segment of Γ into an initial bond and n β 2segments; every such -segment is an ( , θ 0 )-interval geodesic. An ( , θ)-interval geodesic Ψ is good if (1) Ψ is contained in Q R,n,θ , (2) Ψ contains no backtrack of 1 2 n −β 4 . Let G 8 be the event in (4.86), and define the event G 9 : there exist n θ-target-directed 2R-geodesics with distinct H + θ 0 ,R -entry points in B θ 0 ,cross , so (4.87) Note that τ ∈ G 9 \G 3 says that every -segment of every θ-target-directed 2R-geodesic is a good ( , θ)-interval geodesic. Our main task is to bound P (G 9 ). There are at most c 1 n (β 0 +β 1 )(d−1) θ 0 -blocks intersecting H θ 0 ,2R ∩ Q R,n,θ , and we denote the jth one (in some arbitrary order) as B 2R,j . When τ ∈ G 9 , there exists a subcollection G of size at least g n = c 2 n 1−(β 0 +β 1 )(d−1) out of the n 2R-geodesics, which for some m all have H + θ 0 ,2R -entry point in block B fat 2R,m . We call such a subcollection G a crowded set (via B cross and B 2R,m ), and fix such a G and m. Let y * be the center of B 2R,m , let θ * = (y * − y)/|y * − y|, and let  Figure 10. Illustration for the proof of Proposition 4.10. The dashed line is a typical geodesic from the crowded set G. The primary θ-slab geodesic crosses the left gray box, which is part of the square tube Q R,n,θ surrounding L θ . The secondary θ * -slab geodesic crosses the right gray box, which is part of the square tube Q * R,n,θ * surrounding L θ * (y). The hash marks bound the named blocks (such as B θ 0 ,home ) in the hyperplanes as shown.
This is a tube around L θ * (y) = Π ∞ y,y * with cross section larger than that of Q R,n,θ in each dimension by a factor 2 √ d − 1. It is straightforward to show that due to this larger cross section we have Q * R,n,θ * ∩ Ω θ 0 (R, 2R) ⊃ Q R,n,θ ∩ Ω θ 0 (R, 2R). Analogously to [s, r], there is a largest interval [s * , r * ] for which Q R,n,θ ∩ Ω θ * (s * , r * ) ⊂ Q R,n,θ ∩ Ω θ 0 (R, 2R); see Figure 10. Since y * ∈ Q R,n,θ it is easily checked that the θ-ratio of y * − y is bounded by c 3 n β 1 ∆ R /R, and hence There is a "nuisance possibility" we must deal with here: by assumption the g n (or more) 2Rgeodesics in the crowded set G have distinct H + θ 0 ,R -entry points, but this does not guarantee they have distinct entry points for the hyperplanes H θ,r (behind H θ 0 ,R ) and H θ * ,r * (ahead of H θ 0 ,R .) However, at least one of the following options must be true: (I) there is a subset G 1 ⊂ G with |G 1 | ≥ g 1/3 n which all have the same H + θ,r -entry point, (II) there is a subset G 2 ⊂ G with |G 2 | ≥ g 1/3 n which all have the same H + θ * ,s * -entry point, (III) there is a subset G 3 ⊂ G with |G 3 | ≥ g 1/3 n which all have distinct H + θ,r -entry points, and which all have distinct H + θ * ,s * -entry points. If (I) occurs in some τ , then since the geodesics in G 1 have distinct H + θ 0 ,R -entry points, uniqueness of finite geodesics means these geodesics must be "disjoint from H θ 0 ,R to H + θ 0 ,2R ", or more precisely, the geodesics Γ[x R (Γ), x 2R (Γ)], Γ ∈ G 1 , are disjoint, except for possibly sharing a starting point x R (Γ). Similarly, if (II) occurs then the geodesics in G 2 must all have disjoint pre-H θ 0 ,R segments, except for possibly sharing the same x R (Γ). In (III) we cannot conclude any such disjointness.
In H θ,s and H θ,r we have the enlarged home θ-blocks B s,θ,home,+ and B r,θ,home,+ , respectively, centered on the line L θ . In H θ * ,s * and H θ * ,r * we can define analogous shifted enlarged home θ *blocks by translating an enlarged θ * -block within the hyperplane so that it is centered on L θ * (y); we denote these shifted enlarged home θ * -blocks byB s * ,θ * ,home,+ andB r * ,θ * ,home,+ , respectively. The analog of the event G 2 for the geodesic segments from H θ 0 ,R to H θ 0 ,2R is the following. G 10 : (geodesic after H θ 0 ,R evading enlarged θ-blocks) There exists a geodesic Γ from B rfat θ 0 ,cross to B fat 2R,m which contains a site in (H rfat θ * ,s * \B rfat s * ,θ * ,home,+ ) ∪ (H fat θ,r \B fat r * ,θ * ,home,+ ). In view of (4.88) we essentially can apply Lemma 4.6 to bound P (G 10 ), the only thing being different for G 10 is that the tube Q * R,n,θ * is fatter by a constant factor 2 √ d − 1. This makes no material difference so we have Consider now τ ∈ G 9 \(G 3 ∪ G 2 ∪ G 10 ) and suppose option (III) occurs. We have the following situation: each Γ ∈ G 3 contains a unique θ-slab geodesic from B rfat s,θ,home,+ to B fat r,θ,home,+ , which we call the primary θ-slab geodesic of Γ and denote Γ pri . Γ also contains a unique θ * -slab geodesic fromB rfat s * ,θ * ,home,+ to B fat r * ,θ * ,home,+ , which we call the secondary θ * -slab geodesic of Γ and denote Γ sec . A site x ∈ H − θ,r is called popular for G 3 (in τ ) if there exist n β 3 −β 4 /8 2R-geodesics Γ ∈ G 3 for which x lies in the primary θ-slab geodesic of Γ. A key observation is that if such a popular site x exists, then since {Γ ∈ G 3 : x ∈ Γ} have distinct H + θ,r -entry points, the n β 3 −β 4 /8 (or more) geodesics {Γ sec : Γ ∈ G 3 , x ∈ Γ} must be disjoint. Based on the preceding discussion we can now restate the options as follows, with option (III) split into two suboptions.
In bounding the probabilities for options (I)-(IIIb) we only use the geodesics Γ pri or Γ sec . This means the original angle θ 0 is no longer involved, except that we have effectively replaced R with r − s (for geodesics Γ pri ) or r * − s * (for geodesics Γ sec .) In view of (4.28) this has negligible effect. In the interest of expositional and notational clarity, we can therefore henceforth assume θ = θ 0 and [r, s] = [0, R] for options (II) and (IIIb), where we deal with geodesics Γ pri .
The most difficult of the options to control is (IIIb) where we must deal with the lack of disjointness; in fact our proof of a probability bound for that case will essentially subsume the simpler proofs for the other 3 cases. Hence we consider the events G 11 : τ ∈ G 9 \G bad and option (IIIb) occurs, and we call G 3b a crowded subset.
Recall that S θ,i (Γ) denotes the ith -segment of Γ. We arbitrarily number the target θ-blocks For Γ ∈ G 3b we say S θ,i (Γ) (or just Γ) makes a transition from j to k if x (i−1) (Γ) is in fattened target θ-block j in H fat (i−1) and x i (Γ) is in fattened target θ 0 -block k in H fat i ; we call this transition the (i, j, k) transition and write S θ,i (Γ) ∈ T i (j, k). For Γ (1) , Γ (2) ∈ G 3b and i ≤ n β 2 , S θ,i (Γ (1) ) and S θ,i (Γ (2) ) are called neighboring if they make the same transition. A given transition (i, j, k) is called sparse if the number of 2R-geodesics in G 3b making that transition is at most n β 3 .
The definitions of "transition" and "neighboring," among others here, also make sense for general ( , θ)-interval geodesics that are not part of a 2R-geodesic.
Claim 1. If τ ∈ G 11 with crowded subset G 3b , then every -segment of every Γ ∈ G 3b making a non-sparse transition has a low-overlap neighbor.
To prove Claim 1, fix τ ∈ G 11 . For fixed i, j, k and Γ ∈ G with S θ,i (Γ) making a non-sparse (i, j, k) transition, suppose S θ,i (Γ) has no low-overlap neighbors. Then (4.90) We must deal with the technical complication that there may be backtracks, meaning that (i) not all projected overlap intervals, for S θ,i (Γ) and some S θ,i (Γ), are necessarily contained in [(i − 1) , i ], and (ii) having two projected overlap intervals intersect in an interval of positive length need not mean that the corresponding overlap segments have nonempty intersection.
Issue (i) is readily dealt with: since we are assuming τ ∈ G c 3 , every O(S θ,i (Γ), S θ,i (Γ)) in (4.90) is contained in [(i − 2) , i ]. It then follows from (4.90) that some point a ∈ [(i − 2) , i ] must be in at least 1 4 n β 3 −β 4 of these intervals. For issue (ii), let F a (Γ) and F a (Γ) be the first and last points, respectively, of Γ in H θ,a . Because we are assuming τ ∈ G c 3 and Γ has no low-overlap neighbors, if a lies in some O(S θ,i (Γ), S θ,i (Γ)), then the corresponding overlap segment S θ,i (Γ) ∩ S θ,i (Γ) must contain either F a (Γ) or F a (Γ), because no segment of Γ lying entirely between F a (Γ) and F a (Γ) can have projected length more than n −β 4 /2. It follows that either F a (Γ) is contained in S θ,i (Γ)∩S θ,i (Γ) for at least 1 8 n β 3 −β 4 of the neighborsΓ in (4.90), or the same is true for F a (Γ). But this makes F a (Γ) or F a (Γ) a popular site for G 3b , a contradiction to τ ∈ G 11 . This proves Claim 1.
Claim 2. If τ ∈ G 11 with crowded subset G 3b , then there exists Γ ∈ G 3b such that every -segment of Γ has a low-overlap neighbor.
To prove Claim 2, note that for each i, the number of possible transitions by the ith -segment of a target-directed 2R-geodesic is at most the square of the number of target θ-blocks in each H θ,i , so the number of (i, j, k) such that -segment i can transition from block j to block k is at Figure 11. Geodesic Γ (dashed curve) for which every -segment of the pre-H θ,R segment has a low-overlap neighbor (gray curves.) The low-overlap neighbors are -segments of other geodesics in the crowded subset G 3b of the crowded group G. Later in the proof we allow similar but more general "low-overlap partners," which are still geodesics but which do not have to be -segments of geodesics in the crowded subset. Of primary interest are the fast -segments of Γ, for which the corresponding partners are (modulo a small-probability event) "forced" by Lemma 4.8 to be disjointly semi-fast.
Our definition of low-overlap neighbor requires that such a neighbor be an -segment of another 2R-goedesic in our specified G 3b . We now loosen this restriction, and say for Γ a 2R-geodesic, any ( , θ)-interval geodesic Ψ is a low-overlap partner of the ith -segment of Γ if the following hold: (a) Ψ is a good ( , θ)-interval geodesic, (b) Ψ and S θ,i (Γ) make the same transition, and (c) the θ-overlap of Ψ and S θ,i (Γ) is at most n −β 4 .
It follows that every low-overlap neighbor is a low-overlap partner, for τ ∈ G 11 . Define the event G 12 : there exist a target-directed 2R-geodesic from B rfat θ,home to H + θ,2R with H + θ,R -entry point in B fat θ,cross for which every -segment in the pre-H θ,R segment has a low-overlap partner; τ / ∈ G bad .
We then conclude from Claim 2 that (4.91) We now bound P (G 12 ). Define Let γ 0 be a θ-slab path from B rfat θ,home to B fat θ,cross . Given x, y ∈ Ω i ∩ V we can define the set of low-overlap constrained paths P i (x, y, γ 0 ) = γ : γ is a path from x to y in Z d with γ ⊂ Ω i , and either γ ∩ γ 0 = ∅ or for some . Note that part of the passage time T (γ) comes from the overlap segment γ [u, v]; in defining T dis,i (γ, γ 0 ) we replace this part of the passage time with the approximation h((v θ 1 − u θ 1 )y θ ), so that it does not depend on the passage times of bonds in the overlap segment. Next define the disjoint passage times T dis,i (x, y, γ 0 ) := inf{T dis,i (γ, γ 0 ) : γ ∈ P i (x, y, γ 0 )}, (4.93) so that T dis,i (x, y, γ 0 ) is not affected by the passage times of the bonds in γ 0 .
The case of interest is the following: given τ / ∈ G bad , if Γ is a θ-slab geodesic from B rfat θ,home to B fat θ,cross , and Ψ = Ψ[x, y] is a low-overlap partner of S θ,i (Γ), then, due to the bound on backtracks in low-overlap partners we have Ψ Since τ / ∈ G 7 , this yields that for all γ ∈ P i (x, y, Γ) (4.94) Since also τ / ∈ G 6 , For Γ a θ-slab geodesic from B rfat θ,home to B fat θ,cross , we say Γ is clean if Γ contains no θ-backtrack of Thus for τ ∈ G 12 we have the following: there exists a clean θ-target-directed geodesic Γ from B rfat θ,home to B fat θ,cross for which every S θ,i (Γ) has a low-overlap partner Ψ i = Ψ i [y i , z i ], and for each fast -segment S θ,i (Γ), so we now want to bound P (G 13 ). Let a 1 , . . . , a N L and b 1 , . . . , b N R be the sites of Z d in J L and J R respectively, and let n 0 = ηn γ 2 β 2 /8.

Defining events
H jk : Γ a j b k is a clean θ-target-directed θ-slab geodesic, N θ (Γ a j b k ) ≥ n 0 , and every fast -segment S θ,i (Γ a j b k ) with 2 ≤ i ≤ n β 2 − 1, has a disjointly semi-fast low-overlap partner, we have (We note here that when τ ∈ H jk , the corresponding Γ a j b k can serve as the special geodesic in the heuristic in Remark 4.1.) Using the last inequality in (4.16), Lemma 4.2 yields that We next bound P (H jk ). Suppose we fix both of the following: (4.99) a clean θ-target-directed θ-slab path γ from a j to b k , and the times τ γ = {τ e : e ∈ γ}.
These determine the set For each such γ we have the event H γ : every fast -segment S θ,i (γ), 2 ≤ i ≤ n β 2 − 1, has a disjointly semi-fast low-overlap partner.
Conditionally on the fixed objects in (4.99), we may view the events H γ , as well as the event {Γ u j v k = γ}, as determined by the unconditioned passage times {τ e : e / ∈ γ}. In this context, {Γ u j v k = γ} is an increasing event (that is, its indicator is an increasing function of {τ e : e / ∈ γ}), whereas H γ is a decreasing event. It follows from the FKG inequality that We claim that In fact, suppose τ ∈ H γ,I \(G 3 ∪ G 6 ∪ G 7 ), i ∈ I, and (i, j, k) ∈ M(γ). Then there are sites u preceding v in S θ,i (γ) and a disjointly semi-fast ( , θ)-interval geodesic Ψ ⊂ Q R,n,θ for which Ψ makes transition (i, j, k), Ψ ∩ γ = γ[u, v], |u θ 1 − v θ 1 | ≤ n −β 4 . Since τ / ∈ G 7 and γ is clean, Thus Ψ is semi-fast. Since τ / ∈ G 3 , Ψ is also good, so τ ∈ F M(γ),I , proving the claim. This shows that Let u ij denote the center point of the jth block in H i , and define the event With (4.100) and (4.101) we obtain from this that for the functions As a comment on the last two expressions, we can view γ, I as parameters in the probability P (H γ,I ) expressed by the function f 0 , with this probability for a given (γ, I) calculated for a random configuration τ . When we calculate the second expectation in (4.105), we can view it as choosing the parameters γ, I randomly using a completely separate independent passage time configuration τ . Our ability to separate the choice of τ from the choice of parameters (functions of τ ) is a consequence of the FKG property in (4.100) and of (4.101). When we next replace f 0 with f 1 in the last line, the parametrization no longer uses the full path γ but rather only the transitions of γ. This formulation means that to bound P (H jk ), instead of summing over M and I in f 1 (M, I) (which involves too much entropy), we are averaging over Γ u j v k .
We split I into odd and even values, I = I odd ∪ I even and define for u, v ∈ Ω i : Fix M ; for each i ≤ n β 2 there exist unique j i , k i for which (i, j i , k i ) ∈ M . The events with i odd are independent, since the regions Ω i are disjoint. Therefore With (1.2) and Proposition 2.1, using u i,k i − u (i−1),j i ∈ H θ, , we obtain Since u i,k i , u (i−1),j i lie in Q R,n,θ , it follows readily from (4.24) that Combining this with (1.9) and (4.107) yields Since (4.106) also holds for I even , and max(I odd , I even ) ≥ |I|/2, this together with the last inequality in (4.16) shows that for |I| ≥ n 0 − 2, Combining this with (4.105) and Lemmas 4.7, 4.8, and 4.9 we see that for |I| ≥ n 0 − 2 and all M , With (4.91), (4.96), (4.97), (4.98), and (4.105) this shows P (τ ∈G 9 \G bad and option (IIIb) occurs) where we have used n β 1 ≥ (log R) 2 .
The preceding proof of (4.109) applies also to options (I) and (II); the proof could even be substantially simplified for these cases since there is no overlap in the relevant geodesics. To deal with option (IIIa) where the number of geodesics in the crowded subset is smaller, we can simply define m by n β 3 −β 4 = g 1/3 m , so m = c 12 n λ with λ < 1, and then repeat the entire proof with n replaced by m. (This has the effect of replacing each β j with λβ j .) We thus conclude using (4.87) and Lemmas 4.5-4.9 that which completes the proof of Proposition 4.10.

Entry point density bound
In this section we prove Theorem 1.5(ii) and Theorem 1.8.
Proof of Theorem 1.5(ii). The orientation of a finite geodesic from x to y is (y − x)/|y − x|. With θ 0 , R fixed, for Γ a θ 0 -slab geodesic from H − θ 0 ,−R to H + θ 0 ,R , the initial orientation of Γ is the orientation of its pre-H θ 0 ,0 segment. The term "initial orientation" also applies to θ-rays from H − θ 0 ,−R which cross H θ 0 ,0 . Let 1 be as in Remark 1.2 and min as in A3, let (R) = (log R) K 1 ∆ R /R, with K 1 to be specified, and for θ ∈ J(θ 0 , min ) define with H + θ 0 ,0 -entry point x and initial orientation in J(θ, (R)) , with H + θ 0 ,0 -entry point x and initial orientation not in J(θ, (R)) , We may think of W R and Y R as the sets of entry points of "normal" and "crooked" θ-rays, respectively, though formally W R is defined in terms of finite geodesics. Let us first bound the density of Y R . Suppose x ∈ Y R , θ ∈ J(θ 0 , min ), and there exists a θ-ray from H − θ 0 ,−R with starting point a ∈ H rfat θ 0 ,−R for which the initial orientation α = (x − a)/|x − a| / ∈ J(θ, (R)). This means ψ αθ ≥ (R), and it is easily checked that this implies But then, defining events F x,j : for some a ∈ Z d with 2 j−1 < |x − a| ≤ 2 j and some θ ∈ J(θ 0 , min ) with , there is a θ-ray from a containing x, we get from Proposition 3.1 that provided K 1 is large, We now turn to our main task, which is bounding the density of W R ; see Figure 12. We keep θ 0 , θ fixed throughout, with ψ θθ 0 < min . Let R > 0 and n = (log R) K 2 , with K 2 to be specified, satisfying (4.20). Define y R,θ = L θ ∩ H θ 0 ,−R , let B θ 0 ,home ⊂ H θ 0 ,0 be the home θ 0 -block as in Section 4, and define the "big block" centered at y R,θ : expressed in θ 0 -coordinates. As after Remark 4.3, we can define fattened versions B fat θ 0 ,home and B rfat −R,θ 0 ,θ,start,big . We want to show that the θ 0 -slab geodesic in the definition of W R must have starting point in B rfat −R,θ 0 ,θ,start,big . To that end we take a ∈ H rfat θ 0 ,−R \B rfat −R,θ 0 ,θ,start,big and x ∈ B fat Figure 12. Illustration for the proof of Theorem 1.5(ii), showing the event x ∈ W R . The corresponding θ 0 -slab geodesic Γ starts at some a ∈ B rfat −R,θ 0 ,θ,start,big (the block centered at y R,θ , delimited by hash marks in the figure), and enters H + θ 0 ,0 at x. The event x ∈ Y R is similar but Γ is a θ-ray and a is far from y R,θ , meaning the initial orientation of Γ is not close to θ. and let α = (x − a)/|x − a|. We will show that ψ αθ > (R), so that indeed a is not the starting point in question.
we have |z − z| ≤ d + √ d − 1n −β 0 ∆ R . It follows that, provided |u| (and hence R) is large, the number N d of such blocks depends only on d, and by (5.16), y must be in one of these blocks (backwards-fattened.) Suppose now that |W u ∩B fat | ≥ N d n; then some B rfat −R,z,j with j ≤ N d contains the starting points y of at least n of the corresponding θ 0 -slab geodesics from the definition ofW u . Let y j be the center of B rfat −R,z,j and ζ j = (z − y j )/|z − y j |. To apply Proposition 4.10 we need to bound ψ θ 0 ζ j ; for this we will use (5.17) ψ θ 0 ζ j ≤ ψ θ 0 ,αuv + ψ αuv,α u0 + ψ α u0 ,αzy + ψ αzy,α zy j The first two angles on the right are bounded by assumption and by (5.12), so we will bound the last two. From the bounds on |z − x| and |y − (x + u)| in the definition ofW u , provided |u| is large we have ψ α u0 ,αzy ≤ c 9 (log |u|) 2/(1−χ 2 ) /|u| ≤ min /8. Since the pairs y, y j and z, z each lie in the same (forwards or backwards) fattened block, we have ψ αzy,α zy j ≤ c 440 n −β 0 ∆ R /R < min /8, again provided |u| (and hence R) is large. Provided we take 4 ≤ min /8 we thus obtain from (5.17) that ψ θ 0 ζ j < 4 + 3 min /2 < min /2, and then from Proposition 4.10 that, provided K is large enough, and therefore Now x ∈ X uv implies z ∈W u for some z with |z − x| ≤ c 5 (log |u|) 2/(1−χ 2 ) , so we have Observe that P (x ∈ X uv ) = P (F x ) has period Λ 0 , in the sense that it takes the same value at x and x + b j for all x and all j ≤ d − 1. Therefore, letting q = |{n : Λ n ⊂B}|, we have from (5.18) Provided |u| (and henceB) is large, we have q|Λ 0 | ≥ |B|/2 so which with (5.19) shows that, provided K is large, This and (5.15) complete the proof.

Nonexistence of bigeodesics
In this section we prove parts (iii) and (iv)(c) of Theorem 1.5. We begin with (iii); the main idea is that all bigeodesics as in (iii) are θ-rays in one direction and (−θ)-rays in the other, for some θ ∈ J(θ 0 , 2 ), and the crossing-point density of such bigeodesics is bounded above by ρ J(θ 0 , 2 ),R for all R, up to a small error term.
We claim (x −i , i ≥ 0) is a (−θ)-ray for every such θ and Γ, a.s. If not, some such θ-ray is a subsequential ϕ-ray for some ϕ = −θ, so there exists i k → ∞ for which x −i k /|x −i k | → ϕ. Letting r k = |x 0 − x −i k |, the distance from x 0 to the ray {x −i k + tθ : t ≥ 0} is then of order ψ ϕ,−θ r k , and it is easily checked that we therefore have for all sufficiently large k But for the events F δ,j : for some x ∈ Z d with 2 j−1 < |x| ≤ 2 j and some θ ∈ J(θ 0 , ) with , there is a θ-ray from x containing 0, for all R > 0, and, by the preceding remark about x ∈ C bi,R− J(θ 0 , ),0 (A), the first lim sup on the right is bounded above by ρ J(θ 0 , ),R . The second lim sup is bounded by c 6 P for some θ ∈ J(θ 0 , ) there exists a θ-ray from 0 which intersects H θ 0 ,−R , which (cf. Lemma 4.7) is readily shown to approach 0 as R → ∞. Since R is arbitrary, provided ≤ 2 it then follows from Theorem 1.5(ii) that ρ bi J(θ 0 , ),0 = 0. Since θ 0 is rationally oriented, periodicity of P (x ∈ C bi J(θ 0 , ),0 (H fat θ 0 ,0 )) then means that we have P (x ∈ C bi J(θ 0 , ),0 (H fat θ 0 ,0 )) = 0 for all sites x ∈ H fat θ 0 ,0 , which proves Theorem 1.5(iii). Then (iv)(c) follows from (iii) and compactness of S d−1 .

Coalescence time bounds
In this section we prove Theorem 1.7. We start with the upper bound on P ((U θ xy ) θ 1 ≥ r), as the lower bound is much simpler.
Let 3 = min( 0 /2, 2 /2, 6 ), where 2 is from Theorem 1.5 and 6 from Lemma 2.2. Let θ ∈ J(θ 0 , 3 ) and let x, y be θ-start sites with second coordinates x 2 < y 2 . Provided |y − x| is large, there existsθ with ψ θθ < 3 (and thus ψ θ 0θ < 2 3 ) for which y − x ∈ Hθ ,0 . We may assume y − x makes an angle of at least π/4 with the horizontal axis; then there is exactly oneθ-start site z k at each integer height k. We have P ((U θ xy ) θ . Now one of y − x, x − y lies in H − θ,0 so is aθ-start site; in case it is y − x then the preceding shows that x, y can be replaced by 0, y − x, or equivalently, we may assume x = 0. The proof is the same in the other case, so we will indeed assume x = 0 and y ∈ Hθ ,0 .
For points u, v ∈ Hθ ,0 we use notation [u, v] for the interval from u to v in Hθ ,0 ; we call such an interval an H-interval. From (2.8) and Proposition 3.1(ii), provided 3 is small we have P (U θ 0y ) θ 1 ≥ r ≤ P (U θ 0y )θ 1 ≥ ≤ P (U θ 0y )θ 1 ≥ r 2 + c 1 e −c 2 Φ(r) , (7.1) so it is sufficient to prove (1.22) with (U θ 0y ) θ 1 replaced by (U θ 0y )θ 1 . For eachθ-start site z letz be its projection horizontally into Hθ ,0 . (Throughout this section, "projection" will mean horizontal projection into Hθ ,0 , unless stated otherwise.) Let Zθ denote the set of allθ-start sites, and Sθ ,θ ⊂ Zθ the subset which are θ-sources. Let r > 1, and for each z ∈ Zθ let V z be the last site of Γ θ z in H − θ,0 (so necessarily V z ∈ Sθ ,θ ) and let W z be the H + θ,r -entry point of Γ θ Vz . A (θ, θ, r)-gap is an open H-interval I in Hθ ,0 with the properties that (i) I contains no projected θ-source V z , and (ii) the endpoints v, w of I are projections of sources v, w ∈ Sθ ,θ with W v = W w . A (θ, θ, r)-entry interval is the closed interval between two successive (θ, θ, r)-gaps. It then follows from planarity of Z 2 that any two θ-sources v, w satisfy W v = W w if and only if v, w lie in the same (θ, θ, r)-entry interval; thus the gaps separate those groups of Hθ ,r Figure 14. Top: The gray θ-rays share W u+y as their common H + θ,r -entry point; the dashed ones similarly share W u . The thick segment in Hθ ,0 is the (θ, θ, r)-gap G u,u+y , separating those gray and dashed θ-rays which are halfspace θ-rays. The hash marks on Hθ ,0 show the corresponding enlarged (θ, θ, r)-gap (G min , G max ); the lowest start point of a gray geodesic is at G min , and the highest start point of a dashed one is at G max . Bottom: The event A uw . The H + θ,r -entry points from u and w are different, but V u = V w . halfspace θ-rays from H − θ,0 which coalesce before crossing Hθ ,r . (See Figure 14.) Equivalently, (7.2) v, w ∈ Sθ ,θ , (U θ vw )θ 1 ≥ r =⇒ there is a (θ, θ, r)-gap between v and w.
(It should be emphasized that this is only true for θ-sources, not generalθ-start sites.) We note thatθ-start sites are periodic in the sense that u is aθ-start site if and only if u + y is one. We now consider translates of the events in (7.1) corresponding toθ-start sites u, u + y in place of 0, y. We have for all u ∈ Zθ: P (U θ u,u+y )θ 1 ≥ r = P (U θ 0y )θ 1 ≥ r − uθ 1 ≥ P (U θ 0y )θ 1 ≥ r + µ √ d , and therefore, averaging over a period, (7.3) P (U θ 0y )θ 1 ≥ r + µ √ d ≤ 1 y 2 y 2 −1 k=0 P (U θ z k ,z k +y )θ 1 ≥ r .
Case (2) is more complicated. We have |G u,u+y | ≤ |V u − V u+y | ≤ |y| + 2(log r) c 4 ; we call any (θ, θ, r)-gap G short if |G| ≤ |y| + 2(log r) c 4 . Then P (U θ u,u+y )θ 1 ≥ r, τ / ∈ A u,u+y , and Case (2) holds ≤ P d(u, G) ≤ |y| + 2(log r) c 4 for some short (θ, θ, r)-gap G . (7.6) We now take u = z k and consider the average as in (7.3). Define events Q k : (U θ z k ,z k +y )θ 1 ≥ r, τ / ∈ A z k ,z k +y , R k : d(z k , G) ≤ |y| + 2(log r) c 4 for some short (θ, θ, r)-gap G in Hθ ,0 . From (7.4), (7.5), and (7.6), [P (Q k ) + P (A z k ,z k +y )] ≤ 1 y 2 y 2 −1 k=0 P (R k ) + 2C 35 e −C 36 (log r) 2 . (7.7) We need to bound the average on the right in (7.7). Let J t = {y ∈ Hθ ,0 : |y| ≤ t} and let N t be the number of short (θ, θ, r)-gaps G intersecting J t . Each corresponding (θ, θ, r)-entry interval (between two such gaps) contains the projection of a θ-source, and the halfspace θ-rays from these sources have different H + θ,r -entry points for each (θ, θ, r)-entry interval. It then follows using Theorem 1.5(ii) that for ρ θ (r) from (1.17), (7.8) lim sup We now use periodicity as in the proof of Theorem 1.8. By the multidimensional ergodic theorem (see [18], Appendix 14.A), we have Combining this with (7.7) we obtain P ((Uθ 0y ) θ 1 ≥ r + µ √ d) ≤ c 6 (log r) C 16 +c 4 |y| ∆ r , which establishes the upper bound in (1.22). Turning to the lower bound, we need to consider the fact that z and V z may lie on opposite sides of a gap. Given a (θ, θ, r)-gap G, we let G max , G min denote respectively the highest (lowest) projectedθ-start site z for which V z lies below (above) G. It is not necessarily true that G min is below G max , but G max is in or above G, G min is in or below G, and G max is at most one vertical unit below G min . It is easily seen that for everyθ-start site z we have it then follows readily from Proposition 3.1(ii) that G min , G max always exist for all G, a.s. We call G + := G ∪ (G min , G max ) an enlarged (θ, θ, r)-gap. We say an enlarged (θ, θ, r)-gap G + is semi-short if |G + | ≤ |y| + (log r) c 4 , and very long otherwise. A key observation is that (7.11) one of z = G min , G max must satisfy |z − V z | ≥ 1 2 |G + | − 1. Hθ ,r G V w k +y V w k Figure 15. The gray curves are θ-rays. The gap G is marked by the hash marks on Hθ ,0 ; the enlarged gap G + is the thickened part of Hθ ,0 . We fix a height k ∈ [0, y 2 ); the row of dots to the left of Hθ ,0 (other than V w k and V w k +y ) are the sites z k +iy, i ∈ Z. w k is the lowest such site for which V w k and V w k +y lie on opposite sides of G. There is such a site w k for each k ∈ [0, y 2 ), creating y 2 occurrences of events F j . Necessarily at least one of w k , w k + y lies in G + .
Let a r = ∆ r log r and define overlapping intervals in Hθ ,0 : I j = [ja r , (j + 2)a r ] , and define events Y j : I j intersects a semi-short enlarged (θ, θ, r)-gap G + .
Suppose I j intersects a semi-short enlarged (θ, θ, r)-gap G + , for some gap G. For each 0 ≤ k < y 2 , consider the points {z k + iy : i ∈ Z}; let w k be the lowest such point for which V z k +(i−1)y and V z k +iy are on opposite sides of G. Then w k ∈ G + , and the points {w k : 0 ≤ k < y 2 } are all distinct. This shows that {u ∈ Zθ : u ∈ G + , V u and V u+y are on opposite sides of G} ≥ y 2 ; see Figure 15. Now, any given semi-short enlarged (θ, θ, r)-gap G + intersects at least one and at most three H-intervals I j . It follows that for the events F k : V z k and V z k +y are on opposite sides of some (θ, θ, r)-gap G for which G + is semi-short we have Hθ ,r H θ,w θ 1 Figure 16. The event that L 0 = [0, 2a r ] intersects no enlarged (θ, θ, r)-gap.
Because |v − u| ≈ |p v − p u | is large, the common H + θ,r -entry point w from u and v must be far from either p u or p v . This remains true when we shift slightly from θ-coordinates (black lines) to θ-coordinates (gray lines), so Γ θ u or Γ θ v must make a large transverse fluctuation to pass through w. and hence by the ergodic theorem, Here the second inequality reflects that when τ ∈ F k , the H + θ,r -entry points W z k = W z k +y , so either coalescence occurs in H + θ,r/2 or both geodesics Γ θ z k , Γ θ z k +y backtrack to H − θ,r/2 after entering H + θ,r . The third inequality reflects that (i) the first probability on the second line is maximized over k when z k , z k + y lie in Hθ ,0 , and (ii) the second probability on the second line can be bounded as in (7.4). (Also in (7.12), for technical convenience in applying the ergodic theorem, we have assumed ∆ r log r is an integer multiple of y 2 , ensuring P (Y j ) is the same for all j. The added technicalities without this assumption are tedious but straightforward, using our assumption |y| ≤ ∆ r .) Next we show P (Y 0 ) is near 1; with (7.12) this will complete the proof of the lower bound in (1.22). We have P (Y c 0 ) ≤ P (I 0 intersects no enlarged (θ, θ, r)-gap) + P (I 0 intersects a very long enlarged (θ, θ, r)-gap). (7.13) Let us consider the first probability on the right of (7.13). Suppose I 0 intersects no enlarged (θ, θ, r)-gap; then I 0 is contained in some (θ, θ, r)-entry interval [u, v]. See Figure 16. This means u, v are θ-sources with |u − v| ≥ 2∆ r log r, and with W u = W v = w for some w. Now the angle betweenθ and Hθ ,r is at least π/4 by (2.9), and it follows by straightforward geometry that the points p u = Lθ(u) ∩ Hθ ,r and p v = Lθ(v) ∩ Hθ ,r satisfy Provided ψ θθ is small, when we change the angle to θ and consider q u = L θ (u) ∩ H θ,w θ 1 and q v = L θ (v) ∩ H θ,w θ 1 , we have via straightforward use of (2.11) and (2.12) that Since w, q u , q v all lie in H θ,w θ 1 follows that We may assume the first entry in the maximum is the larger one. Then using (2.8), for some c 10 , c 11 , LetK r = log 2 (2∆ r log r) , and define events E k : for some u, v ∈ Zθ with I 0 ⊂ [u, v] and 2 k < |u − v| ≤ 2 k+1 , sup w∈Γ θ u D θ (w − u) ≥ D * (2 k ). (7.14) The preceding together with Proposition 3.1(ii) then shows that P (I 0 intersects no enlarged (θ, θ, r)-gap) ≤ We now turn to the last probability in (7.13). Suppose I 0 intersects a very long enlarged (θ, θ, r)gap (f , g). Then by (7.11), one of z = f or g satisfies |z − V z | ≥ 1 3 |f − g| ≥ 1 3 (|y| + (log r) c 4 ) and D θ (V z − z) ≥ c 14 Φ(|z − V z |). Letting K(r, y) = log 2 ( 1 3 (|y| + (log r) c 4 )) and defining events M k : for someθ-start site z with d(z, I 0 ) ≤ 2 k we have D θ (V z − z) ≥ c 14 Φ(2 k−1 /3), we see that if 2 k−1 < |f − g| ≤ 2 k then τ ∈ M k . It follows using Proposition 3.1(ii) that P (I 0 intersects a very long enlarged (θ, θ, r)-gap) ≤ ≤ e −c 18 (log r)(log log r)/2 . (7.16) With (7.13) and (7.14) this shows that P (Y 0 ) ≥ 1/2; with (7.12) this completes the proof of the lower bound in (1.22).
The general outline of this proof, and in particular the use of gaps, is analogous to ([7], Section 6), with added complications due to the undirected nature of paths here, which means not all start sites are sources, and backtracking may occur after coalescence.