Subgaussian rates of convergence of means in directed first passage percolation
aa r X i v : . [ m a t h . P R ] M a y SUBGAUSSIAN RATES OF CONVERGENCE OF MEANS IN DIRECTEDFIRST PASSAGE PERCOLATION
KENNETH S. ALEXANDER
Abstract.
We consider directed first passage percolation on the integer lattice, with timeconstant µ and passage time a n from the origin to ( n, , . . . , Ea n − nµ = O ( n / (log log n ) / log n ). Introduction
The question of the order of the fluctuations (i.e. the standard deviation) for the passagetime over a distance n in first passage percolation (FPP) has attracted considerable interest.FPP is believed to be one of a family of models in which these fluctuations are of order n / in two dimensions, with a Tracy-Widom distributional limit under corresponding scaling. Inthree dimensions the conjectured order is n χ with χ near 1/4. Little is known rigorously inthis regard, however, and even physicists do not agree on the behavior of the exponent χ asthe dimension becomes large. For directed last passage percolation in two dimensions withgeometrically distributed passage times, the n / scale and the Tracy-Widom distributionallimit were proved in [9], but the “exact solution” methods used there do not seem conduciveto use with more general passage time distributions. (A more probabilitistic proof for ge-ometrically distributed passage times is in [8].) For longest increasing subsequences, whichcan be transformed into an FPP-like problem, analogous results were established in [5]. Forfirst passage percolation, fluctuations are known to be subgaussian, but barely—Benjamini,Kalai and Schramm [7] showed that for Bernoulli-distributed passage times, the fluctuationsare O (( n/ log n ) / ), and Bena¨ım and Rossignol [6] proved exponential bounds, on the scale( n/ log n ) / , for the fluctuations under more general passage time distributions, improvingKesten’s exponential bounds [10] which are on scale n / .Here by an exponential bound on scale c n we mean that, letting a n be the passage timefrom the origin to ( n, , ..., P ( | a n − Ea n | > tc n ) decaying exponentially in t ,uniformly in n .Besides deviations from the mean, it is of interest to understand deviations from theasymptotic value nµ , where µ is the time constant. This means that, letting a n be thepassage time from the origin to ( n, , ..., Ea n − nµ . Subadditivity of Date : June 23, 2018.2010
Mathematics Subject Classification.
Primary 60K35.
Key words and phrases. first passage percolation, time constant, subadditivity.Research supported by NSF grant DMS-0804934. Ea n ensures that this difference is nonnegative. The best known bound, in [2], is of order n / log n , obtained with the help of Kesten’s scale- n / exponential bounds on fluctuations.In [3], analogous results are established for a wider class of models, and the proofs thereshow that for FPP in two or three dimensions, if one has an exponential bound on somescale c n , then(1.1) Ea n − nµ = O ( c n log n ) . For the proven scale c n = ( n/ log n ) / from [6], though, this approach obviously does notlead to a subgaussian (i.e. o ( n / )) bound on Ea n − nµ . Our aim here is to establish sucha subgaussian bound by other methods, showing that in the directed case, for a reasonablywide class of passage time distributions, we have(1.2) Ea n − nµ = O (cid:18) n / log log n (log n ) / (cid:19) . To formalize things, we write sites of Z d +1 as ( n, x ) with n ∈ Z , x ∈ Z d . Let L d +1 be theeven sublattice of Z d +1 : L d +1 = { ( n, x ) ∈ Z d +1 : n + x + · · · + x d is even } . Sites ( n, x ) and ( n + 1 , y ) in L d +1 are adjacent if the Euclidean distance | y − x | = 1. A bond h ( n, x ) , ( n + 1 , y ) i is a pair of adjacent sites, and B d +1 denotes the set of all bonds in L d +1 . We assign i.i.d. nonnegative passage times { ω b : b ∈ B d +1 } to the bonds in B d +1 , withdistribution ν . A (directed lattice) path from ( n, x ) to ( n + m, y ) in L d +1 is a sequence ofsites (( n, x (0) ) , ( n + 1 , x (1) ) , . . . , ( n + m, x ( m ) )) in which consecutive sites are adjacent, or itmay be viewed as the corresponding sequence of bonds; which one should be clear from thecontext. The passage time for a path γ is T ( γ ) = X b ∈ γ ω b . For sites ( n, x ) , ( n + m, y ) in L d +1 , we then define T (( n, x ) , ( n + m, y )) = min { T ( γ ) : γ is a lattice path from ( n, x ) to ( n + m, y ) } and for n even, a n = T ((0 , , ( n, . A geodesic is a path which achieves this minimum; the geodesic is a.s. unique when ω b hasa continuous distribution. The time constant µ = µ ( ν, d ) is(1.3) µ = lim n Ea n n = inf n Ea n n = lim n a n n a.s. , the existence of the first limit (taken through even n ) being a consequence of subadditivityand the second being a consequence of Kingman’s subadditive ergodic theorem [11].In order to use the exponential bound of [6], we need to assume that ν is a nearly gammadistribution, as defined in [6]. To state the definition, let F and Φ be the d.f.’s of ν and UBGAUSSIAN RATES OF CONVERGENCE IN FPP 3 the standard normal distribution, respectively. Assume ν has a density f , and let ϕ be thedensity of the standard normal. Define I = { t ≥ f ( t ) > } and on I define the functionΥ( y ) = ϕ ◦ Φ − ( F ( y )) f ( y ) . The distribution ν is said to be nearly gamma if I is an interval, f restricted to I is continuous,and for some positive A , Υ( y ) ≤ A √ y for all y ∈ I. For a standard normal variable ξ , F − (Φ( ξ )) has distribution ν , and the nearly gammaproperty ensures that the map F − ◦ Φ is nice enough that a log Sobolev inequality for ξ translates into useful information about ν ; see [6]. Most common continuous distributionsare nearly gamma—a sufficient condition for the property, from [6], is as follows. Let a < b be the (possibly infinite) endpoints of I . Suppose that for some α > − f ( x )( x − a ) α stays bounded away from 0 and ∞ as x ց a, and either (i) b < ∞ and for some β > − f ( x )( b − x ) β stays bounded away from 0 and ∞ as x ր b, or (ii) b = ∞ and f ( x ) R ∞ x f ( u ) du stays bounded away from 0 and ∞ as x ր ∞ . Then ν is nearly gamma.We can now state our main result. Theorem 1.1.
Suppose ν is nearly gamma and R e tx ν ( dx ) < ∞ for some t > . Then foreven n , (1.4) Ea n − nµ = O (cid:18) n / log log n (log n ) / (cid:19) . This bound is likely not sharp—an exponential bound on the expected scale n / wouldlead to (1.1) with c n = n / , for example.2. Proof of Theorem 1.1
We consider paths { ( i, x ( i ) ) : 0 ≤ i ≤ kn } from (0 ,
0) to ( kn, n to be specified and k ≥
1. In general we take n sufficiently large, and then take k large,depending on n ; we tacitly take n to be even, throughout. The simple skeleton of such a pathis { ( jn, x ( jn ) ) : 0 ≤ j ≤ k } . The number of possible simple skeletons is at most (2 n ) dk , so ifwe have a probability associated to each skeleton which is bounded by some quantity e − r n k ,then in order for the corresponding bound (2 n ) dk e − r n k on the sum to be small, we should KENNETH S. ALEXANDER have r n at least of order log n . This principle, in a different context, underlies the log n factor which appears in (1.1). In (1.4) we essentially want to replace this log n with log log n .Since (2 n ) dk is unacceptably large for this, it requires entropy reduction—the replacementof a sum over all simple skeletons with the sum over a small subclass which can be shown toapproximate the full class of simple skeletons appropriately. This entropy reduction is themain theme of our proof.Our main technical tool is the following. It is proved in [6] for undirected FPP, but theproof for the directed case is the same. The result there carries the additional hypothesisthat t ≤ m , but that is readily removed—the proof is in Section 3. Theorem 2.1.
Suppose ν is nearly gamma and R e tx ν ( dx ) < ∞ for some t > . Then thereexist C , C > such that for all m ≥ and ( m, x ) ∈ L d with | x | ≤ m , (2.1) P (cid:12)(cid:12) T ((0 , , ( m, x )) − ET ((0 , , ( m, x )) (cid:12)(cid:12) > t (cid:18) m log m (cid:19) / ! ≤ C e − C t . If the exponential bound (2.1) could be improved to some scale c m ≪ ( m/ log m ) / , thenthe rate on the right side of (1.4) could improved to O ( c n log log n ), with the proof virtuallyunchanged.Let H m = ( { m } × Z d ) ∩ L d . A routine extension of Theorem 2.1 to sums of passage timesis as follows. We postpone the proof to Section 3. Lemma 2.2.
Let ν, C , C be as in Theorem 2.1, let n max ≥ and let ≤ s < t ≤ s
0) to ( kn,
0) for which the simple skeleton is a CGskeleton. We proceed as follows. Given the geodesic Γ from (0 ,
0) to ( kn, n ≤ dn/ϕ ( n ) of eachhyperplane H jn , and in such a way that with high probability the passage times of Γ and ˜Γare close. (Here n is an even integer to be specified, cf. Lemma 2.3.) Once this is done, wewould hope to achieve entropy reduction by effectively considering only CG paths. What weactually do is slightly different—˜Γ is not necessarily a true CG path but rather a sequence ofsegments each (with certain exceptions) having CG points as endpoints, with one segmentnot required to connect to the next one. But the effect is the same.To this end, given a site w = ( jn ± n , y ( jn ± n ) ) ∈ H jn ± n , let z ( jn ) be the site in u n Z d closestto y ( jn ± n ) in ℓ norm (breaking ties by some arbitrary rule), and let π jn ( w ) = ( jn, z ( jn ) ),which may be viewed as the projection into H jn of the CG approximation to w within thehyperplane H jn ± n . Given the geodesic Γ = { ( i, x ( i ) ) } from (0 ,
0) to ( kn, d j = d j (Γ) = ( jn, x ( jn ) ) , ≤ j ≤ k,e j = ( jn + n , x ( jn + n ) ) , ≤ j ≤ k − ,f j = ( jn − n , x ( jn − n ) ) , ≤ j ≤ k. We say a sidestep occurs in block j if either | x (( j − n + n ) − x (( j − n ) | ∞ > h n or | x ( jn ) − x ( jn − n ) | ∞ > h n . Let E ex = E ex (Γ) = { ≤ j ≤ k : x ( jn ) − x (( j − n ) is excessive } , E side = E side (Γ) = { ≤ j ≤ k : j / ∈ E ex and a sidestep occurs in block j } , E = E ex ∪ E side and let e ′ j − = π ( j − n ( e j − ) , f ′ j = π jn ( f j ) , j / ∈ E . KENNETH S. ALEXANDER
Define the tuples R j = R j (Γ) = ( d j − , d j ) if j ∈ E ex , ( d j − , e j − , f j , d j ) if j ∈ E side , ( e ′ j − , f ′ j ) if j / ∈ E , and define the CG-approximate skeleton of Γ to be S CG (Γ) = {R j : 1 ≤ j ≤ k } . Note E side (Γ) is a function of S CG (Γ). Let C denote the class of all possible CG-approximateskeletons of paths from (0 ,
0) to ( kn, B ⊂ { , . . . , k } let C B denote the class of allCG-approximate skeletons in C with E = B . S CG (Γ) is thus a sequence of tuples of sites; ineach tuple the first and last sites will be used as the endpoints of a path which approximatesone of the k segments of Γ.For a CG-approximate skeleton in C B , if R , . . . , R j − are specified and j ∈ B , then thereare at most (2 h n u − n + 1) d ≤ (3 ϕ ( n )) d possible values of R j ; if j / ∈ B there are at most(2 n ) d . It follows that the number of CG-approximate skeletons satisfies(2.5) |C B | ≤ (3 ϕ ( n )) d ( k −| B | ) (2 n ) d | B | . Note that the factor ϕ ( n ) in place of n in (2.5) represents the entropy reduction discussedabove. For each j / ∈ E let g j (Γ) be the path from e ′ j − to f ′ j , via e j − and f j , obtained fromΓ by replacing the segment of Γ from d j − to e j − with the geodesic from e ′ j − to e j − , andreplacing the segment of Γ from f j to d j with the geodesic from f j to f ′ j . For j ∈ E let g j (Γ)be the segment of Γ from d j − to d j . Then define the collection of paths˜Γ = { g j (Γ) : 1 ≤ j ≤ k } . Note that for j / ∈ E , g j (Γ) has CG points as endpoints. We define the passage time in thenatural way: T (˜Γ) = k X j =1 T ( g j (Γ)) . Let b nk = ⌊ k log log n log n ⌋ . Taking k sufficiently large (depending on n ), we have12 < P (cid:18) T (Γ) < knµ + 64 dkn / ψ ( n ) (cid:19) ≤ P (cid:18) T (Γ) < knµ + 64 dkn / ψ ( n ) , |E (Γ) | > b nk (cid:19) + P (cid:18) T (˜Γ) − T (Γ) > dkn / ψ ( n ) (cid:19) + P (cid:18) T (˜Γ) < knµ + 128 dkn / ψ ( n ) , |E (Γ) | ≤ b nk (cid:19) , (2.6)where the first inequality follows from the definition of µ . We will show that the first twoprobabilities on the right side are small, and then, since the third probability cannot also be UBGAUSSIAN RATES OF CONVERGENCE IN FPP 7 small, we will see that we must have ET ( v ′ j − , w ′ j ) < nµ + 256 dn / ψ ( n ) for some j ≤ n, v ′ j − ∈ H ( j − n , w ′ j ∈ H jn , which in turn leads easily to (1.4). For the second probability on the right side of (2.6) wehave P (cid:18) T (˜Γ) − T (Γ) > dkn / ψ ( n ) (cid:19) ≤ X B ⊂{ ,...,k } X { R j }∈C B P (cid:18) T (˜Γ) − T (Γ) > dkn / ψ ( n ) , E (Γ) = B, S CG (Γ) = { R j } (cid:19) . (2.7)Given B ⊂ { , . . . , k } and { R j } ∈ C B , for some v ′ j − , w ′ j , p j − , v j − , w j , p j we can express { R j } as R j = ( v ′ j − , w ′ j ) , if j / ∈ B, ( p j − , p j ) if j ∈ B and R j is a 2-tuple , ( p j − , v j − , w j , p j ) if j ∈ B and R j is a 4-tuple . Then let I B ( { R j } ) be the set of all tuple collections { ( p j − , v j − , w j , p j ) : j / ∈ B } compatiblewith { R j } and having the following properties: p j − ∈ H ( j − n , | p j − − v ′ j − | ∞ ≤ h n ,v j − ∈ H ( j − n + n , π ( j − n ( v j − ) = v ′ j − ,w j ∈ H jn − n , π jn ( w j ) = w ′ j ,p j ∈ H jn , | p j − w j | ∞ ≤ h n . Here by “compatible with { R j } ” we mean that the choice of { R j } specifies values p i forcertain i (specifically, for i = j and i = j − j ∈ B ); the tuple collections in I B ( { R j } ) must use these same values p i . Note that p j − , v j − , w j , p j are possible values ofthe variables d j − , e j − , f j , d j , respectively, and(2.8) |I B ( { R j } ) | ≤ (4 h n + 1) d | B | ≤ n d | B | . Then let T skel ( { R j } ) = X j / ∈ B T ( v ′ j − , w ′ j ) + X j ∈ BR j a 2-tuple T ( p j − , p j )+ X j ∈ BR j a 4-tuple (cid:2) T ( p j − , v j − ) + T ( v j − , w j ) + T ( w j , p j ) (cid:3) . (2.9) KENNETH S. ALEXANDER
Then P (cid:18) T (˜Γ) − T (Γ) > dkn / ψ ( n ) , E (Γ) = B, S CG (Γ) = { R j } (cid:19) ≤ X { ( p j − ,v j − ,w j ,p j ): j / ∈ B }∈ I B ( { R j } ) ( P X j / ∈ B (cid:2) T ( v ′ j − , v j − ) − T ( p j − , v j − ) (cid:3) > dkn / ψ ( n ) + P X j / ∈ B (cid:2) T ( w j , w ′ j ) − T ( w j , p j )) (cid:3) > dkn / ψ ( n ) ) . (2.10)It may be noted that we have modified Γ into ˜Γ so that we need only sum over CG-approximate skeletons, rather than over all simple skeletons, and we thereby achieved entropyreduction, but now in (2.10), the number of terms in the sum effectively increases the entropyback to its original level. However we have the advantage that all the path lengths involvedin (2.10) are at most n , not n as in the simple skeleton, so Lemma 2.2 will give a betterbound, which will be sufficient to overcome the additional entropy.To bound the first probability on the right side of (2.10), we have P X j / ∈ B (cid:2) T ( v ′ j − , v j − ) − T ( p j − , v j − ) (cid:3) > dkn / ψ ( n ) ≤ P X j / ∈ B T ( v ′ j − , v j − ) > ( k − | B | ) n µ + 16 dkn / ψ ( n ) + P X j / ∈ B T ( p j − , v j − ) < ( k − | B | ) n µ − dkn / ψ ( n ) ≤ P X j / ∈ B T ( π ( j − n ( v j − ) , v j − ) > ( k − | B | ) n µ + 16 dkn / ψ ( n ) + P X j / ∈ B [ T ( p j − , v j − ) − ET ( p j − , v j − )] < − dkn / ψ ( n ) . (2.11)To bound the first probability on the right side of (2.11), we need an upper bound for ET ( π ( j − n ( v j − ) , v j − ), supplied by the next lemma. We postpone the proof to Section 3. UBGAUSSIAN RATES OF CONVERGENCE IN FPP 9
Lemma 2.3.
Provided n is sufficiently large, there exists an even integer n ≤ dn/ϕ ( n ) such that for every v ∈ H n we have ET ( π ( v ) , v ) ≤ n µ + 8 dn / ψ ( n ) . It follows from Lemmas 2.2 (with n max = n ) and 2.3 that provided n is large, P X j / ∈ B T ( π ( j − n ( v j − ) , v j − ) > ( k − | B | ) n µ + 16 dkn / ψ ( n ) ≤ P X j / ∈ B (cid:2) T ( π ( j − n ( v j − ) , v j − ) − ET ( π ( j − n ( v j − ) , v j − ) (cid:3) > dkn / ψ ( n ) ≤ k +1 exp − C dkn / ψ ( n ) (cid:18) log n n (cid:19) / ! ≤ exp (cid:0) − k ( dϕ ( n )) / log log n (cid:1) , (2.12)and the same bound holds for the second probability in (2.11), so (2.11) yields P X j / ∈ B (cid:2) T ( v ′ j − , v j − ) − T ( p j − , v j − ) (cid:3) > dkn / ψ ( n ) ≤ (cid:0) − k ( dϕ ( n )) / log log n (cid:1) . This bound also holds for the last probability in (2.10), which with (2.8) and (2.10) showsthat P (cid:18) T (˜Γ) − T (Γ) > dkn / ψ ( n ) , E (Γ) = B, S CG (Γ) = { R j } (cid:19) ≤ n dk exp (cid:0) − k ( dϕ ( n )) / log log n (cid:1) . Then from (2.5) and (2.7), provided n is large, P (cid:18) T (˜Γ) − T (Γ) > dkn / ψ ( n ) (cid:19) ≤ · k (2 n ) dk n dk exp (cid:0) − k ( dϕ ( n )) / log log n (cid:1) < . (2.13)We turn now to the first probability on the right side of (2.6). Here we use the non-coarse-grained analogs of the R j , given by V j = V j (Γ) = ( ( d j − , d j ) if j / ∈ E side , ( d j − , e j − , f j , d j ) if j ∈ E side , and define the augmented skeleton of Γ to be S aug (Γ) = {V j , ≤ j ≤ k } . Note that E side (Γ) and E ex (Γ) are functions of S aug (Γ). Let C + aug denote the class of allaugmented skeletons for which |E | ≥ b nk . For a given { V j , ≤ j ≤ k } ∈ C + aug , for some E side ⊂ { , . . . , k } and p j , v j , w j we can write { V j , ≤ j ≤ k } ∈ C + aug as V j = ( ( p j − , p j ) , j / ∈ E side , ( p j − , v j − , w j , p j ) , j ∈ E side , and we then define T skel ( { V j } ) = X j / ∈ E side T ( p j − , p j ) + X j ∈ E side (cid:2) T ( p j − , v j − ) + T ( v j − , w j ) + T ( w j , p j ) (cid:3) , where we take p = 0. (Note this definition is consistent with (2.9).) For j ∈ E side , asidestep occurs either from p j − to v j − or from w j to p j . In the former case, writing v j − as(( j − n + n , y ) for some y ∈ Z d , let b j = ( jn, y ). From the definitions of “sidestep” and h n we have using (2.4) that nµ + n / θ ( n ) ≤ ET ( p j − , b j ) ≤ ET ( p j − , v j − ) + ET ( v j − , b j ) ≤ ET ( p j − , v j − ) + ( n − n ) µ + C n / log n, (2.14)so assuming n is large, ET ( p j − , v j − ) ≥ n µ + 12 n / θ ( n ) , and hence E (cid:2) T ( p j − , v j − ) + T ( v j − , w j ) + T ( w j , p j ) (cid:3) ≥ nµ + 12 n / θ ( n ) . The last bound holds similarly if the sidestep instead occurs from w j to p j , so ET skel ( { V j } ) ≥ knµ + 12 b nk n / θ ( n ) . Therefore by Lemma 2.2 (with n max = n ) and (2.3), provided n is large, P (cid:18) T (Γ) < knµ + 64 dkn / ψ ( n ) , |E (Γ) | > b nk (cid:19) ≤ X { V j }∈C + aug P (cid:18) T skel ( { V j } ) − ET skel ( { V j } ) < − b nk n / θ ( n ) (cid:19) ≤ (2 n ) dk k +1 exp (cid:18) − C kθ ( n ) log log n n ) / (cid:19) < . (2.15) UBGAUSSIAN RATES OF CONVERGENCE IN FPP 11
Now we consider the third probability on the right side of (2.6). From (2.6), (2.13) and(2.15) we have 14 < (cid:18) T (˜Γ) < knµ + 128 dkn / ψ ( n ) , |E (Γ) | ≤ b nk (cid:19) ≤ X B ⊂{ ,...,k }| B |≤ b nk X { R j }∈C B P (cid:18) T skel ( { R j } ) ≤ knµ + 128 dkn / ψ ( n ) (cid:19) . (2.16)From here we proceed by contradiction. Suppose that(2.17) s ( n, x ) ≥ dn / ψ ( n ) for all x ∈ H n . Then for the probability on the right of (2.16) we have from Lemma 2.2 (with n max = n )that P (cid:18) T skel ( { R j } ) ≤ knµ + 128 dkn / ψ ( n ) (cid:19) ≤ P (cid:18) T skel ( { R j } ) − ET skel ( { R j } ) ≤ − dkn / ψ ( n ) (cid:19) ≤ k +1 e − dk log log n , (2.18)and then using (2.5) and (2.16),14 < k +1 (3 ϕ ( n )) dk (2 n ) db nk e − dk log log n ≤ e − dk log log n , (2.19)which is clearly false. Therefore the inequality in (2.17) must fail for some x , and then s (2 n, ≤ s ( n, x ) + s ( n, − x ) ≤ dn / ψ ( n )provided n is sufficiently large, which completes the proof of Theorem 1.1. Note that (2.19)shows the purpose of entropy reduction—if ϕ ( n ) were replaced by n , we would have to havelog n in place of log log n in the definition of ψ ( n ).3. Proofs of Lemmas and Theorem 2.1
Proof of Theorem 2.1.
As mentioned in the introduction, the theorem is proved in [6] underthe additional hypothesis that t ≤ m , so we assume t > m . Let µ = E ( ω b ), let γ ( m,x ) bea (nonrandom) path from (0 ,
0) to ( m, x ) and write T ( m,x ) for T ((0 , , ( m, x )). Let m , m satisfy m log m = 4 µ , m = 2( m log m ) / µ . We can always choose C , C so that C e − C t ≥ t ≤ m , so we may assume t > m .If m ≥ m , then since t > m we have(3.1) 2 µ m ≤ t (cid:18) m log m (cid:19) / , while if m < m then (3.1) follows from t > m . Either way, since T ( m,x ) ≥ ET ( m,x ) ≤ mµ , we then have T ( m,x ) − ET ( m,x ) ≥ − t (cid:18) m log m (cid:19) / . Further, by (3.1), T ( m,x ) − ET ( m,x ) > t (cid:18) m log m (cid:19) / = ⇒ T ( γ ( m,x ) ) ≥ T ( m,x ) ≥ µ m = ⇒ T ( γ ( m,x ) ) − µ m ≥ T ( γ ( m,x ) ) ≥ t (cid:18) m log m (cid:19) / , (3.2)so, letting I denote the rate function of the variable ω b − Eω b , P T ( m,x ) − ET ( m,x ) > t (cid:18) m log m (cid:19) / ! ≤ exp (cid:18) − I (cid:18) t m log m ) / (cid:19) m (cid:19) . (3.3)Since I is convex, there exists C > I ( x ) ≥ C x for all x ≥ /
2, and we have t m log m ) / ≥ (cid:18) m log m (cid:19) / ≥ , so by (3.3), P T ( m,x ) − ET ( m,x ) > t (cid:18) m log m (cid:19) / ! ≤ e − C t , which completes the proof. (cid:3) Proof of Lemma 2.2.
We have P r X j =1 ( T j − ET j ) + > a ! ≤ e − λa Y j E exp ( λ ( ET j − T j ) + ) . (3.4) UBGAUSSIAN RATES OF CONVERGENCE IN FPP 13
By Theorem 2.1, for each j , letting r j = t j − s j , for λ ≤ C (cid:16) log n max n max (cid:17) / , E exp ( λ ( ET j − T j ) + ) ≤ Z ∞ P (cid:18) ET j − T j > log tλ (cid:19) dt ≤ Z ∞ C exp − C log tλ (cid:18) log r j r j (cid:19) / ! dt = 1 + C C λ (cid:16) log r j r j (cid:17) / − . (3.5)Letting λ = C (cid:16) log n max n max (cid:17) / , since r j ≤ n max we obtain E exp ( λ ( ET j − T j ) + ) ≤ P r X j =1 ( T j − ET j ) + > a ! ≤ r exp − C a (cid:18) log n max n max (cid:19) / ! . The same bound holds for P (cid:16)P rj =1 ( ET j − T j ) + > a (cid:17) . (cid:3) For the proof of Lemma 2.3 we need the following. We say ( m, x ) is slow if s ( m, x ) ≥ n / ψ ( n ) , and fast otherwise. A path (( l, x (0) ) , ( l + 1 , x (1) ) , . . . , ( l + m, x ( m ) )) is clean if every increment( t − s, x ( t ) − x ( s ) ) with 0 ≤ s < t ≤ m is fast.Given a path γ = { ( m, x ( m ) ) } from (0 ,
0) to ( n, x ∗ ), let τ j = τ j ( γ ) = min { m : x ( m )1 = ju n } , ≤ j ≤ ϕ ( n ) . The climbing skeleton of γ is C ( γ ) = { ( τ j , x ( τ j ) ) : 1 ≤ j ≤ ϕ ( n ) } . A climbing segment of γ is a segment of γ from ( τ j − , x ( τ j − ) ) to ( τ j , x ( τ j ) ) for some j . A climbing segment is short if τ j − τ j − ≤ n/ϕ ( n ), and long otherwise. (Note n/ϕ ( n ) is the average length of the climbingsegments in γ .) Since the total length of γ is n , there can be at most ϕ ( n ) / γ , so there are at least ϕ ( n ) / J s ( γ ) = { j ≤ ϕ ( n ) : the j th climbing segment of γ is short } , J l ( γ ) = { j ≤ ϕ ( n ) : the j th climbing segment of γ is long } ,J l ( γ ) = (cid:0) ∪ j ∈J l ( γ ) ( τ j − , τ j ) (cid:1) ∩ (cid:22) nϕ ( n ) (cid:23) Z . If no short climbing segment of γ is clean, then for each j ∈ J s ( γ ) there exist α j ( γ ) <β j ( γ ) in [ τ j − , τ j ] for which the increment of γ from ( α j , x ( α j ) ) to ( β j , x ( β j ) ) is slow. (If α j , β j are not unique we make a choice by some arbitrary rule.) We can reorder the values { τ j , j ≤ ϕ ( n ) } ∪ { α j , β j : j ∈ J s ( γ ) } ∪ J l ( γ ) into a single sequence { σ j , ≤ j ≤ N ( γ ) } with ϕ ( n ) ≤ N ( γ ) ≤ ϕ ( n ), such that at least ϕ ( n ) / σ j − σ j − , x ( σ j ) − x ( σ j − ) ) , j ≤ N ( γ ) , are slow. The augmented climbing skeleton of γ is then the sequence A ( γ ) = { ( σ j , x ( σ j ) ) : 1 ≤ j ≤ N ( γ ) } . Lemma 3.1.
Provided n is large, there exists a path from (0 , to ( n, x ∗ ) containing a shortclimbing segment which is clean. Note that Lemma 3.1 is a purely deterministic statement, since the property of being cleandoes not involve the configuration { ω b } .Translating the segment obtained in Lemma 3.1 to begin at the origin, we obtain a path α ∗ from (0 ,
0) to some site ( m ∗ , y ∗ ), with the following properties:(3.6) m ∗ ≤ nϕ ( n ) , y ∗ = u n and α ∗ is clean . Proof of Lemma 3.1.
Let D denote the event that the geodesic γ ∗ from (0 ,
0) to ( n, x ∗ ) doesnot contain a short climbing segment which is clean. We will show that P ( D ) <
1, which issufficient to prove the lemma.Since x ∗ is unexcessive, provided n is large it follows from Theorem 2.1 that(3.7) P (cid:0) T ∗ < nµ + 2 n / θ ( n ) (cid:1) > . Let A ∗ be the set of all augmented climbing skeletons of paths from (0 ,
0) to ( n, x ∗ ), so |A ∗ | ≤ (2 n ) d +1) ϕ ( n ) and P (cid:0) D ∩ { T ∗ < nµ + 2 n / θ ( n ) } (cid:1) (3.8) ≤ X { ( σ j ,x ( σj ) ) }∈A ∗ P (cid:0) D ∩ { A ( γ ∗ ) = { ( σ j , x ( σ j ) ) }} ∩ { T ∗ < nµ + 2 n / θ ( n ) } (cid:1) ≤ X { ( σ j ,x ( σj ) ) }∈A ∗ P X j T (( σ j − , x ( σ j − ) ) , ( σ j , x ( σ j ) )) < nµ + 2 n / θ ( n ) ! . Here the sum over j has at most 4 ϕ ( n ) terms. In each { ( σ j , x ( σ j ) ) } ∈ A ∗ there are at least ϕ ( n ) / X j ET (( σ j − , x ( σ j − ) ) , ( σ j , x ( σ j ) )) > nµ + ϕ ( n )2 n / ψ ( n ) . UBGAUSSIAN RATES OF CONVERGENCE IN FPP 15
Then by (2.3) and Lemma 2.2 (with n max = 2 n/ϕ ( n )), letting T j = T (( σ j − , x ( σ j − ) ) , ( σ j , x ( σ j ) )),provided n is large, P X j T j < nµ + 2 n / θ ( n ) ! ≤ P X j ( T j − ET j ) < − n / ϕ ( n )4 ψ ( n ) ! ≤ ϕ ( n )+1 exp (cid:18) − C ϕ ( n ) / (log n ) / ψ ( n ) (cid:19) = 2 ϕ ( n )+1 exp (cid:18) − ϕ ( n ) / log log n (cid:19) . (3.9)Then by (3.7) and (3.8), provided n is large, P (cid:0) D ∩ { T ∗ < nµ + 2 n / θ ( n ) } (cid:1) ≤ (2 n ) d +1) ϕ ( n ) ϕ ( n )+1 exp (cid:18) − ϕ ( n ) / log log n (cid:19) ≤ exp (cid:18) − ϕ ( n ) / log log n (cid:19) < P (cid:0) T ∗ < nµ + 2 n / θ ( n ) (cid:1) , (3.10)which shows P ( D ) <
1, as desired. (cid:3)
Proof of Lemma 2.3.
We use the path α ∗ = { ( i, a ( i ) ) , i ≤ m ∗ } satisfying (3.6). Write v ∈ H n as ( n , ˆ v ) with ˆ v ∈ Z d . We may assume that π ( v ) = 0, and then from symmetry that0 ≤ ˆ v l ≤ u n / l ≤ d . Let w ∈ Z d have all coordinates even, with | w l − ˆ v l | ≤ l ≤ d . Define ζ and ξ j on Z d by ζ ( i, x , . . . , x d ) = ( i, x , − x , . . . , − x d ) ,ξ j ( i, x , . . . , x d ) = ( i, x j , x , . . . , x j − , x , x j +1 , . . . , x d ) , ≤ j ≤ d. Then α ∗ and ζ ◦ α ∗ are clean paths, the latter the reflection of the former through the plane { x : x = · · · = x d = 0 } , with both paths ending at sites with first transverse coordinate u n .When we add the two paths together we obtain a “path” which stays in that plane: α ∗ ( i ) + ζ ◦ α ∗ ( i ) = (2 i, a ( i )1 , , . . . , , i ≤ m ∗ , while composing both maps with ξ j interchanges the roles of the first and j th transversecoordinates:(3.11) ξ j ◦ α ∗ ( i ) + ξ j ◦ ζ ◦ α ∗ ( i ) = (2 i, , . . . , , a ( i )1 , , . . . , , i ≤ m ∗ , j ≤ d. We put “path” in quotes because (3.11) is not a true path, as consecutive sites are notadjacent; nonetheless we refer to it as the jth symmetrized path . Since a ( m ∗ )1 = u n and w j ∈ [0 , u n + 1] is even for all j ≤ d , we can define i j = min { i ≤ m ∗ : 2 a ( i )1 = w j } , which isthe time when the j th symmetrized path reaches height w j in the j th transverse coordinate.We then have(3.12) ξ j ◦ α ∗ ( i j ) + ξ j ◦ ζ ◦ α ∗ ( i j ) = (2 i j , , . . . , , w j , , . . . , , j ≤ d. Now define η ( i, x ) = ( i, − x ) , ( i, x ) ∈ Z d +1 . Then η ◦ α ∗ is clean, and(3.13) α ∗ ( m ∗ − i j ) + η ◦ α ∗ ( m ∗ − i j ) = (2( m ∗ − i j ) , , . . . , , j ≤ d. If we append the “horizontal path” α ∗ + η ◦ α ∗ with endpoint (3.13) to the j th symmetrizedpath with endpoint (3.12), we obtain a “path” of fixed length 2 m ∗ . Summing the incrementsof all the symmetrized and horizontal paths we obtain d X j =1 (cid:2) ξ j ◦ α ∗ ( i j ) + ξ j ◦ ζ ◦ α ∗ ( i j ) + α ∗ ( m ∗ − i j ) + η ◦ α ∗ ( m ∗ − i j ) (cid:3) = (2 dm ∗ , w ) , which expresses (2 dm ∗ , w ) as a sum of 4 d fast increments. Since | ˆ v − w | ≤ d , (2 d, ˆ v − w ) isfast provided n is large, so (2 d ( m ∗ + 1) , ˆ v ) is given as a sum of 4 d + 1 fast increments. Itfollows that s (2 d ( m ∗ + 1) , ˆ v ) ≤ (4 d + 1) n / ψ ( n ) . Taking n = 2 d ( m ∗ + 1) completes the proof. (cid:3) In Lemmas 2.3 and 3.1 we have effectively bounded the difference between a subadditivequantity, in this case of form ET ((0 , , ( n , u )), and its asymptotic approximation n µ byexpressing ( n , u ) as a sum of a bounded number of increments of a clean path. Earlier usesof a similar idea in other contexts appear in [1] and [4]. References [1] Alexander, K.S. (1990). Lower bounds on the connectivity function in all directions for Bernoulii per-colation in two and three dimensions.
Ann. Probab. Ann. Appl.Probab. Ann. Probab. Ann. Probab. J. Amer. Math. Soc. Ann. Inst. Henri Poincar´e Probab. Stat. Ann. Probab. Ann. Probab. Ann.Math. (2)
Ann. Appl. Probab. UBGAUSSIAN RATES OF CONVERGENCE IN FPP 17 [11] Kingman, J.F.C. (1968). The ergodic theory of subadditive stochastic processes.
J. Roy. Statist. Soc.Ser. B Department of Mathematics KAP 108, University of Southern California, Los Angeles,CA 90089-2532 USA
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