Subdiffusive concentration in first-passage percolation

We prove exponential concentration in i.i.d. first-passage percolation in $Z^d$ for all $d \geq 2$ and general edge-weights $(t_e)$. Precisely, under an exponential moment assumption $E e^{\alpha t_e}<\infty$ for some $\alpha>0$) on the edge-weight distribution, we prove the inequality $$ P(|T(0,x)-E T(0,x)| \geq \lambda \sqrt{\frac{|x|}{log |x|}}) \leq ce^{-c' \lambda}, |x|>1 $$ for the point-to-point passage time $T(0,x)$. Under a weaker assumption $E t_e^2(\log t_e)_+<\infty$ we show a corresponding inequality for the lower-tail of the distribution of $T(0,x)$. These results extend work of Benaim-Rossignol to general distributions.


The model
Under minimal assumptions on (t e ), the passage time T (0, x) is asymptotically linear in x, but the lower order behavior has resisted precise quantification. If d = 1, the passage time is a sum of i.i.d. variables, so its fluctuations are diffusive, giving χ = 1/2, where χ is the (dimension-dependent) conjectured exponent given roughly by Var T (0, x) ≍ x 2χ 1 . For d = 2, the minimization in the definition of T is expected [14] to create subdiffusive fluctuations, with a predicted value χ = 1/3. Subdiffusive behavior is expected in higher dimensions as well.
In this paper, we prove an exponential version of subdiffusive fluctuations for T (0, x) under minimal assumptions on the law of (t e ). This result follows up on work done by the authors (extending the work of [2,5]) in [8], in which it was shown that given only that the distribution of t e has 2 + log moments. Our concentration inequalities apply to a nearly optimal class of distributions: for the upper tail inequality in (1.4) we require that t e has exponential moments and for the lower tail inequality (1.5), that t e has 2 + log moments. In contrast to existing work on subdiffusive concentration listed below, our methods do not rely on any properties of the distribution other than the tail behavior.
(1.5) Remark 1.2. If (1.1) fails, then T (0, x) itself is sublinear in x and the model has a different character (see [15,Theorem 6.1] and [7,21] for more details). Because T (0, x) is bounded below by the minimum of the 2d weights of edges adjacent to 0, it is necessary to assume (1.2) to obtain an upper-tail exponential concentration inequality. For the lower tail, our methods require 2 + log moments and this is the same assumption made in [8] for a sublinear variance bound.
The strategy of the proof is to use a relation, stated in Lemma 2.2, between bounds on Var e λT (0,x) and exponential concentration. To obtain the required variance bound (Theorem 2.3), we follow the method of Benaïm-Rossignol, applying the Falik-Samorodnisky inequality (Lemma 2.4) to the variable e λFm , where F m is an averaged version of the passage time. From here, bounding the variance follows a broadly similar outline to that given in [8]: representing the passage times as a push-foward of Bernoulli sequences and the bound follows after a careful analysis of discrete derivatives. The main complications arise in giving these bounds and are dealt with using estimates on greedy lattice animals in Section 5.

Preliminary results
We will need a couple of results on the length of geodesics. By Proposition 1.3 below, condition (1.1) ensures that P(∃ a geodesic from x to y) = 1 for all x, y ∈ Z d , (1.6) where a geodesic is a path γ from x to y that has T (γ) = T (x, y).
Proposition 1.3 (Kesten). Assuming (1.1), there exist a, C 1 > 0 such that for all n ∈ N, As a consequence, we state a bound used in work of one of the authors and N. Kubota [9]. For this, let G(0, x) be the maximal number of edges in any self-avoiding geodesic from 0 to x. An application of Borel-Cantelli to (1.7) implies under (1.1), lim inf

Proof. Defining
A m = ∃ self-avoiding γ from 0 with #γ ≥ m but T (γ) < a#γ and summing (1.7) over n, one has, for some C 2 > 0, For x ∈ Z d , assume Y x ≥ n ≥ 1 and let γ be any self-avoiding geodesic from 0 to x with length G(0, x) = Y x . Then because Y x = 0, G(0, x) > (1/a)T (0, x) and so with #γ ≥ n. So A n occurs and (1.9) completes the proof.
We can state a couple of relevant consequences of this proposition.

There exists C 3 such that
For β < C 2 /2, the second term is bounded in x. On the other hand letting γ x be a deterministic path from 0 to x of length x 1 , the first term is bounded by So we conclude for x = 0 and some C 4 ≥ 1, For the remainder of the paper we assume (1.1).

Setup for the proof
Instead of showing concentration for T (0, x), we use an idea from [2]: to show it for T (z, z + x), where z is a random vertex near the origin. So, given x ∈ Z d , fix ζ with 0 < ζ < 1/4 and define where T z = T (z, z + x) (this particular randomization was used by both [1] and [19]). For λ ∈ R we define G = G λ = e λFm . (2.1) Below are the concentration inequalities for F m analogous to (1.4). In the next subsection, we will show why they suffice to prove Theorem 1.1.
This theorem is a consequence on a bound for Var e λFm , and this is what we focus on from Section 3 onward. The link between a variance bound and concentration is given by the following lemma from [5,Lemma 4.1] (which itself is a version of [17,Corollary 3.2]). We have split the statement from [5] into upper and lower deviations.
Lemma 2.2. Let X be a random variable and K > 0. Suppose that Then Taking K = C x 1 log x 1 for x 1 > 1 and X = F m in the previous lemma shows that to prove Theorem 2.1, it suffices to show the following variance bound.
The proof of this bound will be broken into several sections below.

Theorem implies Theorem 1.1
Assume first that we have the concentration bound for some b, c > 0 and that (1.2) holds. We will derive from (2.4) the corresponding estimate for the passage time T = T (0, x): and note that EF m = ET . If both events {|F m − EF m | < λ/2} and {|T (0, x) − F m | < λ/2} occur, then the triangle inequality implies that we have the bound This results in the estimate By subadditivity, we can write Repeating the argument for (1.12) (bounding T (0, z) by the passage time of a deterministic path), we have for α ≥ 0 and each z ∈ B m Here α is from (1.2). We now obtain a bound for the second term on the right in (2.6). Let M > 0. First, by the triangle inequality: The last quantity is bounded by Choosing 2M = λ x 1/2 1 /(log x 1 ) 1/2 and adjusting constants, we find the bound Combined with (2.4) in (2.6), this shows (2.5).
We now move to proving that under assumption (1.3), if we prove Theorem 2.1, then there exist b ′′ , c ′′ > 0 such that (log x 1 ) 1/2 and S ′ = e∈x+Bm t e , then This means that However the event on the right implies that for any z ∈ B m , T (z, z + x) ≤ T (0, x) + 4ES. Therefore

Now we can bound 4ES by
as long as λ ≥ C 7 x dζ 1 cx . To finish, we simply choose ζ = d/4, so that x dζ 1 /c x ≤ C 8 for x 1 > 1 and some C 8 > 0. This implies giving the bound C 9 e −C 10 λ .

Falik-Samorodnitsky and entropy
Enumerate the edges of E d as e 1 , e 2 , . . . and write e λFm as a sum of a martingale difference sequence: We have written F k for the sigma-algebra σ(t e 1 , . . . , t e k ), with F 0 trivial. In particular if F ∈ L 1 (Ω, P), To prove concentration for F m , we bound the variance of G; the lower bound comes from the proof of [10, Theorem 2.2]. (2.9) In the above lemma, we have used Ent to refer to entropy: We will need some basic results on entropy. This material is taken from [8,Section 2], though it appears in various places, including [4]. By Jensen's inequality, Ent X ≥ 0. There is a variational characterization of entropy [18, Section 5.2] that we will use. Proposition 2.6. We have the formula The second fact we need is a tensorization for entropy. For an edge e, write Ent e X for the entropy of X considered only as a function of t e (with all other weights fixed).

Bound on influences
To bound the sum ∞ k=1 (E|V k |) 2 we start with a simple lemma from [8,Lemma 5.2]. For a given edge-weight configuration (t e ) and edge e, let (t e c , r) denote the configuration with value t f if f = e and r otherwise. Let T z (t e c , r) be the variable T z = T (z, z + x) in the configuration (t e c , r) and define Geo(z, z + x) as the set of edges in the intersection of all geodesics from z to z + x.
has the following properties almost surely.
We need one more lemma from [8] bounding the length of geodesics. Let G be the set of all finite self-avoiding geodesics.
With these two tools we can bound the influences in the denominator of the logarithm of (2.9). The following proof is very similar to the one of Benaim-Rossignol [5,Theorem 4.2].
This inequality holds for any λ for which the left side is defined.
m be the variable F m with the edge weight t e k replaced by an independent copy t ′ e k . Then we can give the upper bound We will use (3.1) when λ > 0 and (3.2) when λ ≤ 0. With these restrictions, the integrands in both cases above are only nonzero when F (k) m > F m . Apply the mean value theorem to get To combine these, when λ > 0, we use F m ≤ e λFm e λt ′ e k , so we obtain for both cases By Cauchy-Schwarz, we have the following two bounds: We will bound these terms using Lemma 3.1. Write is the variable T z in the configuration in which the k-th edge-weight t e k is replaced by the independent copy t ′ e k . By convexity of the function x → (x + ) 2 , we obtain the bound By translation invariance, the final probability equals P(e k − z ∈ Geo(0, x)): We have used the assumption Et 2 e < ∞ and Lemma 3.2 to bound the expectation above. After incorporating the factor λ 2 Ee 2λFm , this is our bound for (3.4).
For (3.3), write The expectation equals By (1.10) and Et 2 e < ∞, the last expression is bounded by We can now finish the proof with this bound and (3.5):

Entropy bound
The purpose of the present section is to give an intermediate upper bound for the sum of entropy terms in the left side of (2.9). Namely we will prove the following inequality, recalling that F is the distribution function of t e .
The constant C λ is determined as follows: We will prove this in a couple of steps. First we use the Bernoulli encoding from [8] to give an upper bound (Lemma 4.3 below) in terms of discrete derivatives relative to Bernoulli sequences. Next we split into two cases, λ ≥ 0 and λ ≤ 0. The first is handled in Proposition 4.4 and the second in Proposition 4.7. These three results will prove Theorem 4.1.

Bernoulli encoding
We will now view our edge variables as the push-forward of Bernoulli sequences. Specifically, for each edge e, let Ω e be a copy of {0, 1} N with the product sigma-algebra. We will construct a measurable map T e : Ω e → R using the distribution function F . To do this, we create a sequence of partitions of the support of µ. Recalling I := inf supp(µ) = inf{x : F (x) > 0}, set Note that by right continuity of F , the minimum above is attained; that is, Let us note two properties of the sequence.
Each ω ∈ Ω e gives us an "address" for a point in the support of µ. Given ω = (ω 1 , ω 2 , . . .) and j ≥ 1, we associate a number T j (ω) by i(ω, j) is just the number between 0 and 2 j − 1 that corresponds to the binary number ω 1 · · · ω j . It will be important to note that if ω i ≤ω i for all i ≥ 1 (written ω ≤ω), then i(ω, j) ≤ i(ω, j) for all j ≥ 1. This, combined with the monotonicity statement (4.2), implies It is well-known that one can represent Lebesgue measure on [0, 1] using binary expansions and Bernoulli sequences. One way to view the encoding T in Lemma 4.2 is a composition of this representation with the right-continuous inverse of the distribution function F . The function T j instead uses an inverse approximated by simple functions taking dyadic values.
Lemma 4.2. For each ω, the numbers (T j (ω)) form a non-decreasing sequence and have a limit T (ω). This map T : Ω e → R ∪ {∞} is measurable and has the following properties.
We now view G = e λFm as a function of sequences of Bernoulli variables, as in [8]. So define Ω B = e Ω e with product sigma-algebra and measure π := e π e , where π e is a product of the form j≥1 π e,j with π e,j uniform on {0, 1}. An element of Ω B will be denoted ω B = ω e,j : e ∈ E d , j ≥ 1 .
Call T e the map from the previous lemma on Ω e and define the product map T := e T e : Ω B → Ω with action T (ω B ) = (T e (ω e ) : e ∈ E d ) .
In what follows, we will consider functions f on the original space Ω as functions on Ω B , through the map T . We will suppress mention of this in the notation and, for instance, write f (ω B ) to mean f • T (ω B ). We will estimate discrete derivatives, so for a function f : where ω e,j,+

Derivative bound: positive exponential
For the next derivative bounds we continue with G = e λFm and set H as the derivative of G; that is, H = λe λFm .
Lemma 4.5. There exists C 18 independent of e such that for λ ∈ [0, α/2), Proof. We consider two types of values of j. Note that when D z,e > I, F (D − z,e ) > 0 and therefore for some j, F (D − z,e ) ≥ 2 −j . So define The event in Ω e listed on the right depends only on ω e,k for k > j, so it is independent (under π e ) of the state of the first j coordinates. Thus the above equals 2 j π e (T e (0, . . . , 0, ω e,j+1 , . . .) < D z,e , ω e,1 , . . . , ω e,j = 0) ≤ 2 j π(T e (ω e ) < D z,e ) = 2 j F (D − z,e ) .
Because N has geometric distribution, this is bounded by C 20 C λ independently of e.
Returning to (4.21), note that by Lemma 3.1, if t e < D z,e then e is in Geo(z, z + x). So applying the bound on E πe [L z N ], we obtain for some C 21
We have now isolated the term from [8, (6.23) Thus we obtain Use the bound H 2 ≥ λ 2 e 2λte e 2λFm(t e c ,I) , which implies H 2 (ω e c , 0) ≤ Eπ e H 2 Eµe 2λte . Combined with Lemma 4.6, this gives an upper bound for the right side of (4.24) when λ ≤ 0: Since λ ≤ 0, H 2 = λ 2 e 2λFm is decreasing in the variable t e . However (1 − log F (t e ))1 [I,Dz,e) is also decreasing in t e . Therefore the Chebyshev association inequality [4, Theorem 2.14] gives an upper bound of C 23 Summing over edges e,

Control by lattice animals
The next step is to use the theory of greedy lattice animals to decouple and control the terms in the expectation of Theorem 4.1. Specifically we will show where C λ is from Theorem 4.1.
The theorem follows from inequalities (5.4) and (5.8), which we now set out to prove. We begin by generating a new set of "lattice animal weights" from a given realization (t e ); set w e := 1 − log(F (t e )) for all e ∈ E d .
For a realization of (t e ), consider the edge greedy lattice animal problem. For a connected subset of edges γ ⊆ E d , define N (γ) = e∈γ w e , and define the random variable N n := max γ:#γ=n 0∈γ N (γ) (here the notation 0 ∈ γ means that 0 is an endpoint of some edge in γ).
where the union is over all lattice animals of size n containing the origin. Now, there exists a constant C 25 such that the number of such lattice animals is bounded by e C 25 n . Therefore, letting (w i ) be a sequence of i.i.d. random variables distributed as w e , P (N n > βn) ≤ e C 25 n P n i=1 w i > βn ≤ e C 25 n−βn/2 Ee n i=1 w i /2 = e C 25 n−βn/2 Ee we/2 n .
We now consider the first-passage model on Z d . For any x ∈ Z d , x = 0, let when we need to allow the starting point to vary as well, write Y z,x := e∈Geo(z,z+x) w e .

The case λ ≥ 0
In this section, we consider the case of upper exponential concentration. For the remainder of this section, assume (1.2) and let λ ∈ [0, α/2). Rephrase the bound from Theorem 4.1: Applying Proposition 2.6 to the expectation on the right-hand side of (5.1) and the fact that Y x = Y z,x in distribution yields a bound for some C 26 > 0 such that the expectation below exists: We focus our efforts on a bound for the second term of (5.2).
Proof. Recall that G(0, x) is the maximal number of edges in a geodesic from 0 to x. We begin by writing where in the last step we have used Cauchy-Schwarz. By Corollary 1.5, there exist In particular, uniformly in x, for some C 28 ∈ (0, ∞), Using Proposition 5.3, we see that if C 26 is chosen sufficiently small, then uniformly in j, Applying these bounds in (5.3), for C 26 sufficiently small, So under (1.2) with λ ∈ [0, α/2), we return to (5.2) and find for some

The case λ ≤ 0
When λ ≤ 0, the problem is again to bound above the term We will break this up differently from before, now using a variant of the idea from [9]. Let C 30 > 0 be arbitrary (to be fixed later, independent of x). Then where we have used the Harris-FKG inequality on the first term (since λ ≤ 0, e 2λFm is a decreasing function of (t e ) whereas T z is increasing) and have defined the new variable We will bound P(Z z,x ≥ n) in what follows. Analogously to the proof of Proposition 1.4, define, for C 31 > 0, 32 n for all n ≥ 1.
Proof. By translation invariance we can consider z = 0. The content of Proposition 5.3 is that there exist constants C 33 , C 34 > 0 such that P (∃ a self-avoiding γ from 0 with #γ = n such that N (γ) > C 33 n) ≤ e −C 34 n , n ≥ 1 .
By (1.7), for small C 31 , the last probability is bounded by e −C 39 n . Therefore P(A ′ n ) ≤ e −C 40 n .
From Lemma 5.5, we can decompose Consider some outcome in (A ′ n ) c such that Z z,x ≥ n > 0. For this outcome, we must have a contradiction for C 30 > C −1 31 . This implies that independent of x, z and n, there exists C 30 such that Ent(V 2 k ) ≤ λ 2 C 46 C λ ( x 1 + 1 + EF m ) Ee 2λFm for x ∈ Z d and 0 ≤ λ ≤ C 44 .
On the other hand, when we assume (1.3), Theorem A.4 gives the upper bound (with C 49 from that theorem) If we further restrict the range of λ we can bound C λ using assumptions (1.3) and (1.2) and find for some
Theorem A.1. Let X be a random variable and let X ′ be an independent copy. Then for λ ∈ R, Ent e λX ≤ E e λX q(λ(X ′ − X) + ) (A.1) where q(x) = x(e x − 1). The proof of this bound follows from (1.11) and the following proposition. Recall that Geo(0, x) is the set of edges in the intersection of all geodesics from 0 to x. EEnt e k e λFm .
Introduce F (k) m as the variable F m evaluated at the configuration in which t e k is replaced by an independent copy t ′ e k . Then we can apply (A.1) conditionally: Convexity of q on [0, ∞) gives the upper bound 1 #B m z∈Bm ∞ k=1 EE e k e λFm q λ(T (k) z − T z ) + . Integrate t ′ e k first and bring the sum inside the integral for Eq(λt e ) #B m z∈Bm E e λFm #Geo(z, z + x) .
To deal with this product, use Proposition 2.6, along with a parameter p > 0, to obtain Ent e λFm ≤ Eq(λt e ) #B m z∈Bm pEnt e λFm + pEe λFm log E exp #Geo(z, z + x) p .
By translation invariance, the expression inside the sum does not depend on z. Therefore if we choose p ≤ (2Eq(λt e )) −1 this is no bigger than 1 2 Ent e λFm + 1 2 Ee λFm log Ee #Geo(0,x)/p .

A.2.2 Negative exponential
The bound given below is similar to the one derived in [9] for T instead of F m . By convexity of x → (x + ) 2 , this is bounded by EE e k e λFm ((T (k) z − T z ) + ) 2 and using Lemma 3.1, by EE e k e λFm (t ′ e k 1 {e k ∈Geo(z,z+x)} ) 2 .
Again, integrate over t ′ e k first to get The variable e λFm is decreasing as a function of the edge-weights (since λ ≤ 0) whereas T z is increasing. So apply the Harris-FKG inequality to the first term for an upper bound of (1/a)Ee λFm ET z = (1/a)Ee λFm ET (0, x) .