ON THE NON-CONVEXITY OF THE TIME CONSTANT IN FIRST-PASSAGE PERCOLATION

We give a counterexample to a conjecture of Hammersley and Welsh (1965) about the convexity of the time constant in (cid:12)rst{passage percolation, as a functional on the space of distribution functions. The present counterexample only works for (cid:12)rst{passage percolation on ZZ d for d large.


Introduction
First-passage percolation was introduced by Hammersley and Welsh (1965). They considered an i.i.d. family of random variables {t(e)}, where e runs through the edges of Z Z d . Here, and throughout this paper, Z Z d is regarded as a graph with edges between any two points u = (u 1 , . . ., u d ) ∈ Z Z d and v = (v 1 , . . . , v d ) ∈ Z Z d if and only if It is assumed throughout that the t(e) are positive (i.e., ≥ 0) and that IE{t(e)} < ∞.
The passage time of a path r = (e 1 , e 2 , . . . , e n ) on Z Z d is defined as t(e i ), and the travel time from u to v is defined as (see also Smythe and Wierman (1978) or Kesten (1986) for more details). Let F be the common distribution of the t(e). It was proven by Kingman (1968) that under condition (2), there exists a constant 0 ≤ µ(F ) = µ(F, d) < ∞, which depends only on F and the dimension d, such that 1 n T (0, nξ 1 ) → µ(F ) w.p.1 and in L 1 .
Here ξ 1 = (1, 0, . . ., 0) is the first coordinate vector; µ is called the time constant. Hammersley and Welsh (1965) (see Conjecture 6.5.5) conjectured that µ is a convex functional on the space of distribution functions on [0, ∞) satisfying (2). That is, they conjectured that for any two such distribution functions F 0 and F 1 , In Remark 4 at the end of the next section we give some pairs F 0 , F 1 with F 0 = F 1 for which (5) holds, even with strict inequality when 0 < α < 1. However, our main result here is an example for which (5) fails. The present example relies on the results of Hara (1990) on the correlation function for percolation in high dimensions. Once one has a single counterexample to (5), simpler examples may well be forthcoming. There is no reason to believe that dimension plays a significant role in this problem and we therefore doubt the general validity of (5) in any dimension ≥ 2.

The Example
Let p c = p c (d) be the critical probability for (Bernoulli) edge percolation on Z Z d (see Grimmett (1989) for a precise definition) and let δ a be the distribution which puts mass 1 on the point a. Take for some 0 < q < 1. Thus F 0 puts masses p c and 1 − p c on the points 0 and 2, respectively, while F 1 puts masses q and 1 − q on the points 1 and 2, respectively. We shall show that there exist some d 0 < ∞ and α 0 (d) > 0 such that (5) fails for d ≥ d 0 and α ≤ α 0 .
To prove this claim we first observe that by virtue of Theorem 6.1 in Kesten (1986) and the inequality p c = p T which is due to Menshikov (1986) and Aizenman and Barsky (1987) (see Ch.3 in Grimmett (1989)). The right hand side of (5) therefore equals αµ(F 1 ), and it suffices to prove that lim sup In fact, we shall show the stronger property for some constant C 1 > 0 and small α. To this end we introduce the distribution function and take for {s p (e) : e ∈ Z Z d } an i.i.d. family of random variables with common distribution function G(p). The joint distribution of all the s p (e), e ∈ Z Z d , is denoted by IP p . IE p will be expectation with respect to IP p . Now so that a random variable with distribution αF 1 + (1 − α)F 0 is stochastically larger than any s (1−α)pc (e). Consequently, (9) will therefore follow if we prove for some constant C 2 > 0 and small p c −p > 0. Here and in several other places we use the well known fact that 0 < p c (d) < 1 (see Broadbent and Hammersley (1957), Hammersley (1957Hammersley ( ), (1959; also Grimmett (1989), Theorem 1.10). We reduce one step further by an appeal to a lower bound for µ(G(p)) of Chayes, Chayes and Durrett (1986). Their Theorem 3.1 with ε = (p c − p)/2 yields Now (10) is immediate from (11) and Theorem 1.1 of Hara (1990) which shows that for d ≥ some d 0 and suitable constants 0 < C i (d) < ∞, Remark 1 The above construction of a counterexample for (5) works as long as This is generally believed to be true for all d ≥ 3, but it is known not to be the case for d = 2 (cf. Kesten (1987)).
Remark 2 It is known (see Kesten (1986), Theorem 6.9) that µ is continuous, in the sense that if F n ⇒ F , then µ(F n ) → µ(F ). One can therefore slightly modify the distributions F 0 and F 1 above to obtain a counterexample to (5) with continuous distributions F 0 , F 1 .

Remark 3
In our construction above we concentrated on the time constant in the first coordinate direction. However, the same example works in any direction. More precisely, define for any vector where nx = stands for ( nx 1 , . . . , nx d ), and T (u, v) is defined by (3) and t(e) have the common distribution F ; the limit in (14) exists with probability 1 and in L 1 by Section 3 of Kesten (1986). If we set F α = αF 1 + (1 − α)F 0 , then by equations (3.11)-(3.15) there, it holds that for all x ∈ IR d µ(F 0 , d, x) = 0, (see also first line on p.160 of Kesten (1986)); |x| = d 1 |x i | as in (1)). Therefore, for d ≥ d 0 , there exists an α 1 (d) > 0 so that for all 0 < α ≤ α 1 and x = 0.
Remark 4 Here is an example of a pair of distribution functions F 0 and F 1 for which (5) does hold. Let F 0 = δ a and F 1 = δ b for some a, b > 0. Since the passage time of any path r of k edges equals ka when the t(e) have distribution F 0 , it is clear that µ(F 0 ) = a. Similarly µ(F 1 ) = b. Moreover, if the t(e) have common distribution F α = αF 1 + (1 − α)F 0 , then IE{t(e)} = αb + (1 − α)a.
Consequently (see Hammersley and Welsh (1965), Theorem 4.1.9), A more interesting, and less singular class of examples for which (5) holds can be constructed by means of van den Berg and Kesten (1993). Let F 1 be a distribution on [0, ∞) which has a finite mean and satisfies the following condition: If r := min(supp(F 1 )), then is the critical probability of oriented bond percolation on (Z Z + ) d . In the terminology of van den Berg and Kesten (1993) this means that F 1 is useful. Let F 0 (x) = F 1 (βx) for some β > 0, β = 1. Then (5) holds for this pair F 0 , F 1 . To prove our last claim, let s and I (α) be two independent random variables such that s has distribution F 1 and Then F 0 is the distribution of s/β and F α = αF 1 + (1 − α)F 0 is the distribution of I (α) s. Finally, we write H α for the distribution of (α + (1 − α)/β)s, i.e., Then F α is more variable than H α in the terminology of van den Berg and Kesten (1993). This follows from Theorem 2.6 there with t = (α + (1 − α)/β)s and t = I (α) s. Therefore, by Theorem (2.9b) in van den Berg and Kesten (1993) µ(F α ) < µ(H α ), 0 < α < 1.