## Nondifferentiability of the time constants of first-passage percolation.(English)Zbl 1029.82017

This paper studies the time constant in the first-passage percolation (FPP) problem on $$\mathbb{Z}^2$$, introduced in 1965 by Hammersley and Welsh, and answer negatively to an open conjecture of this field. The model studied here on $$\mathbb{Z}^2$$ is this of the general FPP problem but the main result concerns the Bernoulli case.
On the lattice $$\mathbb{Z}^2$$, one considers a random field $$x=(x(e))_{e \in\mathbb{Z}^2}$$ of i.i.d. random variables, with a marginal distribution $$F$$ of finite mean. When $$e=(u,v)$$, this random variable $$x(e)$$ can be viewed as the amount of time needed to go from $$u$$ to $$v$$. In the simplest Bernoulli case, this amount is either $$0$$ or $$1$$ with probability $$p$$ and $$1-p$$. To study the stream along the lattice, one also considers paths along edges $$\gamma = \{v_0,e_1,v_1,\dots,e_n,v_n\}$$ and defines the passage time of $$\gamma$$ to be $$\tau(\gamma)=\sum_{e \in \gamma} x(e)$$.
Hammersley and Welsh had first studied the first passage time (FPT) $$a_{0,n}$$ from the origin and the point $$u(n,0)$$ of the horizontal axis, and proved by subbadditivity that there exists a finite constant $$\mu(F)$$ such that the ratio $$a_{0,n}/n$$ converges (a.s. and in $$L^1$$) to $$F$$ when $$n$$ goes to infinity. This so-called time constant $$\mu(F)$$ has been studied intensively and is a central object of this paper. Depending on the configuration, different paths can lead to the FPT $$a_{0,n}$$ (these paths are then called routes.
Another object of interest, because it provides information on the FPT, is the length $$N_n$$ of the shortest route. This length behaves differently depending on the percolation regime of the model. In the supercritical case, Zhang and Zhang (1984) proved that $$N_n/n$$ converges a.s. and in $$L^1$$ to another finite time constant $$\lambda(F)$$, but this problem is still open in other regimes. Hammersley and Welsh had already studied this question and transformed it into a problem of smoothness of the first time constant $$\mu(F \otimes t)$$ of the shifted distribution $$F \otimes t$$ as a function of the shift parameter $$t$$ (For the Bernoulli case, the shifted Bernoulli r.v.’s $$x'(e)=x(e)+t$$ take values $$t$$ and $$t+1$$ with probability $$p$$ and $$1-p$$). This approach has later on been carried on by Smythe and Wierman (1978) and Kesten (1980) who obtained a new criterion for this convergence in the subcritical case : the ratio $$N_n/n$$ must converge with probability one if the time constant $$\mu(F \otimes t)$$ is differentiable at $$t=0$$.
The main result of this paper closes the door of this approach : Theorem 1 proves that this differentiability condition does not hold for the basic Bernoulli percolation problem in the subcritical case ($$F(0)=p<1/2$$). According to the authors, it is likely that the convergence still holds for subcritical percolation but the proof must be more subtle : a conjecture of Kesten about the divergence in the critical case suggests that the analysis of the sub- and super critical case may be different.
The proof of the main theorem (Theorem 1) is achieved after a careful preparation relying on combinatorial, topological and geometrical techniques. The FPP $$a_{0,n}$$ is first related to the length $$N_n$$ of the shortest route and graph theory is used to restrict the attention on finite volume analysis of the dual lattice. The non-differentiability of the time constant is proved after some probabilistic comparaisons between the longest and the shortest route at $$t=0$$. The authors also indicate in their conclusion alternative routes to prove the convergence of $$N_n /n$$ in the subcritical case.

### MSC:

 82B43 Percolation 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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### References:

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