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Lower bounds for fluctuations in first-passage percolation for general distributions. (English. French summary) Zbl 1434.60280
Summary: In first-passage percolation (FPP), one assigns i.i.d. weights to the edges of the cubic lattice $$\mathbb{Z}^d$$ and analyzes the induced weighted graph metric. If $$T(x,y)$$ is the distance between vertices $$x$$ and $$y$$, then a primary question in the model is: what is the order of the fluctuations of $$T(0,x)$$? It is expected that the variance of $$T(0,x)$$ grows like the norm of $$x$$ to a power strictly less than 1, but the best lower bounds available are (only in two dimensions) of order $$\log \|x\|$$. This result was found in the ’90s and there has not been any improvement since. In this paper, we address the problem of getting stronger fluctuation bounds: to show that $$T(0,x)$$ is with high probability not contained in an interval of size $$o(\log \|x\|)^{1/2}$$, and similar statements for FPP in thin cylinders. Such statements have been proved for special edge-weight distributions, and here we obtain such bounds for general edge-weight distributions. The methods involve inducing a fluctuation in the number of edges in a box whose weights are of “hi-mode” (large).

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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