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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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