Almost sure bounds for higher-order derivatives of first-passage percolation with respect to the environment

2025/04/11 by Matic, Ivan, Radoicic, Rados, Stefanica, Dan · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2504.08938

Abstract

The variance of the first-passage percolation is bounded by the L²-norms of the derivatives with respect to the environment. In this paper we prove that the environment derivatives of orders k∈\2,3,4\ are bounded below by -\binomk-2\lceil\frack-22\rceil and above by \binomk-2\lceil\frack-22\rceil. We believe that these are the bounds for all k. We provide examples of environments that show that these extreme values can be attained.

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Cited by

• Higher-order derivatives of first-passage percolation with respect to the environment

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