Superlinearity of geodesic length in 2$D$ critical first-passage percolation

First-passage percolation is the study of the metric space $(\mathbb{Z}^d,T)$, where $T$ is a random metric defined as the weighted graph metric using random edge-weights $(t_e)_{e\in \mathcal{E}^d}$ assigned to the nearest-neighbor edges $\mathcal{E}^d$ of the $d$-dimensional cubic lattice. We study the so-called critical case in two dimensions, in which $\mathbb{P}(t_e=0)=p_c$, where $p_c$ is the threshold for two-dimensional bond percolation. In contrast to the standard case $(<p_c)$, the distance $T(0,x)$ in the critical case grows sub linearly in $x$ and geodesics are expected to have Euclidean length which is superlinear. We show a strong version of this super linearity, namely that there is $s>1$ such that with probability at least $1-e^{-\|x\|_1^c}$, the minimal length geodesic from $0$ to $x$ has at least $\|x\|_1^s$ number of edges. Our proofs combine recent ideas to bound $T$ for general critical distributions, and modifications of techniques of Aizenman-Burchard to estimate the Hausdorff dimension of random curves.

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