Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
139 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Superlinearity of geodesic length in 2$D$ critical first-passage percolation (1610.02593v1)

Published 8 Oct 2016 in math.PR

Abstract: 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 $1-e{-|x|_1c}$, the minimal length geodesic from $0$ to $x$ has at least $|x|_1s$ 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.

Summary

We haven't generated a summary for this paper yet.