Uniqueness and dimension for the geodesic of the critical long-range percolation metric

Jian Ding; Zherui Fan; Lu-Jing Huang

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arXiv · 2506.10511

Uniqueness and dimension for the geodesic of the critical long-range percolation metric

Jian Ding·Zherui Fan·Lu-Jing Huang

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Abstract

By recent works of B\"aumler [2] and of the authors of this paper [5], the (limiting) random metric for the critical long-range percolation was constructed. In this paper, we prove the uniqueness of the geodesic between two fixed points, for which an important ingredient of independent interest is the continuity of the metric distribution. In addition, we establish the Hausdorff dimension of the geodesics.

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Jian Ding, Zherui Fan, Lu-Jing Huang. 2025-06-12. Uniqueness and dimension for the geodesic of the critical long-range percolation metric. https://arxiv.org/abs/2506.10511

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