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

2025/06/12 by Jian Ding, Ding, Jian, Zherui Fan +3
Mathematics · #60K35 #82B27 #82B43 #Advanced Combinatorial Mathematics #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2506.10511

openalex publication_date 2025/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By recent works of Bäumler [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.

Citations

• Polynomial lower bound on the effective resistance for the one-dimensional critical long-range percolation
• The polynomial growth of the infinite long-range percolation cluster
• Uniqueness of the critical long-range percolation metrics
• Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice
• Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice
• Distances in critical long range percolation
• The diameter of a long range percolation graph
• Randomized Distributed Edge Coloring via an Extension of the Chernoff--Hoeffding Bounds

Related

• Uniqueness of the critical long-range percolation metrics
• Superlinearity of geodesic length in 2D critical first-passage percolation
• Ergodicity of the number of infinite geodesics originating from zero
• Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape
• Scaling limit of first passage percolation geodesics on planar maps