Local limits of random spanning trees in random environment

2024/10/22 by Luca Makowiec, Makowiec, Luca · 1 citation
Computer Science · Mathematics · #05C05 (Secondary) #60K35 (Primary) 82B41 #82B44 #Combinatorics (math.CO) #Data Management and Algorithms #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2410.16836

openalex publication_date 2024/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the edge overlap and local limit of the random spanning tree in random environment (RSTRE) on the complete graph with n vertices and weights given by exp(-βω_e) for ω_e uniformly distributed on [0,1]. We show that for β growing with β= o(n/log n), the edge overlap is (1+o(1)) β, while for β much larger than n log² n, the edge overlap is (1-o(1))n. Furthermore, there is a transition of the local limit around β= n. When β= o(n/ log n) the RSTRE locally converges to the same limit as the uniform spanning tree, whereas for β larger than n log^λn, where λ= λ(n) → ∞ arbitrarily slowly, the local limit of the RSTRE is the same as that of the minimum spanning tree.

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• Repeat times and a two-weight UST model

Related

• Random spanning trees in random environment
• Observables of random spanning trees in random environment
• The scaling limit of the minimum spanning tree of the complete graph
• The local limit of uniform spanning trees
• The local weak limit of the minimum spanning tree of the complete graph