Geometry of the minimal spanning tree in the heavy-tailed regime: new universality classes

• 成果类型: Article
• 署名作者: Bhamidi, Shankar; Sen, Sanchayan
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill; Indian Institute of Science (IISC) - Bangalore
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01259-3
• 发表日期: 2024
• 页码: 729-804
• 关键词: critical random graphs; continuum random trees; SCALING LIMITS; percolation; THEOREM; GROWTH; rates

摘要

A well-known open problem on the behavior of optimal paths in random graphs in the strong disorder regime, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade (Braunstein et al. in Phys RevLett 91(16):168701, 2003; Braunstein et al. in Int J Bifurc Chaos 17(07):2215-2255,2007; Chen et al. in Phys Rev Lett 96(6):068702, 2006; Wu et al. in Phys Rev Lett96(14):148702, 2006) is as follows: for a large class of random graph models with degree exponent tau is an element of (3,4), distances in the minimal spanning tree (MST) on the giant component in the supercritical regime scale liken n((tau-3)/(tau-1)). The aim of this paper is to make progress towards a proof of this conjecture. We consider a supercritical in homogeneous random graph model with degree exponent tau is an element of(3,4)that is closely related to Aldous's multiplicative coalescent, and show that the MST constructed by assigning i.i. d. continuous weights to the edges in its giant component, endowed with the tree distance scaled by n(-(tau-3)/(tau-1)), converges in distribution with respect to the Gromov-Hausdorff topology to a random compact real tree. Further, almost surely, every point in this limiting space either has degree one (leaf), or two, or infinity (hub),both the set of leaves and the set of hubs are dense in this space, and the Minkowski dimension of this space equals(tau-1)/(tau-3). The multiplicative coalescent, in an asymptotic sense, describes the evolution of the component sizes of various near-critical random graph processes. We expect the limiting spaces in this paper to be the candidates for the scaling limit of the MST constructed for a wide array of other heavy-tailed random graph models.