CLUSTER-SIZE DECAY IN SUPERCRITICAL KERNEL-BASED SPATIAL RANDOM GRAPHS

• 成果类型: Article
• 署名作者: Jorritsma, Joost; Komjathy, Julia; Mitsche, Dieter
• 署名单位: University of Oxford; Delft University of Technology; Pontificia Universidad Catolica de Chile; Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/24-AOP1742
• 发表日期: 2025
• 页码: 1537-1597
• 关键词: long-range percolation; large deviations; largest component; finite cluster; phase; MODEL

摘要

We consider a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation and the age-dependent random connection model. We assume that the model is supercritical: there is an infinite component. We identify the stretch-exponent zeta is an element of (0, 1) of the decay of the cluster-size distribution. That is, with |C(0)| denoting the number of vertices in the component of the vertex at 0 is an element of R-d, we prove P(k < |C(0)| < infinity) = exp( -Theta(k(zeta))) as k -> infinity. The value of zeta undergoes several phase transitions with respect to three main model parameters: the Euclidean dimension d, the power-law tail exponent tau of the degree distribution and a long-range parameter alpha governing the presence of long edges in Euclidean space. In this paper we present the proof for the region in the phase diagram where the model is a generalization of continuum scale-free percolation and/or hyperbolic random graphs: zeta in this regime depends both on tau, alpha. We also prove that the second-largest component in a box of volume n is of size Theta((log n)(1/zeta)) with high probability. We develop a deterministic algorithm, the cover expansion, as new methodology. This algorithm enables us to prevent too large components that may be de-localized or locally dense in space.