2022/06/29 by Céline Kerriou, Peter Mörters, Kerriou, Céline +1
Mathematics · #Ball (mathematics) #Combinatorics #Discrete mathematics #FOS: Mathematics #Finite set #Geometry #Geometry and complex manifolds #Graph #Jump #Markov Chains and Monte Carlo Methods #Mathematical analysis #Mathematics #Physics #Probability (math.PR) #Quantum mechanics #Random graph #Stochastic processes and statistical mechanics #Torus #Vertex (graph theory)
paper · pdf · doi:10.48550/arxiv.2206.14627
openalex publication_date 2022/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a large deviation principle for the sum of n independent heavy-tailed random variables, which are subject to a moving cut-off boundary at location n. Conditional on the sum being large at scale n, we show that a finite number of summands take values near the cut-off boundary, while the remaining variables still obey the law of large numbers. This generalises the well-known single-big-jump principle for random variables without cut-off to a situation where just the minimal necessary number of jumps occur. As an application, we consider a random graph with vertex set given by the lattice points of a torus with sidelength 2N+1. Every vertex is the centre of a ball with random radius sampled from a heavy-tailed distribution. Oriented edges are drawn from the central vertex to all other vertices in this ball. When this graph is conditioned on having an exceptionally large number of edges we use our main result to show that, as N goes to infinity, the excess outdegrees condense in a fixed, finite number of randomly scattered vertices of macroscopic outdegree. By contrast, no condensation occurs for the indegrees of the vertices, which all remain microscopic in size.