Scaling limits and generic bounds for exploration processes

Paola Bermolen, Matthieu Jonckheere (Corresponding author), Jaron Sanders

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

We consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its neighboring nodes become blocked. Given an initial number of vertices N growing to infinity, we study statistical properties of the proportion of explored (active or blocked) nodes in time using scaling limits. We obtain exact limits for homogeneous graphs and prove an explicit central limit theorem for the final proportion of active nodes, known as the jamming constant, through a diffusion approximation for the exploration process which can be described as a unidimensional process. We then focus on bounding the trajectories of such exploration processes on random geometric graphs, i.e., random sequential adsorption. As opposed to exploration processes on homogeneous random graphs, these do not allow for such a dimensional reduction. Instead we derive a fundamental relationship between the number of explored nodes and the discovered volume in the spatial process, and we obtain generic bounds for the fluid limit and jamming constant: bounds that are independent of the dimension of space and the detailed shape of the volume associated to the discovered node. Lastly, using coupling techinques, we give trajectorial interpretations of the generic bounds.

Original languageEnglish
Pages (from-to)989-1018
Number of pages30
JournalJournal of Statistical Physics
Volume169
Issue number5
DOIs
Publication statusPublished - 1 Dec 2017
Externally publishedYes

Keywords

  • Random graphs
  • Random sequential adsorption
  • Scaling limits

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