Local limits of spatial Gibbs random graphs

Author(s)

Endo, Eric O. ^{[1, 2, 3]} ; Valesin, Daniel ^[1]

Affiliation

^[1] Univ Groningen, Johann Bernoulli Inst, Nijenborgh 9, NL-9747 AG Groningen - Netherlands

^[2] NYU Shanghai, NYU ECNU Inst Math Sci, 1555 Century Ave, Shanghai - Peoples R China

^[3] Univ Sao Paulo, Inst Math & Stat, Rua Matao 1010, Sao Paulo, SP - Brazil

• Document type: Journal article
• Source: ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS; v. 17, n. 1, p. 51-63, 2020.
• Web of Science Citations: 0

Abstract

We study the spatial Gibbs random graphs introduced in Mourrat and Valesin (2018) from the point of view of local convergence. These are random graphs embedded in an ambient space consisting of a line segment, defined through a probability measure that favors graphs of small (graph-theoretic) diameter but penalizes the presence of edges whose extremities are distant in the geometry of the ambient space. In tourrat and Vilesin (201 ) these graphs were shown to exhibit threshold behavior with respect to the various parameters that define them; this behavior was related to the formation of hierarchical structures of edges organized so as to produce a small diameter. Here we prove that, for certain values of the underlying parameters, the spatial Gibbs graphs may or may not converge locally, in a manner that is compatible with the aforementioned hierarchical structures. (AU)

• FAPESP's process: 15/14434-8 - Phase Transitions in Spin Models with General External Fields
• Grantee: Eric Ossami Endo
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• FAPESP's process: 14/10637-9 - Combinatorial Problems in Ferromagnetic Models
• Grantee: Eric Ossami Endo
• Support Opportunities: Scholarships in Brazil - Doctorate