Local limits of spatial Gibbs random graphs

TL;DR

This paper investigates the local convergence properties of spatial Gibbs random graphs embedded in a line segment, revealing conditions under which they exhibit hierarchical structures and convergence behavior.

Contribution

It provides new insights into the local limits of spatial Gibbs random graphs and characterizes their convergence based on parameter regimes, extending previous threshold results.

Findings

Graphs can or cannot converge locally depending on parameters.

Hierarchical edge structures influence local convergence.

Threshold behavior relates to graph diameter and edge organization.

Abstract

We study the spatial Gibbs random graphs introduced in [MV16] 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 [MV16] 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.