Scaling of components in critical long-range geometric random graphs on the 2-dim torus
Goriachkin, Vasilii LU
and Turova, Tatyana
LU
(2026)
In Stochastic Processes and their Applications
196.
- Abstract
We consider random graphs on the set of N2 vertices placed on the discrete 2-dimensional torus. The edges between pairs of vertices are independent, and their probabilities decay with the distance ρ between these vertices as (Nρ)−1. This is a versatile example of an inhomogeneous random graph that is not of rank 1. Here, we study the critical phase: the main result is the weak limit of the size of the largest connected component rescaled with (N2)−2/3 described by a diffusion process. This completes the proof that in all regimes, the model is within the same universality class as the Erdős-Rényi graph.
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/52bb5187-ad25-43ba-96c7-28ec5f958bbc
- author
- Goriachkin, Vasilii
LU
and Turova, Tatyana
LU
- organization
- publishing date
- 2026-06
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Critical long-range geometric random graphs, Phase transition
- in
- Stochastic Processes and their Applications
- volume
- 196
- article number
- 104927
- publisher
- Elsevier
- external identifiers
-
- scopus:105032357019
- ISSN
- 0304-4149
- DOI
- 10.1016/j.spa.2026.104927
- language
- English
- LU publication?
- yes
- additional info
- Publisher Copyright: © 2026
- id
- 52bb5187-ad25-43ba-96c7-28ec5f958bbc
- date added to LUP
- 2026-05-07 13:43:56
- date last changed
- 2026-05-07 13:45:02
@article{52bb5187-ad25-43ba-96c7-28ec5f958bbc,
abstract = {{<p>We consider random graphs on the set of N<sup>2</sup> vertices placed on the discrete 2-dimensional torus. The edges between pairs of vertices are independent, and their probabilities decay with the distance ρ between these vertices as (Nρ)<sup>−1</sup>. This is a versatile example of an inhomogeneous random graph that is not of rank 1. Here, we study the critical phase: the main result is the weak limit of the size of the largest connected component rescaled with (N<sup>2</sup>)<sup>−2/3</sup> described by a diffusion process. This completes the proof that in all regimes, the model is within the same universality class as the Erdős-Rényi graph.</p>}},
author = {{Goriachkin, Vasilii and Turova, Tatyana}},
issn = {{0304-4149}},
keywords = {{Critical long-range geometric random graphs; Phase transition}},
language = {{eng}},
publisher = {{Elsevier}},
series = {{Stochastic Processes and their Applications}},
title = {{Scaling of components in critical long-range geometric random graphs on the 2-dim torus}},
url = {{http://dx.doi.org/10.1016/j.spa.2026.104927}},
doi = {{10.1016/j.spa.2026.104927}},
volume = {{196}},
year = {{2026}},
}