A1716
Title: Connectivity transitions in large-scale simplicial complexes via local limits
Authors: Taegyu Kang - Georgia Institute of Technology (United States) [presenting]
Souvik Dhara - Georgia Institute of Technology (United States)
Abstract: Higher-dimensional connected components in random simplicial complexes are investigated. In a d-dimensional simplicial complex, d-dimensional connectivity is defined by incidence between (d-1)- and d-dimensional simplices. The phase transition of the largest d-dimensional connected component is identified in terms of the parameter that governs the number of d-dimensional simplices incident to a typical (d-1)-dimensional simplex. When the parameter is larger than the threshold 1/d, a large d-dimensional component, called the giant component, emerges. Below the threshold, all d-dimensional components are small. Additionally, when the parameter is larger than 1/d, the proportion of 0-dimensional simplices in the d-dimensional giant component converges to 1, meaning the giant occupies almost all vertices. The model considered is the multi-parameter random simplicial complex and its special case, the Linial-Meshulam random simplicial complex. The primary machinery is the local-weak convergence of random simplicial complexes. Local-weak convergence in probability of the multi-parameter random simplicial complex is established and the local information is leveraged for the formation of the giant component and its concentration.
