Giants through higher-order paths in random simplicial complexes
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913088693862400 |
|---|---|
| author | Dhara, Souvik Kang, Taegyu |
| author_facet | Dhara, Souvik Kang, Taegyu |
| contents | We investigate the giant component formed via high-dimensional paths in the multi-parameter random simplicial complex (MRSC) model. For a $d$-dimensional simplicial complex, we define $d$-dimensional connectivity through incidence between $(d-1)$- and $d$-dimensional simplices. The phase transition of the largest $d$-dimensional connected component is determined in terms of the parameter $λ$ that governs the number of $d$-simplices incident to a typical $(d-1)$-simplex. In the subcritical regime, we show that the largest component contains $Θ(\log n)$ many $(d-1)$-simplices with high probability in the MRSC model. In the supercritical regime, we determine the asymptotic proportion of $1$-simplices in the giant component in dimension $2$, for $λ_c < λ< \barλ$, where $\barλ > 4$ is an explicit constant. In particular, for Linial-Meshulam complexes, this result holds throughout the entire supercritical regime. Additionally, we show that the number of vertices in the giant component undergoes a discontinuous phase transition in $d$-dimensional Linial-Meshulam complexes, in the sense that the asymptotic proportion of vertices in the giant jumps from $0$ to $1$. Our approach is based on local-weak convergence. We establish local-weak convergence in probability for the MRSC model and prove the concentration result via a refined analysis of the breadth-first exploration process, which tracks contributions from newly discovered and previously explored vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03151 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Giants through higher-order paths in random simplicial complexes Dhara, Souvik Kang, Taegyu Probability 60C05, 05E45, 05C80 We investigate the giant component formed via high-dimensional paths in the multi-parameter random simplicial complex (MRSC) model. For a $d$-dimensional simplicial complex, we define $d$-dimensional connectivity through incidence between $(d-1)$- and $d$-dimensional simplices. The phase transition of the largest $d$-dimensional connected component is determined in terms of the parameter $λ$ that governs the number of $d$-simplices incident to a typical $(d-1)$-simplex. In the subcritical regime, we show that the largest component contains $Θ(\log n)$ many $(d-1)$-simplices with high probability in the MRSC model. In the supercritical regime, we determine the asymptotic proportion of $1$-simplices in the giant component in dimension $2$, for $λ_c < λ< \barλ$, where $\barλ > 4$ is an explicit constant. In particular, for Linial-Meshulam complexes, this result holds throughout the entire supercritical regime. Additionally, we show that the number of vertices in the giant component undergoes a discontinuous phase transition in $d$-dimensional Linial-Meshulam complexes, in the sense that the asymptotic proportion of vertices in the giant jumps from $0$ to $1$. Our approach is based on local-weak convergence. We establish local-weak convergence in probability for the MRSC model and prove the concentration result via a refined analysis of the breadth-first exploration process, which tracks contributions from newly discovered and previously explored vertices. |
| title | Giants through higher-order paths in random simplicial complexes |
| topic | Probability 60C05, 05E45, 05C80 |
| url | https://arxiv.org/abs/2605.03151 |