PreprintCC BY 4.0arXiv
Eigenvalue Distribution of the Laplacian on Random Complexes
Abstract
We study the empirical eigenvalue distribution of a standardized up Laplacian of the Linial-Meshulam model Yq+1(n, p). We first give a definition of a Gaussian--semicircle law: 𝒩(0, σ2)boxplus SC(sσ2), and give a combinatorial formula of its moments in terms of pairing partitions. In addition to that, we also prove that the limiting empirical eigenvalue distribution of this standardized up Laplacian follows a Gaussian--semicircle law, in the sense of almost surely weak convergence.
Related
- Eigenvalue comparison theorems for the Witten-Laplacian and the weighted p-Laplacian on complete manifolds with a modified Ricci curvature bounded from below
- Sharp lower bound of the first eigenvalue of the p-Laplacian under integral Ricci curvature
- An endpoint exception to a conjectured equality characterization for Laplacian eigenvalue products
- Non-radial minimizers for the first eigenvalue of the Laplacian in cones and a related overdetermined problem
- The least signless Laplacian eigenvalue of {C3,C5}-free graphs
BibTeX
@misc{dong2026eigenvalue,
title = {{Eigenvalue Distribution of the Laplacian on Random Complexes}},
author = {Rui Dong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31919},
eprint = {2609.31919},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31919},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Eigenvalue Distribution of the Laplacian on Random Complexes
AU - Rui Dong
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31919
UR - https://arxiv.org/abs/2609.31919
AB - We study the empirical eigenvalue distribution of a standardized up Laplacian of the Linial-Meshulam model Y_q+1(n, p). We first give a definition of a Gaussian--semicircle law: 𝒩(0, σ^2)boxplus SC(sσ^2), and give a combinatorial formula of its moments in terms of pairing partitions. In addition to that, we also prove that the limiting empirical eigenvalue distribution of this standardized up Laplacian follows a Gaussian--semicircle law, in the sense of almost surely weak convergence.
ER -