Title: The rank of sparse symmetric matrices over arbitrary fields
| Speaker: | Noela Müller |
| Affiliation: | TU/e Eindhoven University of Technology |
| Location: | MC 5501 |
Abstract: Consider a sequence of sparse Erdös-Rényi random graphs (G_{n,d/n})_n on n vertices with edge probability d/n. Moreover, we equip the edges of G_{n,d/n} with prescribed non-zero edge weights chosen from an arbitrary field F. We show that the normalised rank of the adjacency matrices of (G_n,d/n)_n converges in probability to a constant, and derive the limiting expression. The result shows that for the general class of sparse symmetric matrices under consideration, the asymptotics of the normalised rank are independent of the edge weights and even the field, in the sense that the limiting constant for the general case coincides with the one previously established by Bordenave, Lelarge and Salez for adjacency matrices of sparse non-weighted Erdös-Rényi graphs over the real numbers. Our proof, which is purely combinatorial in its nature, is based on an intricate extension of the novel perturbation approach from Coja-Oghlan, Ergür, Gao, Hetterich and Rolvien to the symmetric setting.
The talk is based on joint work with Remco van der Hofstad and Haodong Zhu.