Local Law for Eigenvalues of Random Regular Bipartite Graphs
Tóm tắt
In this paper, we study the local law for eigenvalues of large random regular bipartite graphs with degree growing moderately fast. We prove that the empirical spectral distribution of the adjacency matrix converges in probability to a scaled down copy of the Marchenko–Pastur distribution on intervals of short length.
Tài liệu tham khảo
Bai, Z.D., Yao, J., Miao, B.: Convergence rates of spectral distributions of large sample covariance matrices. SIAM J. Matrix Anal. Appl. 25(1), 105–127 (2003)
Dumitriu, I., Johnson, T.: The Marcenko–Pastur law for sparse random bipartite biregular graphs. Random Struct. Algorithms 48(2), 313–340 (2016)
Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral statistics of Erdős-Rényi graphs II: eigenvalue spacing and the extreme eigenvalues. Commun. Math. Phys. 314(3), 587–640 (2012)
Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral statistics of Erdős-Rényi graphs I: local semicircle law. Ann. Probab. 41(3B), 2279–2375 (2013)
Friedman, J.: A proof of Alon’s second eigenvalue conjecture. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pp. 720–724. ACM (2003)
Götze, F., Tikhomirov, A.: Rate of convergence in probability to the Marchenko–Pastur law. Bernoulli 10(3), 503–548 (2004)
Guionnet, A., Zeitouni, O.: Concentration of the spectral measure for large matrices. Electron. Commun. Probab. 5, 119–136 (2000)
McKay, B., Wang, X.: Asymptotic enumeration of 0–1 matrices with equal row sums and equal column sums. Linear Algebra Appl. 373, 273–288 (2003)
McKay, B.D.: The expected eigenvalue distribution of a large regular graph. Linear Algebra Appl. 40, 203–216 (1981)
Tran, L.V., Vu, V.H., Wang, K.: Sparse random graphs: eigenvalues and eigenvectors. Random Struct. Algorithms 42(1), 110–134 (2013)