7 papers
An Alon-Boppana Bound for the Non-Backtracking Operator
Theo McKenzie
For any fixed , we prove a lower bound on the th largest modulus of an eigenvalue of the non-backtracking matrix . Specifically, consider any deterministic or random famil…
The Bethe-Hessian down to the Percolation Threshold
Dingding Dong, Theo McKenzie
The Bethe-Hessian is a symmetric matrix for which the negative spectrum has been observed to encode the informative structure of sparse stochastic block models. We prove that, in t…
The Gaussian Wave for Graphs of Finite Cone Type
Amir Dembo, Theo McKenzie
We show that for any infinite tree of finite cone type satisfying a mild expansion condition, the only typical process on its vertices with covariance induced by the Green's functi…
Arbitrary Spectral Edge of Regular Graphs
Dingding Dong, Theo McKenzie
We prove that for each and , the set of limit points of the first eigenvalues of sequences of -regular graphs is \[ \{(μ_1,\dots,μ_k): d=μ_1\geq \dots\geq…
Ramanujan Property and Edge Universality of Random Regular Graphs
Jiaoyang Huang, Theo McKenzie, Horng-Tzer Yau
We consider the normalized adjacency matrix of a random -regular graph on vertices with any fixed degree and denote its eigenvalues as $λ_1=d/\sqrt{d-1}\geq λ_2\ge…
Optimal Eigenvalue Rigidity of Random Regular Graphs
Jiaoyang Huang, Theo McKenzie, Horng-Tzer Yau
Consider the normalized adjacency matrices of random -regular graphs on vertices with fixed degree , and denote the eigenvalues as $λ_1=d/\sqrt{d-1}\geq λ_2\geqλ_3\…
The Spectral Edge of Constant Degree Erdős-Rényi Graphs
Ella Hiesmayr, Theo McKenzie
We show that for an Erdős-Rényi graph on vertices with expected degree satisfying , the largest eigenvalues can be precisely determined…