Theo McKenzie

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20232026

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math.PR2026

An Alon-Boppana Bound for the Non-Backtracking Operator

Theo McKenzie

For any fixed $k$ , we prove a lower bound on the $k$ th largest modulus of an eigenvalue of the non-backtracking matrix $B$ . Specifically, consider any deterministic or random famil…

math.PR2026

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…

math.PR2024

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 $d$ -regular graph on $N$ vertices with any fixed degree $d \geq 3$ and denote its eigenvalues as $λ_1=d/\sqrt{d-1}\geq λ_2\ge…

math.PR2024

Optimal Eigenvalue Rigidity of Random Regular Graphs

Jiaoyang Huang, Theo McKenzie, Horng-Tzer Yau

Consider the normalized adjacency matrices of random $d$ -regular graphs on $N$ vertices with fixed degree $d \geq 3$ , and denote the eigenvalues as $λ_1=d/\sqrt{d-1}\geq λ_2\geqλ_3\…

math.PR2023

The Spectral Edge of Constant Degree Erdős-Rényi Graphs

Ella Hiesmayr, Theo McKenzie

We show that for an Erdős-Rényi graph on $N$ vertices with expected degree $d$ satisfying $lo g^{- 1/9} N \leq d \leq lo g^{1/40} N$ , the largest eigenvalues can be precisely determined…