2024/09/17 by Debapratim Banerjee, Banerjee, Debapratim, Soumendu Sundar Mukherjee +3 · 1 voice · 1 citation
Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Statistics Theory (math.ST) #math-ph #math.CO #math.PR #math.ST
paper · pdf · doi:10.48550/arxiv.2409.11381
openalex publication_date 2024/09/17 · arxiv published 2024/09/17 · arxiv updated 2025/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the largest eigenvalue of a Gaussian random symmetric matrix Xn, with zero-mean, unit variance entries satisfying the condition sup(i, j) ≠ (i', j')|𝔼[Xij Xi'j']| = O(n-(1 + ε)), where ε > 0. It follows from Catalano et al. (2024) that the empirical spectral distribution of n-1/2 Xn converges weakly almost surely to the standard semi-circle law. Using a Füredi-Komlós-type high moment analysis, we show that the largest eigenvalue λ1(n-1/2 Xn) of n-1/2 Xn converges almost surely to 2. This result is essentially optimal in the sense that one cannot take ε = 0 and still obtain an almost sure limit of 2. We also derive Gaussian fluctuation results for the largest eigenvalue in the case where the entries have a common non-zero mean. Let Yn = Xn + \fracλ√(n)1 1^\top. When ε ≥ 1 and λ≫ n1/4, we show that n1/2(λ1(n-1/2 Yn) - λ- \frac1λ) \xrightarrowd √(2) Z, where Z is a standard Gaussian. On the other hand, when 0 < ε < 1, we have Var((1)/(n)∑i, jXij) = O(n1 - ε). Assuming that Var((1)/(n)∑i, j Xij) = σ2 n1 - ε (1 + o(1)), if λ≫ nε/4, then we have nε/2(λ1(n-1/2 Yn) - λ- \frac1λ) \xrightarrowd σZ. While the ranges of λ in these fluctuation results are certainly not optimal, a striking aspect is that different scalings are required in the two regimes 0 < ε < 1 and ε ≥ 1.