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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.14259 |
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| _version_ | 1866915398331400192 |
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| author | Nagel, Leonhard |
| author_facet | Nagel, Leonhard |
| contents | We study how eigenvectors of random regular graphs behave when projected onto fixed directions. For a random $d$-regular graph with $N$ vertices, where the degree $d$ grows slowly with $N$, we prove that these projections follow approximately normal distributions. Our main result establishes a Berry-Esseen bound showing convergence to the Gaussian with error $O(\sqrt{d} \cdot N^{-1/6+\varepsilon})$ for degrees $d \leq N^{1/4}$. This bound significantly improves upon previous results that had error terms scaling as $d^3$, and we prove our $\sqrt{d}$ scaling is optimal by establishing a matching lower bound. Our proof combines three techniques: (1) refined concentration inequalities that exploit the specific variance structure of regular graphs, (2) a vector-based analysis of the resolvent that avoids iterative procedures, and (3) a framework combining Stein's method with graph-theoretic tools to control higher-order fluctuations. These results provide sharp constants for eigenvector universality in the transition from sparse to moderately dense graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14259 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp Square Root Bounds for Edge Eigenvector Universality in Sparse Random Regular Graphs Nagel, Leonhard Probability Discrete Mathematics Mathematical Physics Combinatorics Spectral Theory 60B20, 05C80, 15B52, 60F05, 05C50, 82B44 G.3; G.2.2; F.2.2; G.1.3; G.2.m We study how eigenvectors of random regular graphs behave when projected onto fixed directions. For a random $d$-regular graph with $N$ vertices, where the degree $d$ grows slowly with $N$, we prove that these projections follow approximately normal distributions. Our main result establishes a Berry-Esseen bound showing convergence to the Gaussian with error $O(\sqrt{d} \cdot N^{-1/6+\varepsilon})$ for degrees $d \leq N^{1/4}$. This bound significantly improves upon previous results that had error terms scaling as $d^3$, and we prove our $\sqrt{d}$ scaling is optimal by establishing a matching lower bound. Our proof combines three techniques: (1) refined concentration inequalities that exploit the specific variance structure of regular graphs, (2) a vector-based analysis of the resolvent that avoids iterative procedures, and (3) a framework combining Stein's method with graph-theoretic tools to control higher-order fluctuations. These results provide sharp constants for eigenvector universality in the transition from sparse to moderately dense graphs. |
| title | Sharp Square Root Bounds for Edge Eigenvector Universality in Sparse Random Regular Graphs |
| topic | Probability Discrete Mathematics Mathematical Physics Combinatorics Spectral Theory 60B20, 05C80, 15B52, 60F05, 05C50, 82B44 G.3; G.2.2; F.2.2; G.1.3; G.2.m |
| url | https://arxiv.org/abs/2507.14259 |