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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2512.20146 |
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| _version_ | 1866915699206651904 |
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| author | Chen, Yiming Chen, Zijun Zhu, Yizhe |
| author_facet | Chen, Yiming Chen, Zijun Zhu, Yizhe |
| contents | We study the limiting spectral distribution of the normalized Laplacian $\mathcal L$ of an Erdős-Rényi graph $G(n,p)$. To account for the presence of isolated vertices in the sparse regime, we define $\mathcal L$ using the Moore-Penrose pseudoinverse of the degree matrix. Under this convention, we show that the empirical spectral distribution of a suitably normalized $\mathcal L$ converges weakly in probability to the semicircle law whenever $np\to\infty$, thereby providing a rigorous justification of a prediction made in (Akara-pipattana and Evnin, 2023). Moreover, if $np>\log n+ω(1)$, so that $G(n,p)$ has no isolated vertices with high probability, the same conclusion holds for the standard definition of $\mathcal L$. We further strengthen this result to almost sure convergence when $np=Ω(\log n)$. Finally, we extend our approach to the Chung-Lu random graph model, where we establish a semicircle law for $\mathcal L$ itself, improving upon (Chung, Lu, and Vu 2003), which obtained the semicircle law only for a proxy matrix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20146 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A semicircle law for the normalized Laplacian of sparse random graphs Chen, Yiming Chen, Zijun Zhu, Yizhe Probability 60B20, 60F05 We study the limiting spectral distribution of the normalized Laplacian $\mathcal L$ of an Erdős-Rényi graph $G(n,p)$. To account for the presence of isolated vertices in the sparse regime, we define $\mathcal L$ using the Moore-Penrose pseudoinverse of the degree matrix. Under this convention, we show that the empirical spectral distribution of a suitably normalized $\mathcal L$ converges weakly in probability to the semicircle law whenever $np\to\infty$, thereby providing a rigorous justification of a prediction made in (Akara-pipattana and Evnin, 2023). Moreover, if $np>\log n+ω(1)$, so that $G(n,p)$ has no isolated vertices with high probability, the same conclusion holds for the standard definition of $\mathcal L$. We further strengthen this result to almost sure convergence when $np=Ω(\log n)$. Finally, we extend our approach to the Chung-Lu random graph model, where we establish a semicircle law for $\mathcal L$ itself, improving upon (Chung, Lu, and Vu 2003), which obtained the semicircle law only for a proxy matrix. |
| title | A semicircle law for the normalized Laplacian of sparse random graphs |
| topic | Probability 60B20, 60F05 |
| url | https://arxiv.org/abs/2512.20146 |