Spectral radius concentration for inhomogeneous random matrices with independent entries
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918116259266560 |
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| author | Han, Yi |
| author_facet | Han, Yi |
| contents | Let $A$ be a square random matrix of size $n$, with mean zero, independent but not identically distributed entries, with variance profile $S$. When entries are i.i.d. with unit variance, the spectral radius of $n^{-1/2}A$ converges to $1$ whereas the operator norm converges to 2. Motivated by recent interest in inhomogeneous random matrices, in particular non-Hermitian random band matrices, we formulate general upper bounds for $ρ(A)$, the spectral radius of $A$, in terms of the variance $S$. We prove (1) after suitable normalization $ρ(A)$ is bounded by $1+ε$ up to the optimal sparsity $σ_*\gg (\log n)^{-1/2}$ where $σ_*$ is the largest standard deviation of an individual entry; (2) a small deviation inequality for $ρ(A)$ capturing fluctuation beyond the optimal scale $σ_*^{-1}$; (3) a large deviation inequality for $ρ(A)$ with Gaussian entries and doubly stochastic variance; and (4) boundedness of $ρ(A)$ in certain heavy-tailed regimes with only $2+ε$ finite moments and inhomogeneous variance profile $S$. The proof relies heavily on the trace moment method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01079 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral radius concentration for inhomogeneous random matrices with independent entries Han, Yi Probability Let $A$ be a square random matrix of size $n$, with mean zero, independent but not identically distributed entries, with variance profile $S$. When entries are i.i.d. with unit variance, the spectral radius of $n^{-1/2}A$ converges to $1$ whereas the operator norm converges to 2. Motivated by recent interest in inhomogeneous random matrices, in particular non-Hermitian random band matrices, we formulate general upper bounds for $ρ(A)$, the spectral radius of $A$, in terms of the variance $S$. We prove (1) after suitable normalization $ρ(A)$ is bounded by $1+ε$ up to the optimal sparsity $σ_*\gg (\log n)^{-1/2}$ where $σ_*$ is the largest standard deviation of an individual entry; (2) a small deviation inequality for $ρ(A)$ capturing fluctuation beyond the optimal scale $σ_*^{-1}$; (3) a large deviation inequality for $ρ(A)$ with Gaussian entries and doubly stochastic variance; and (4) boundedness of $ρ(A)$ in certain heavy-tailed regimes with only $2+ε$ finite moments and inhomogeneous variance profile $S$. The proof relies heavily on the trace moment method. |
| title | Spectral radius concentration for inhomogeneous random matrices with independent entries |
| topic | Probability |
| url | https://arxiv.org/abs/2501.01079 |