The dimension-free structure of nonhomogeneous random matrices
The main result of the paper is the following proposition (Theorem 1.1). Let \(X\) be an \(n \times n\) symmetric matrix with \(X_{ij}=b_{ij}g_{ij}\), where \(b_{ij}\ge0\) and \(g_{ij}\) are i.i.d. standard Gaussian random variables for \(i\ge j\). Then \[\left(\mathbf{E}\|X\|^p_{S_p}\right)^{1/p}\asymp\left(\sum\limits_i\left(\sum\limits_j b_{ij}^2\right)^{\frac{p}{2}}\right)^{\frac{1}{p}}+\sqrt{p}\left(\sum\limits_i\max\limits_j b_{ij}^p\right)^{\frac{1}{p}}\] and \[\mathbf{E}\|X\|_{S_p}\asymp\left(\sum\limits_i\left(\sum\limits_j b_{ij}^2\right)^{\frac{p}{2}}\right)^{\frac{1}{p}}+\max\limits_{i\le e^p}\max\limits_j b_{ij}^\ast \sqrt{\log i}+\sqrt{p}\left(\sum\limits_{i\ge e^p}\max\limits_j b_{ij}^\ast\right)^{\frac{1}{p}}\] for all \(2\le p<\infty\) and \[\mathbf{E}\|X\|_{S_\infty}\asymp \max\limits_i\sqrt{\sum\limits_j b_{ij}^2}+\max\limits_{ij} b_{ij}^\ast \sqrt{\log i}\] for \(p=\infty\). Here the matrix \(\left(b_{ij}^\ast\right)\) is obtained by permuting the rows and columns of the matrix \(\left(b_{ij}\right)\) such that \(\max_j b_{1j}^\ast\ge \max_j b_{2j}^\ast\ge\cdots\ge \max_j b_{nj}^\ast\), the constants in the estimates are universal (independent of \(n, p, \left(b_{ij}\right)\)) and \(S_p\) denotes the \(p\)-Schatten class. Theorem 1.1 settles, in the case \(p=\infty\), a conjecture of the first author [\textit{R. Latala}, Proc. Am. Math. Soc. 133, No. 5, 1273--1282 (2005; Zbl 1067.15022)], and provides a complete characterization of the class of infinite matrices with independent Gaussian entries that define bounded operators on \(\ell_2\).
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