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Bounding second eigenvalue of random rank-1 matrix
Consider a fixed set of $n$ points $\mathcal{X}$ on a bounded domain $\Omega \subset \mathbb{R}^d$ and a rank-1 matrix $\mathbf{R}$ with independent entries $R_{ij} \sim \mathcal{N}(0,\lambda_i/n)$ independently for all $(i,j)$ (so $\lambda_i \in [0,1]$ is fixed).
What are the typical bounds on the second eigenvalue of $\mathbf{R}$ as a function of $n$ and $\Omega$? Do the bounds depend on $n$, $\Omega$, and the $\lambda_i$?
More precisely, do there exist constants $C_1,C_2$ and an explicit function $g(n,\Omega,\lambda_1,\ldots,\lambda_n)$ such that $C_1 \leq \lambda_2 \leq C_2$ on average over all matrices $\mathbf{R}$ with (independent) entries $\mathcal{N}(0,\lambda_i/n)$ for all $i=1,\ldots,n$ and with $\lambda_2 \geq g(n,\Omega,\lambda_1,\ldots,\lambda_n)$?
A:
For $\lambda_i$ fixed, the matrix $\mathbf{R}$ is a random variable with a multivariate Gaussian distribution with non-zero mean, so the eigenvalues are also random variables. However, it will be almost surely independent and not too concentrated around their expectation. This makes $\lambda_2$ have an almost surely
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