The structured Gaussian matrix norm conjecture

Let $X=(X_{ij})_{1\leq i,j\leq n}$ be the centered Gaussian random matrix considered in this section, with entry standard deviations $b_{ij}$, and assume without loss of generality that its rows and columns have been permuted so that

$$\max_j b_{1j}\geq \max_j b_{2j}\geq\cdots\geq\max_j b_{nj}>0.$$

Here $\|X\|$ denotes the operator norm and $\asymp$ denotes equivalence up to universal constant factors. Structured Gaussian matrix norm conjecture.

$$\mathbf{E}\|X\|\asymp \|\mathbf{E}X^2\|^{1/2}+\mathbf{E}\max_{i,j\leq n}|X_{ij}|$$

and equivalently,

$$\mathbf{E}\|X\|\asymp \max_{i\leq n}\sqrt{\sum_{j=1}^n b_{ij}^2}+\max_{i,j\leq n}b_{ij}\sqrt{\log i}.$$

The conjecture identifies the average variance scale and the largest-entry fluctuation as the only mechanisms that can force a large expected norm. The paper explains that its dimension-free form is not accessible by the moment method and suggests that random-process methods may be essential; its status is presented as unresolved.