Spectral Norms of Gaussian Matrices with Variance Profiles

Definitions and mathematical statement

For a finite dimension $n\geq 1$, under the conventional real-symmetric interpretation, let $X=(X_{ij})_{i,j=1}^{n}$ satisfy $X_{ji}=X_{ij}$, with $\{X_{ij}:1\leq i\leq j\leq n\}$ independent Gaussian random variables. Write \[ \mu_{ij}={\mathbb{E}} X_{ij},\qquad \operatorname{Var}(X_{ij})=b_{ij}^{2},\qquad b_{ij}\geq 0, \] and extend the variance profile symmetrically by $b_{ji}=b_{ij}$. Zero variances are permitted by the supplied arbitrary-variance formulation. Define \[ \|X\|_{\mathrm{op}}=\sup_{\substack{v\in{\mathbb{R}}^n\\ \|v\|_2=1}}\|Xv\|_2 =\max_{1\leq k\leq n}|\lambda_k(X)|, \] where $\lambda_1(X)\leq\cdots\leq\lambda_n(X)$ are the eigenvalues.

Requested determination. For arbitrary prescribed variance profiles $(b_{ij}^{2})$, determine or obtain sharp bounds for $\|X\|_{\mathrm{op}}$, with particular attention to the spectral edge, in terms of that profile.

Unresolved qualifications. The source does not explicitly specify the scalar field; the definitions above state the conventional real-symmetric interpretation. It does not require $\mu_{ij}=0$ or otherwise prescribe the means, so a centered formulation cannot be imposed without clarification. It also does not specify whether the desired bounds concern expectation, probability tails, another distributional quantity, or an asymptotic limit. The phrase "spectral edge" is not defined: it may concern an extreme eigenvalue or an edge of a limiting spectral distribution. No sequence of profiles, normalization, limiting regime, constants, or meaning of sharpness is supplied. Thus this is a research request with unresolved hypotheses and target quantities, rather than a fully quantified conjectural inequality.

Short English statement

Determine sharp spectral-norm estimates for a symmetric Gaussian matrix with independent upper-triangular entries and arbitrary prescribed variances. The source emphasizes the spectral edge but leaves centering, the precise probabilistic target, and any asymptotic regime unspecified. Its supplied status is partially solved; this statement-only draft requires human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete record for permanent ID $30002890$, titled Spectral Norms of Gaussian Matrices with Variance Profiles; source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2015; no individual proposer is identified.
• Supplied source citation: Probabilistic Techniques in Modern Statistics (2016), Oberwolfach Reports, pp. 1449--1498, DOI 10.4171/owr/2015/26. https://doi.org/10.4171/owr/2015/26. The background describes these reports as records of mathematical workshops and their open-problem discussions. The relationship between the citation's 2016 date and the supplied 2015 proposed year is unresolved.
• Supplied literature reference: Latała--van Handel--Youssef, Inventiones Mathematicae. https://doi.org/10.1007/s00222-018-0817-x. No article title or particular theorem is supplied.
• Supplied literature reference: Oberwolfach report 26. https://doi.org/10.4171/owr/2015/26.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem partially solved and states that sharp spectral-norm estimates are known for the extracted structured Gaussian setting, while the broader random-matrix question remains model-dependent and is not a single completely solved problem. This is supplied dated curation, not an independently verified assessment. The source itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review of its cleanup. These are supplied editorial assertions, not independent mathematical verification of this draft.