Sharp norm growth for graph matrices

Determine, for every fixed shape α, functions f(α) and g(α) such that the spectral norm of the associated graph matrix satisfies E||M_α|| = Θ( n^{f(α)} (log n)^{g(α)} ), thereby establishing tight polynomial and polylogarithmic growth rates depending only on α.

Background

Graph matrices arise in Sum-of-Squares lower bounds via pseudo-calibrated moment matrices and capture polynomial dependencies on local random inputs. Existing moment-method analyses give bounds with extra polylog factors in many cases; iterated matrix concentration can remove them for some shapes.

A complete characterization of f and g would clarify when noncommutative effects produce extra polylog factors, sharpen SoS lower bounds, and improve understanding of matrix chaos beyond the linear case.

References

We conjecture that given a fixed shape $\alpha$, the norm of the associated graph matrix grows as a product of a polynomial and a polylogarithmic factor (depending only on $\alpha$). There are functions $f$ and $g$ depending only on the shape $\alpha$, such that \begin{equation}\label{eq:sharp_gmtx_bound} \mathbb{E}|M_\alpha| = \Theta\left(n{f(\alpha)}\,(\log n){g(\alpha)}\right). \end{equation}

eq:sharp_gmtx_bound:

E∥Mα∥=Θ(nf(α) (log⁡n)g(α)).\mathbb{E}\|M_\alpha\| = \Theta\left(n^{f(\alpha)}\,(\log n)^{g(\alpha)}\right).

— Randomstrasse101: Open Problems of 2025  (2603.29571 - Bandeira et al., 31 Mar 2026) in Conjecture [Sharp graph matrix bounds], Section “Sharp Bounds for Graph Matrices (PNN)” (Entry 14)