Strong convergence of directed Kikuchi matrices for multiple variables

Prove that the strong-convergence statement for tuples of independent directed Kikuchi matrices extends to every number of variables \(k\ge 1\) and every parameter \(\alpha>0\) in the double limit \(n\to\infty\), followed by \(\ell\to\infty\) with \(r=r(\ell)\) satisfying \(r^2/\ell\to\alpha\).

Background

Theorem 1.4 proves weak convergence of tuples of independent directed and undirected Kikuchi matrices to the $e^{-\alpha}$-Gaussian system, and proves strong convergence only for a single directed Kikuchi matrix, corresponding to $k=1$.

The authors identify the natural extension as multivariate strong convergence for all $k\ge1$ and $\alpha>0$. They explicitly exclude $\alpha=0$ from this formulation because the associated “1-Gaussian system” has unbounded creation-derived operators, so strong convergence is not meaningful in that case.