Universality for Products of Random Matrices with i.i.d. Entries and the Fuss--Catalan Number

Let $(w_{ij}{)}_{i,j\ge 1}$ be a single infinite array of independent identically distributed real- or complex-valued entries of mean zero, variance ${\sigma }^2$, and finite fourth moment. Set $W_n=(w_{ij}{)}_{1\le i,j\le n}$ and $X_n=n^{-1\mathrm{/}2}{W}_n$. For every fixed $k\ge 1$, we identify the almost sure limiting operator norm of several fixed products built from this family. Define the $k$-th freeness coefficient by $${\gamma }_k:=\sqrt{\frac{(k+1{)}^{k+1}}{k^k}}\mathrm{.}$$ Then we prove $$\mathrm{\parallel }{X}_n^k\mathrm{\parallel }\to {\sigma }^k{\gamma }_k\text{almost surely}\mathrm{.}$$ The same limit holds for products sampled with replacement from any fixed finite pool of independent copies of $X_n$; in particular, it holds for the product of $k$ independent copies. Thus, the freeness coefficient captures the non-commuting characteristic between large random matrices %powers and independent or fixed-pool sampled products under the finite fourth moment assumption. The improvement of the classical Bai–Yin-type power estimate from the scale ${\sigma }^k(k+1)$ to ${\sigma }^k\sqrt{k+1}$ is a direct corollary of our result. The main technical challenge is to prove the upper bound using a high-moment expansion of %the upper bound is proved by a high-moment expansion of $\text{E}\text{Tr}((X_n^k{X}_n^{\ast k}{)}^m)$. The leading zero-defect trace words are tree-like and are counted by the Fuss–Catalan number $$F_{k,m}=\frac{1}{km+1}\left(\genfrac{}{}{0px}{}{(k+1)m}{m}\right)\mathrm{.}$$ The combinatorial tool helps to devise a defect-sensitive global enumeration: if $L=km$ and $$r=(L+1-v)+(L-q),$$ then the number of admissible word classes with defect $r$ is at most $F_{k,m}(Cm{)}^{Dr}$. This polynomial-in-$m$ loss, with degree proportional to the defect, is summable in the logarithmic moment range.