Almost sure behavior of the critical points of random polynomials

Let $(Z_k{)}_{k\ge 1}$ be a sequence of independent and identically distributed complex random variables with common distribution $\mu$ and let $P_n(X):={\prod }_{k=1}^n(X-Z_k)$ the associated random polynomial in $\mathbb{C}[X]$. In [Kab15], the author established the conjecture stated by Pemantle and Rivin in [PR13] that the empirical measure ${\nu }_n$ associated with the critical points of $P_n$ converges weakly in probability to the base measure $\mu$. In this note, we establish that the convergence in fact holds in the almost sure sense. Our result positively answers a question raised by Z. Kabluchko and formalized as a conjecture in the recent paper [MV22].