A total variation threshold for the first coordinates of uniform parking functions via Lorentzian polynomials

Let $π_n$ be a uniform parking function of size $n$. We determine the scale at which its first $k$ coordinates cease to be close in total variation to $k$ independent uniform variables on $\{1,\ldots,n\}$. The distance is comparable to $1\wedge \sqrt{k\log(k+1)/n}$, uniformly for $n\ge2$ and $1\le k\le n$, and tends to one above the scale $n/\log n$. At the critical scale $k\simλn/\log n$, it converges to $2Φ(\sqrt{λ/π})-1$, where $Φ$ is the standard normal distribution function. A key ingredient is the Lorentzian property of the occupation generating polynomial of parking functions. Together with a strengthened Han inequality, this yields a relative entropy estimate for approximation by the product of the one-coordinate marginals.