My question is about the asymptotic behavior of the ratio between the largest and second largest values of $n$ independent chi-square random variables.

Let $X_1, \ldots, X_n$ be $n$ independent and identically distributed random variables with distribution $\chi_1^2$. Let $X_{(n)}$ be the largest and $X_{(n-1)}$ be the second largest of these $n$ random variables. I was wondering what is the asymptotic order of $X_{(n)}/ X_{(n-1)}$. My conjecture is that this quantity has the order of $1 + O(1/ \log n)$.

Can we show that \begin{equation} \frac{X_{(n)}} {X_{(n-1)}} - 1 \asymp \frac{C}{\log n} \end{equation} with high probability, where $C$ is a constant?