Expected behavior of the maximum ranking generated in optimal assignment
Frieze and Sorkin showed that in the optimal solution of the assignment problem with weights chosen uniformly in $[0,1]$, the maximal value is $\Theta\bigl(\frac{\log n}{n}\bigr)$.
This suggests that the answer in your case is $\Theta(\log n)$.
I found this reference in the survey of Krokhmal and Pardalos mentioned in my answer to your original question. I suggest consulting the survey on your own for any further questions.