It can be shown that the maximum eigenvalue of a row-stochastic matrix is $1$, adapting, e.g. What are possible eigenvalues of a 0-1 (boolean) matrix?
A contrario no general result holds regarding its minimum eigenvalue. But, is it still true that no general result holds if one adds the condition that the matrix, in addition to being row-stochastic, is a permutation matrix ? Also, note that only their real parts is of importance to me.
I did a simulation by randomly generating $300$ permutation matrices (incidentally, sometime not of full rank and that can be dimensionally reduced thus, not the point here). The chart of their Real parts follows.
None of their real part is lower than $-1$.