  Let $A_n$ be the sum of $d$ permutation matrices of size $n\times n$, each
drawn uniformly at random and independently. We prove that the normalized
characteristic polynomial $\frac{1}{\sqrt{d}}\det(I_n - z A_n/\sqrt{d})$
converges when $n\to \infty$ towards a random analytic function on the unit
disk. As an application, we obtain an elementary proof of the spectral gap of
random regular digraphs. Our results are valid both in the regime where $d$ is
fixed and for $d$ slowly growing with $n$.
