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Let $A_n$ be the sum of $d$ permutation matrices of size $n\times n$, eachdrawn uniformly at random and independently. We prove that the normalizedcharacteristic 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 unitdisk. As an application, we obtain an elementary proof of the spectral gap ofrandom regular digraphs. Our results are valid both in the regime where $d$ isfixed and for $d$ slowly growing with $n$.

The characteristic polynomial of sums of random permutations and regular digraphs