let $p$ be a uniform random permutation matrix of size $n$ and let $\chi_n(z)= \det(zi - p)$ denote its characteristic polynomial. we prove a law of large numbers for the maximum modulus of $\chi_n$ on the unit circle, specifically, \[ \sup_{|z|=1}|\chi_n(z)|= n^{x_c + o(1)} \] with probability tending to one as $n\to \infty$, for a numerical constant $x_c\approx 0.677$. the main idea of the proof is to uncover an approximate branching structure in the distribution of (the logarithm of) $\chi_n$, viewed as a random field on the circle, and to adapt a well-known second moment argument for the maximum of the branching random walk. unlike the well-studied \emph{cue field} in which $p_n$ is replaced with a haar unitary, the distribution of $\chi_n(z)$ is sensitive to diophantine properties of the argument of $z$. to deal with this we borrow tools from the hardy--littlewood circle method in analytic number theory. based on joint work with ofer zeitouni.