© 2020 Wiley Periodicals LLC Let PN be a uniform random N × N permutation matrix and let χN(z) = det(zIN − PN) denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of χN on the unit circle, specifically, (Formula presented.) with probability tending to 1 as N → ∞, for a numerical constant x0 ≈ 0.652. The main idea of the proof is to uncover a logarithmic correlation structure for the distribution of (the logarithm of) χN, 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 CUE field in which PN is replaced with a Haar unitary, the distribution of χN(e2πit) is sensitive to Diophantine properties of the point t. To deal with this we borrow tools from the Hardy-Littlewood circle method. © 2020 Wiley Periodicals LLC.