Nicholas Cook - University of California, Los Angeles (UCLA), Mathematics Let P be a uniform random permutation matrix of size N and let χN(z)=det(zI−P) denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of χN on the unit circle, specifically, sup|z|=1|χN(z)|=Nxc+o(1) with probability tending to one as N→∞, for a numerical constant xc≈0.677. The main idea of the proof is to uncover an approximate branching structure in 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 \emph{CUE field} in which PN is replaced with a Haar unitary, the distribution of χN(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.