Maximum of the Characteristic Polynomial for a Random Permutation Matrix

Let P_N be a uniform random N × N permutation matrix and let χ_N(z) = det(zI_N − P_N) denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of χ_N on the unit circle, specifically, 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) χ_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 CUE field in which P_N is replaced with a Haar unitary, the distribution of χ_N(e^2πit) 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