Maximum of the Characteristic Polynomial for a Random Permutation Matrix. Issue 8 (18th May 2020) - Cook, Nicholas; Zeitouni, Ofer
Abstract : 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, sup ∣ z ∣ = 1 ∣ χ N z ∣ = N x 0 + o 1 with probability tending to 1 as N → ∞, for a numerical constant x 0 ≈ 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 2 π i t ) 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