Maximum of the Characteristic Polynomial for a Random Permutation Matrix. Issue 8 (18th May 2020)

• Record Type: Journal Article
• Authors: Cook, Nicholas; Zeitouni, Ofer
• Is Part Of: Communications on pure and applied mathematics. Volume 73:Issue 8(2020:Aug.)
• Page Start: 1660
• Page End: 1731
• Publication Date: 2020-05-18
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.21899
• Languages: English
• ISSNs: 0010-3640

Abstract

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