# Maximum of the Characteristic Polynomial for a Random Permutation Matrix

Journal Article

© 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.

### Full Text

### Duke Authors

### Cited Authors

- Cook, N; Zeitouni, O

### Published Date

- August 1, 2020

### Published In

### Volume / Issue

- 73 / 8

### Start / End Page

- 1660 - 1731

### Electronic International Standard Serial Number (EISSN)

- 1097-0312

### International Standard Serial Number (ISSN)

- 0010-3640

### Digital Object Identifier (DOI)

- 10.1002/cpa.21899

### Citation Source

- Scopus