## Maximum of the Characteristic Polynomial for a Random Permutation Matrix

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.

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- General Mathematics
- 4904 Pure mathematics
- 4901 Applied mathematics
- 0102 Applied Mathematics
- 0101 Pure Mathematics

### Citation

*Communications on Pure and Applied Mathematics*,

*73*(8), 1660–1731. https://doi.org/10.1002/cpa.21899

*Communications on Pure and Applied Mathematics*73, no. 8 (August 1, 2020): 1660–1731. https://doi.org/10.1002/cpa.21899.

*Communications on Pure and Applied Mathematics*, vol. 73, no. 8, Aug. 2020, pp. 1660–731.

*Scopus*, doi:10.1002/cpa.21899.

## Published In

## DOI

## EISSN

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- General Mathematics
- 4904 Pure mathematics
- 4901 Applied mathematics
- 0102 Applied Mathematics
- 0101 Pure Mathematics