Irreducibility of the characteristic polynomials of random tridiagonal matrices

TL;DR

Under the assumption of the Riemann hypothesis for certain Dedekind zeta functions, the paper proves that the characteristic polynomials of specific large random tridiagonal matrices are almost surely irreducible with Galois groups being symmetric or alternating, extending prior results.

Contribution

It establishes irreducibility and Galois group structure of characteristic polynomials of large random tridiagonal matrices under number-theoretic assumptions, linking random matrix theory and algebraic number theory.

Findings

Characteristic polynomials are irreducible with high probability

Galois group is either symmetric or alternating

Results extend previous work on random polynomials and matrices

Abstract

Conditionally on the Riemann hypothesis for certain Dedekind zeta functions, we show that the characteristic polynomial of a class of random tridiagonal matrices of large dimension is irreducible, with probability exponentially close to one; moreover, its Galois group over the rational numbers is either the symmetric or the alternating group. This is the counterpart of the results of Breuillard--Varj\'u (for polynomials with independent coefficients), and with those of Eberhard and Ferber--Jain--Sah--Sawhney (for full random matrices). We also analyse a related class of random tridiagonal matrices for which the Galois group is much smaller.