Irreducibility of the characteristic polynomials of random tridiagonal matrices
Lior Bary-Soroker, Daniele Garzoni, Sasha Sodin
TL;DR
This work analyzes the irreducibility and Galois group of the characteristic polynomials $P_n$ of large random tridiagonal matrices $H_n$ with i.i.d. integer entries, conditioned on the extended Riemann Hypothesis for zeta functions associated to roots of $P_n$. The authors encode $P_n$ via transfer matrices and link root data to random walks on $\mathrm{PSL}_2(p)$, employing BV-type averaging over primes and modern strong approximation results to establish that $P_n$ is irreducible with probability $1 - C e^{-c n}$ and that $\mathrm{Gal}(P_n/\mathbb{Q})$ is typically $S_n$ or $A_n$. They prove a stretched-exponential bound first and upgrade to an exponential bound using Bourgain–Gamburd–Sarnak and Golsefidy–Srinivas, combined with a mixing analysis on $\mathrm{PSL}_2(p)$ and Goursat-type group arguments, and they also treat a constant-diagonal extension (the Dyson spike case) where the Galois group concentrates near a wreath product, with irreducibility for even $n$ and a linear factor times an irreducible piece for odd $n$ under nondegenerate tails. The results extend BV’s program from independent-coefficient models to a sparse random-matrix family, align with known results for full random matrices, and highlight intricate symmetry phenomena via the spectral/arithmetic interplay. Overall, the paper advances understanding of when high-degree random integer polynomials arising from matrix models are irreducible and how their Galois groups typically behave, under a conditional RH framework.
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--Varjú (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.