The Characteristic Polynomial of a Random Matrix
نویسندگان
چکیده
Form an $n \times n$ matrix by drawing entries independently from $\{\pm1\}$ (or another fixed nontrivial finitely supported distribution in $\mathbf{Z}$) and let $\phi$ be the characteristic polynomial. Conditionally on extended Riemann hypothesis, with high probability is irreducible $\mathrm{Gal}(\phi) \geq A_n$.
منابع مشابه
On the Characteristic Polynomial of a Random Unitary Matrix
We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N ×N unitary matrix, as N → ∞. First we show that lnZ/ √ 1 2 lnN , evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex normal random variables. This leads to a refinement of a recent central limit theorem due to Keating and Sn...
متن کاملThe characteristic polynomial of a random permutation matrix
We establish a central limit theorem for the logarithm of the characteristic polynomial of a random permutation matrix. With this result we can obtain a central limit theorem for the counting function for the eigenvalues lying in some interval on the unit circle.
متن کاملThe characteristic polynomial of a random unitary matrix: a probabilistic approach
In this article, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin-Fourier transform of such a random polynomial, first obtained by Keating and Snaith in [8] using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we ...
متن کاملSome Properties of the Characteristic Polynomial of a Nonnegative Matrix*
In this paper, we extend and generalize some of the well known spectral properties of a nonnegative matrix by establishing analogous properties for the zeros of derivatives of the characteristic polynomial of such a matrix.
متن کاملOn a conjecture regarding characteristic polynomial of a matrix pair
For n-by-n Hermitian matrices A(> 0) and B, define
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Combinatorica
سال: 2022
ISSN: ['0209-9683', '1439-6912']
DOI: https://doi.org/10.1007/s00493-020-4657-0