Random-matrix factors $g_G(k)$ in the moments of characteristic polynomials of unitary, orthogonal and symplectic matrices

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Numbers

$G$

$k$

$g_G(k)$

$U$

24024

$U$

701149020

$U$

1671643033734960

$U$

475073684264389879228560

$U$

22081374992701950398847674830857600

$U$

220381378415074546123953914908618547085974856000

$U$

10:

599868742615440724911356453304513631101279740967209774643120000

$U$

11:

551174774146602245392053367437420001094064949488253888377205259492780127332520000

$U$

12:

207241394414418553004822541258946187207183288761903149578574090088120492477229663925938231681424000000

$O$

286

$O$

33592

$O$

23178480

$O$

108995910720

$O$

10:

3973186258569120

$O$

11:

1257987096462161167200

$O$

12:

3830793890438041335187545600

$USp$

1/2

$USp$

1/4

$USp$

1/4

$USp$

3/4

$USp$

143/16

$USp$

4199/8

$USp$

1448655/8

$USp$

1703061105/4

$USp$

124162070580285/16

$USp$

10:

39312096764442536475/32

$USp$

11:

7482019317261799482788175/4

$USp$

12:

480669499374355203785613321154875/16

Definition

For positive integer $k$ and compact symmetry type $G\in\{U,O,USp\}$, $f_G(k)$ is the leading factor in the conductor-normalised CFKRS random-matrix moment [4] in (1). This table stores $g_G(k)=e_G(k)!f_G(k)$, where the exponent $e_G(k)$ is given in (2) [2].

Parameters

$G$

— compact symmetry type

$k$

— moment parameter ($k\geq1$)

Formulas

(1)

Let $\Lambda_A(z)$ be the characteristic polynomial of $A$, and let $X$ be the conductor-normalised matrix-size variable. The CFKRS Haar averages satisfy $\langle |\Lambda_A(1)|^{2k}\rangle_U\sim f_U(k)X^{k^2}$, $\langle \Lambda_A(1)^k\rangle_O\sim f_O(k)X^{k(k-1)/2}$, and $\langle \Lambda_A(1)^k\rangle_{USp}\sim f_{USp}(k)X^{k(k+1)/2}$ [2] [3].

(2)

$g_G(k)=e_G(k)!f_G(k)$, with $e_U(k)=k^2$, $e_O(k)=k(k-1)/2$, and $e_{USp}(k)=k(k+1)/2$. Equivalently, $f_G(k)=g_G(k)/e_G(k)!$.

(3)

$g_U(k)=(k^2)!\prod_{j=0}^{k-1}\frac{j!}{(k+j)!}$. Thus $g_U(1),g_U(2),g_U(3),g_U(4),g_U(5)=1,2,42,24024,701149020$ [1] [2].

(4)

$g_O(k)=2^{k-1}\left(k(k-1)/2\right)!\prod_{j=1}^{k-1}\frac{j!}{(2j)!}$ [3] [2].

(5)

$g_{USp}(k)=\left(k(k+1)/2\right)!\prod_{j=1}^{k}\frac{j!}{(2j)!}$ [3] [2].

Comments

(6)

The parameter is $k$. In the unitary rows it is half the exponent on the absolute value: $k=1$ is the second unitary moment and $k=2$ is the fourth unitary moment. In the orthogonal and symplectic rows it is the exponent of $\Lambda_A(1)^k$ in (1).

(7)

The product formulas (3), (4), and (5) are the exact CFKRS factors for positive integer $k$. The half-integer values of $k$ held by the table of arithmetic factors $a_k$ and the table of leading constants $c_k$ are not included here.

(8)

The factorial scaling $g_G(k)=e_G(k)!f_G(k)$ is the CFKRS integer-moment convention. For the entries here it makes the unitary and orthogonal rows integral, while the symplectic rows remain rational.

Programs

(P1)

Sage

from sage.arith.misc import factorial
from sage.misc.misc_c import prod
from sage.rings.rational_field import QQ

def g(G, k):
    if G == 'U':
        return QQ(factorial(k*k)) * prod(
            QQ(factorial(j)) / QQ(factorial(k+j))
            for j in range(k))
    if G == 'O':
        return QQ(2)**(k-1) * QQ(factorial(k*(k-1)//2)) * prod(
            QQ(factorial(j)) / QQ(factorial(2*j))
            for j in range(1, k))
    if G == 'USp':
        return QQ(factorial(k*(k+1)//2)) * prod(
            QQ(factorial(j)) / QQ(factorial(2*j))
            for j in range(1, k+1))
    raise ValueError(G)

print(g('U', 5))

References

[1]

J. P. Keating and N. C. Snaith, Random matrix theory and $\zeta(1/2+it)$, Communications in Mathematical Physics 214 (2000), 57-89. (doi)

[2]

J. Brian Conrey, David W. Farmer, Jon P. Keating, Michael O. Rubinstein and Nina C. Snaith, Integral moments of L-functions, Proceedings of the London Mathematical Society 91 (2005), 33-104. (arXiv)

[3]

Madan Lal Mehta and Jean-Marie Normand, Moments of the characteristic polynomial in the three ensembles of random matrices, Journal of Physics A: Mathematical and General 34 (2001), 4627-4639. (arXiv) (doi)

Links

[4]

Wikipedia: Random matrix

Similar tables

Arithmetic factors $a_k$ in the moments of the Riemann zeta function — stores the arithmetic factor paired with $f_U(k)$ in the zeta moment leading constant

Leading constants $c_k$ of the moments of the Riemann zeta function — stores $c_k=a_k f_U(k)$ in the same unitary normalisation

Data properties

Entries are of type: rational number

How well the digits are known: exact

Table is complete: no (it holds $G=U,O,USp$ for every positive integer $1\leq k\leq12$)

How they were obtained:

The generator evaluates the factorial products in (3), (5), and (4) over $\mathbb Q$.

Before the draft was filled, the unitary rows were checked against the printed values $1,2,42,24024,701149020$, and the symplectic and orthogonal rows were checked by extracting the leading terms from the corresponding CFKRS random-matrix moment formulas after the conductor normalisation.