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Morris integrals for circular ensembles with a coalescing charge
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Numbers
$\beta$
$n$
$k$ 
$Z_{\beta,n}(k)$
1
1
0:
1
1
1
1:
1.273239544735162686151070106980114896275677165923651589981338752471174381073812280720910422130024688
1
1
2:
2
1
1
3:
3.395305452627100496402853618613639723401805775796404239950236673256465016196832748589094459013399167
1
1
4:
6
1
2
0:
1.273239544735162686151070106980114896275677165923651589981338752471174381073812280720910422130024688
1
2
1:
1.909859317102744029226605160470172344413515748885477384972008128706761571610718421081365633195037031
1
2
2:
4.244131815783875620503567023267049654252257219745505299937795841570581270246040935736368073766748959
1
2
3:
11.14084601643267350382186343607600534241217520183195141233671408412277583439585745630796619363771602
1
2
4:
32.08563652732609969100696669589889538614706458127602006752973656227359440306006947416694263767662213
1
3
0:
1.909859317102744029226605160470172344413515748885477384972008128706761571610718421081365633195037031
1
3
1:
3.242277876554808686204142822711284444939480789511889563059489020613539870791534118118696765619898804
1
3
2:
9.549296585513720146133025802350861722067578744427386924860040643533807858053592105406828165975185157
1
3
3:
36.31351221741385728548639961436638578332218484253316310626627703087164655286518212292940377494286660
1
3
4:
160.4281826366304984550348334794944769307353229063801003376486828113679720153003473708347131883831106
1
4
0:
3.242277876554808686204142822711284444939480789511889563059489020613539870791534118118696765619898804
1
4
1:
6.079271018540266286632767792583658334261526480334792930736541913650387257734126471472556435537310257
1
4
2:
22.69594513588366080342899975897899111457636552658322694141642314429477909554073882683087735933929163
1
4
3:
119.1537119633892192180022487346397033515259190145619414424362215075475902515888788408621061365312810
1
4
4:
770.3652234694225438421043346762011841176206355880249601829345912977770733000685064650023515112879557
1
5
0:
6.079271018540266286632767792583658334261526480334792930736541913650387257734126471472556435537310257
1
5
1:
12.38458922234860384681806823240845641973632108063158146607938133078427972908626798267855928687315969
1
5
2:
56.73986283970915200857249939744747778644091381645806735354105786073694773885184706707719339834822906
1
5
3:
396.3068551151553230981781834370706054315622745802106069145402025850969513307605754457138971799411102
1
5
4:
3611.086985012918174259864068794693050551346729318867000857505896708330031094071124054698522709162293
1
6
0:
12.38458922234860384681806823240845641973632108063158146607938133078427972908626798267855928687315969
1
6
1:
27.09128892388757091491452425839349841817320236388158445704864666109061190737621121210934844003503683
1
6
2:
148.6150706681832461618168187889014770368358529675789775929525759694113567490352157921427114424779163
1
6
3:
1341.018801732434760288268950790478171699573517012138430623908009723985289415122454999412747781734323
1
6
4:
16697.96472578944615803841828249585881278162833700012369669388585713457601187374246007432036421555446
1
7
0:
27.09128892388757091491452425839349841817320236388158445704864666109061190737621121210934844003503683
1
7
1:
63.07419497278055540215470039317725680739001458875348614042440993980618347859099801030071457502647740
1
7
2:
406.3693338583135637237178638759024762725980354582237668557296999163591786106431681816402266005255525
1
7
3:
4625.440964670574062824678028832998832541934403175255650297790062252453455096673187422052402168608343
1
7
4:
76706.27545909526828848898398521535142121560517309431823168753815621195855454500442596640917311520328
1
8
0:
63.07419497278055540215470039317725680739001458875348614042440993980618347859099801030071457502647740
1
8
1:
155.2216516908271480599900829988346554244363640270105322987006963362417796543450341659744147744792217
1
8
2:
1156.360241167643515706169507208249708135483600793813912574447515563113363774168296855513100542152086
1
8
3:
16277.57720731140692655762670381112753217589337429917115372374635579388795975231591620518362935038772
1
8
4:
351764.7853631971574778167640927495612148141113614781922051469342342990852601019959034470851849226645
1
9
0:
155.2216516908271480599900829988346554244363640270105322987006963362417796543450341659744147744792217
1
9
1:
401.5427964584000068765593136315403728997332080619153118236818872136063507926706043408187653735533007
1
9
2:
3414.876337198197257319781825974362419337600008594231710571415319397319152395590751651437125038542878
1
9
3:
58464.63116434304100122703606475227829420115509381486940152808277830108467541283999202321223838936058
1
9
4:
1615919.482762186942163720760051068296830552324066790445442393729138811422913593543681460047568238490
1
10
0:
401.5427964584000068765593136315403728997332080619153118236818872136063507926706043408187653735533007
1
10
1:
1086.988898225278143615060954479130775076230910886356683960201358746207816794221596906982048452626708
1
10
2:
10440.11270791840017879054215442004969539306340960979810741572906755376512060943571286128789971238582
1
10
3:
214317.9777667506739827695181914686178191968612630933261874863678994606412112606915234932938865762327
1
10
4:
7454240.473453737727656447098255915482510647274461395848694830554233388296115137098982959560394643473
2
1
0:
1
2
1
1:
2
2
1
2:
6
2
1
3:
20
2
1
4:
70
2
2
0:
2
2
2
1:
6
2
2
2:
40
2
2
3:
350
2
2
4:
3528
2
3
0:
6
2
3
1:
24
2
3
2:
300
2
3
3:
5880
2
3
4:
148176
2
4
0:
24
2
4
1:
120
2
4
2:
2520
2
4
3:
98784
2
4
4:
5588352
2
5
0:
120
2
5
1:
720
2
5
2:
23520
2
5
3:
1693440
2
5
4:
197588160
2
6
0:
720
2
6
1:
5040
2
6
2:
241920
2
6
3:
29937600
2
6
4:
6727406400
2
7
0:
5040
2
7
1:
40320
2
7
2:
2721600
2
7
3:
548856000
2
7
4:
224471126880
2
8
0:
40320
2
8
1:
362880
2
8
2:
33264000
2
8
3:
10464854400
2
8
4:
7427953653120
2
9
0:
362880
2
9
1:
3628800
2
9
2:
439084800
2
9
3:
207774927360
2
9
4:
245797739066880
2
10
0:
3628800
2
10
1:
39916800
2
10
2:
6227020800
2
10
3:
4297163270400
2
10
4:
8181798866841600
4
1
0:
1
4
1
1:
6
4
1
2:
70
4
1
3:
924
4
1
4:
12870
4
2
0:
6
4
2
1:
90
4
2
2:
5880
4
2
3:
594594
4
2
4:
75088728
4
3
0:
90
4
3
1:
2520
4
3
2:
623700
4
3
3:
340107768
4
3
4:
286634153520
4
4
0:
2520
4
4
1:
113400
4
4
2:
83243160
4
4
3:
191326078944
4
4
4:
854610753110400
4
5
0:
113400
4
5
1:
7484400
4
5
2:
13772959200
4
5
3:
111125856441600
4
5
4:
2197697290912094400
4
6
0:
7484400
4
6
1:
681080400
4
6
2:
2778808032000
4
6
3:
68335990604481600
4
6
4:
5178593567360018520000
4
7
0:
681080400
4
7
1:
81729648000
4
7
2:
673166245752000
4
7
3:
45087014663729438400
4
7
4:
11627377494092525545380000
4
8
0:
81729648000
4
8
1:
12504636144000
4
8
2:
193106317163760000
4
8
3:
32137409863871295984000
4
8
4:
25538669075752507769094000000
4
9
0:
12504636144000
4
9
1:
2375880867360000
4
9
2:
64798349247856224000
4
9
3:
24827302376175404064960000
4
9
4:
55881710617742376215448600000000
4
10
0:
2375880867360000
4
10
1:
548828480360160000
4
10
2:
25161956396245468800000
4
10
3:
20810038775287338436860000000
4
10
4:
123393096727843916328663147840000000
Definition
The symmetric Morris integral [1] [4] is $Z_{\beta,n}(k)=\frac{1}{(2\pi)^n}\int_{[-\pi,\pi]^n}\prod_{i=1}^n|1+e^{i\theta_i}|^{\beta k}\prod_{i<j}|e^{i\theta_i}-e^{i\theta_j}|^\beta\,d\theta_1\cdots d\theta_n$, with labelled angles and positive real absolute values.
Parameters
$\beta$
—   Dyson exponent ($\beta\in\{1,2,4\}$)
$n$
—   number of remaining particles ($n\geq1$)
$k$
—   number of coalescing particles ($k\geq0$)
Formulas
(1)
With $c=\beta/2$, $Z_{\beta,n}(k)=\prod_{j=0}^{n-1}\frac{\Gamma(1+2kc+jc)\Gamma(1+(j+1)c)}{\Gamma(1+kc+jc)^2\Gamma(1+c)}$ [1], equation (1.18), and [2], equation (6). Here $\Gamma$ is the Gamma function.
(2)
$Z_{\beta,n}(0)=\Gamma(1+n\beta/2)/\Gamma(1+\beta/2)^n$. In particular, $Z_{2,n}(0)=n!$ [2].
(3)
$Z_{\beta,1}(k)=\Gamma(1+\beta k)/\Gamma(1+\beta k/2)^2$.
(4)
$Z_{\beta,n}(1)=Z_{\beta,n+1}(0)$ by rotational invariance. In particular, $Z_{2,n}(1)=(n+1)!$.
(5)
$Z_{2,n}(k)=n!\det[\binom{2k}{k+i-j}]_{i,j=0}^{n-1}$, where a binomial coefficient is zero if its lower index lies outside $\{0,\ldots,2k\}$.
(6)
$\frac{Z_{2,n}(k)}{n!}=\frac{G(n+1)G(n+2k+1)G(k+1)^2}{G(n+k+1)^2G(2k+1)}$, where $G$ is the Barnes $G$-function.
Comments
(7)
The domain is the full Cartesian product, with no division by $n!$. A simultaneous half-turn changes $|1+e^{i\theta}|$ to $|1-e^{i\theta}|$ without changing the integral. For $c=\beta/2$, this is $M_n(kc,kc,c)$ in [1]; the equal Morris exponents remove the complex phase.
(8)
The exponents $\beta=1,2,4$ describe the circular orthogonal, unitary and symplectic ensembles, respectively [2]. In the symplectic case, $n$ counts each doubly represented eigenvalue once. The circular orthogonal and symplectic ensembles consist of symmetric and selfdual unitary matrices, respectively.
(9)
When $k$ particles of an ensemble of $n+k$ particles coalesce, the remaining $n$ angles acquire the factor $|1+e^{i\theta}|^{\beta k}$. Thus this integral enters the local $k$-point correlation coefficient [1], equation (4.12). For $\beta=2$, $Z_{2,n}(k)/n!$ is $\mathbb E|\det(I+U)|^{2k}$ for a Haar-distributed $n\times n$ unitary matrix $U$.
(10)
Integer and half-integer Gamma reduction expresses every selected value as a positive rational times an integer power of $\pi$. The rational values are stored exactly.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from math import factorial
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField


def morris(beta, n, k, digits=100):
    if beta not in (1, 2, 4) or n < 1 or k < 0:
        raise ValueError('Require beta in {1,2,4}, n >= 1, k >= 0')
    q, e = QQ(1), 0
    for j in range(n):
        factors = ((2+2*k*beta+j*beta, 1), (2+(j+1)*beta, 1),
                   (2+k*beta+j*beta, -2), (2+beta, -1))
        for a, s in factors:
            if a % 2 == 0:
                q *= QQ(factorial(a//2-1))**s
            else:
                m = (a-1)//2
                q *= (QQ(factorial(2*m))/QQ(4**m*factorial(m)))**s
                e += s
    if e == 0:
        return q
    R = RealBallField(numberdb.bits(digits, losing=96))
    return R(q)*R.pi()**(e//2)


if __name__ == '__main__':
    print(morris(1, 3, 2))
References
[1]
P. J. Forrester and S. O. Warnaar, The importance of the Selberg integral, Bulletin of the American Mathematical Society 45 (2008), 489–534, equations (1.17), (1.18) and (4.12). (arXiv) (doi)
[2]
S. Nishigaki, Eigenphase distributions of unimodular circular ensembles, Progress of Theoretical and Experimental Physics (2024), 021A02, equations (2) and (6) in the published version. (arXiv) (doi)
[3]
P. J. Forrester, Meet Andréief, Bordeaux 1886, and Andreev, Kharkov 1882–83, Random Matrices: Theory and Applications 8 (2019), 1930001, equations (1.7) and (3.1). (arXiv)
Links
Similar tables
Random-matrix moment factors —   asymptotic moment factors; the unitary case is related to the finite-size characteristic-polynomial moments represented here
Values of the Barnes $G$-function —   the unitary Morris integrals have the Barnes quotient in (6)
Values of the Gamma function —   the factors of the Morris product in (1)
Selberg integrals for classical Jacobi ensembles —   normalization integrals for the corresponding classical ensembles on an interval
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds $\beta=1,2,4$, $1\leq n\leq10$ and $0\leq k\leq4$. These noncanonical finite bounds cover all three classical circular ensembles, the uncharged Dyson case and the first four coalescing-particle counts across small finite systems before extending dimension or charge order. The attached argument-selection.md explains each parameter and the coverage choices)
How they were obtained:

The Gamma product is reduced exactly to a rational times an integer power of $\pi$. The 103 integer values are stored exactly. The remaining 47 values have 100 significant digits certified by 429-bit real ball arithmetic, with at least 424 retained relative bits. Independent 640-bit enclosures lie inside every serialized decimal interval.

more

All 150 entries agree exactly, as rational coefficients and powers of $\pi$, with a separate evaluation of the defining angular integral. This uses a Fourier determinant for $\beta=2$, a skew Fourier moment Pfaffian for $\beta=1$ and a confluent-Vandermonde Pfaffian for $\beta=4$. The integration identities are described in [3]. Known controls, the single-charge identity, and the unitary Barnes quotient were checked across the selected range. The attached checks.py and independent-checks.md reproduce and explain these checks; numerical-evidence.json records the measurements.



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