Mehta integrals for Gaussian ensembles

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Numbers

$\beta$

$n$

$Z_{\beta,n}$

2.506628274631000502415765284811045253006986740609938316629923576342293654607841974946595838378057266

7.089815403622064109192669933364580731190197824489548512855231159411645138364128725499802626954178662

26.65729762895019748209528594036416219168773013625414133851237364173860875859684346345196160858859142

150.7964473723100754462068823974161384414641311700050794067973404307751875017380319341456716164216193

1417.464895115017776986158139502048408808051684451886047243157695736621958788149005598353033681434914

24055.17694631297789163041368275337023269885736602361369445587953734450070535847743355660760950194069

791403.2002390489391734674101616105598769128351255080043218065220967575403052709561432833116510460169

53722230.67600959683403990141052586585973952861349307153769314332464094797739221797870725299112082351

7953454434.834633973519864452229800052351488543129097835527891405414586041056163938475198295416801804

10:

2699491951070.298975159450373939308457333537023168554306601526882747625028664192796137894780335571983

11:

2198095141979198.628684974316378187705063807711921493368755406070916566993828303698200254976678138248

12:

4476349384089691157.057630766642199709601823167466197488572591102632390824236008185843337157566930016

13:

23692003194135031736218.77529414531313074388707448587177605238391252791680963943199286372996503415546

14:

337735966548279094272152660.0614547466875994682734264570018399651679387640530467757614417724338992512

15:

13406529928176040201257377424063.83566489361279462313427372061651612953008332635632792779477899392052

16:

1528909921630212058336051921056795201.804222656352048549605560631064892596036514715770242666918004935

17:

515869550602246632056716842772145307593104.2398147152386090435118413935762624804463315301121962660644

18:

529478001069139379894204748254342010539364039138.4320006008710622404028842373671034163300961287459268

19:

1697186314713490736276750893814644937223380076372591061.221673932922867307158829586328412940851206204

20:

17419574702699749432466458845995280555057647267019101803344562.84020472300581947964326707565620059847

21:

586284546677046374714277872969096018765975859762494071834774317370964.3652222374006065806574055294398

22:

66192557094497148940012475308187664981618823294401677034480910300674119499872.66250571583860073215825

23:

25619927633512371574386990498104276980064569363396584613933209245504080991616360542402.62339713957457

24:

34710419005581755934885427752168398491531469227955729257218171164582511921064093020755132755139.94744

25:

1.679339910116998451375147878078281419114593929874633477077444714298185510595920321787954167392130748e104

26:

2.957766727401772572315607795446903598573498487523057160253325165822107016124461728452125816900054736e114

27:

1.931863515566695360822746827734059254887265055249179545369237403343669682864388783148410032361407859e125

28:

4.763539668962608196398695917830235310789448681769889276752612451948768335284910582399030355406696364e136

29:

4.511389349587320051379212984356871201834753057353702558299836069948673685858630571636747559696699356e148

30:

1.668610278810819161269289216126458524917696340031080035863628096805973242562143102286868012527953949e161

31:

2.449441865406270703526634561481102698701048201680924992383610394699569868095278455720535786730893407e174

32:

1.449545071729293045928533282195493916960653177467414992038947158728692807607796786009317823156866910e188

33:

3.510976235013273808199032635107279897621910418920633377306074774667998200625122094747983153754151438e202

34:

3.532168380686452479847891770465843282059409902353490761807227732370699870062361988528846534366785625e217

35:

1.497185503087276748551688313818293781567723308587378919528756585962709213510480151502886748036253072e233

36:

2.711200444571691744471101268046367011016299371181571574103040978800094554454910721367973662878126825e249

37:

2.126021099100927744773344864416871746682476706541901352479672765839394628755741985165588830303500046e266

38:

7.314879646252440312220189580496742709205957230740415561424132897772511121063125776627580072933447446e283

39:

1.118530264631841483395493109936531609137162931635827686337312743000350642244653921554010065175542711e302

40:

7.696926686130123258537886014868794538436744836469698690209896851344807912725920648049444026536966972e320

41:

2.412746895620406063724941058829800712431991528298956483921435531062681485410283565892292005984764387e340

42:

3.486588317319302233276039659835658243812484026466202696710903837625126816975425444677996414105742543e360

43:

2.349814320209968569156114586174076557612984562141079761683446652650090994319790464685347277665062820e381

44:

7.470421940025080219891590823894489163632246658273436641024195379387183323616716139163980173920147044e402

45:

1.132819003102184652668179551355490548584593746032591992519590848255761213826073024894576774184471842e425

46:

8.283234331647170451115700364562297153157843829550012499832075923416426370335297455945958061462971541e447

47:

2.951774683334160384125480371546240018924143719416050537383017210080762813770317494759936531559352384e471

48:

5.180048991972984954501112469093642245412995464211495974972171257493651107256782294724539396839999257e495

49:

4.522548235242911857672901977953538908334334051701212637116166663843294865205627164151247183765239967e520

50:

1.984147160637731007304040640042906327799336240084749040394047023509503375648770617963170676905834695e546

2.506628274631000502415765284811045253006986740609938316629923576342293654607841974946595838378057266

12.56637061435917295385057353311801153678867759750042328389977836923126562514483599451213930136846827

188.9953193486690369314877519336064545077402245935848062990876927648829278384198674131137378241913219

11369.78427005494112889733363185732610788138171714139080162817848140549163542692443993804296108377217

3419978.727332940786463568152802265580586177991166623071966305870137119386092849530947595135007476714

6172283071.129891883123297944621579496768098403541058494199498964357082288082261251791837881075713102

77976961096207.08735226564271569183261661647425064713824381539298391759317245179907495648299480517862

7880917179887306188.499904750191575776661635050691081009312976921303566709151897154205108251240999061

7168523785846478767440475.512046334495219576977295643336903776810668530696767477546304837296590850634

10:

65205270015261796752312607663946.18499431982016696590318165957380415703040095634719971979907131454305

11:

6524216283935030587089613777470453534010.091124496197842278238055890074489361998371711073297785660833

12:

7833489184466272680676924967432932273368170648672.775111920554318931058983748088563116020306700013362

13:

122271572817909177159223565399426263893748327279944880812110.5569582973496803952339280203458488530752

14:

26719220559615096599500129049281398913754044645439673851117309370859815.69745476649337007794053424933

15:

87581691826670075381629206759766078908115977964573499400654625863716062670527198868.87281212435432539

16:

4593279344272791728691386343957983825839164408836136020417174212203244694297790980023623917033575.307

17:

4.095258378848439176541954708005239947591832247778486734766293770714645622690898416408764059147438515e111

18:

6.572222562250978821339338765285578734062551933274260117022535897427724165228537104059290689161802933e127

19:

2.003995847945076211447864782823910957302559875289230431205673288750485302168217381687380838722406089e145

20:

1.222113012924391824273580599267686870203987350177430015083851571777809630292494431220370207795684986e164

21:

1.565111253883268876719836220473022315600308149054968724801980412285906010585241234353638502258408016e184

22:

4.409625840228191436334201969546789634727100926855961901163939250681957564691880901649161791327413654e205

23:

2.857499107868175383743972407696301463930125777211133754600088696767672549444771997320562506740341440e228

24:

4.444078358019532808660267195986187602244354981201015457562986049946615139454036624971261412531163404e252

25:

1.727894892236017379788761331860526287059922893965932258977720043366868109031045075352834398638014420e278

26:

1.746732021138162954503950730648499067368288380743653154531071907317880310129189124319536681617418556e305

27:

4.767591172311988847987014646731542092182016277890694434132388396630984355789126507863721669699284235e333

28:

3.643592197971509961702452672488381961501501552838907776230905029679201018062351472462535982183504410e363

29:

8.075297254603878717952314537791175919846988197882104013515033461650954051899100558966909085466480820e394

30:

5.369186962238865401365616157702214260856136762706102814152469774238449570134316505707436956744457947e427

31:

1.106675332837129156367691064144156725117148932942463761470646564820703162892124020981911546397059669e462

32:

7.299311726254971513217074655331539250672641087384717125147717191756527994452147883814008840017108797e497

33:

1.588757202019602788093040737317687167130003341072106831094438662683466741933966823944488429297053091e535

34:

1.175742103029530605310815021449754952760493156483741868704406353728638502786953924394872564484214222e574

35:

3.045332048708701787283085598839025189873048942170964122540898018587212267054816454318977308448962638e614

36:

2.839616795790961291957537093311942237851111615354761601698624591316289504105633914852684636221015740e656

37:

9.796851918437506161846315741043295745097728147021588977733056773983017818308872023430100024229761276e699

38:

1.284390094767071433991583880104537330584281128749236803434760974511687796888590991730868951402507809e745

39:

6.567074733957591208316019255240924625311928109148255935744921884639540268552533433395415573348644089e791

40:

1.343095707047018336703388022899781018394788271688420883115117962502612294579822279171745966484011794e840

41:

1.126226702238880799245275858509173512609645150171035127599502888118655291201692277304861232168671863e890

42:

3.966376802754637342131229931868057362737631882842639629074723535019312135551153392313110002758708248e941

43:

6.006625763149482790333801558444311586500196432890764986292271492654148279851565957488825637173490659e994

44:

4.002394158499651991065910051234219218486959136767934128796797168095334220621778242012678319220971944e1049

45:

1.200111649076124702288046339975804763513668809062073145815112905217539845603022541583382359132235862e1106

46:

1.655317392734088784823880652392051350828494219280653376180343149435272966923641985713834075034376925e1164

47:

1.073096462495600832571446580730037691350801171169890040304581552525792621106354821685116282962010662e1224

48:

3.339161969662617052782888847619672701508614121816513551911389225473470478596867872663123364840781578e1285

49:

5.091342199114322890879273236978989101971814464404357124787728201450954842247532727124512015436851620e1348

50:

3.881477691706784484712299059725771928762267247970403853028124614752498867004641726690055900193300254e1413

2.506628274631000502415765284811045253006986740609938316629923576342293654607841974946595838378057266

75.39822368615503772310344119870806922073206558500253970339867021538759375086901596707283580821080963

68038.31496552085329533559069609832362278648085369053026767156939535785402183115226872094561670887587

3438222763.264614197378553690273655415023329831263556578412361172777020670553101950637264191431732705

15637127695239677.16265793510223996528692721459487525667247467673435750932359970279615731786465306569

9387585063542463989405724.477525222405390882659298070995554986576067723681828193038957726206665802418

1025704299267018003175015924982431153.695701620739286067745319802862410788034852981050567521890720966

26896867786472176828818174683942539879706036239295.67307022383763525154960887549921712507987720868786

215825455889600317545615364326932260722798776178679694584116584991.6605622562889947113134732359970544

10:

658092925774986091156007096855067689174948354749283932487887489713672538075725909190.6409910317470362

11:

9.270726165851173155543032346807582229307464906916463508138792816019339721301830562705673232820585776e104

12:

7.209071982444389075395344730498090014608463532583494914240119030952610507544343096432496187620441861e128

13:

3.643831847355671112225947661389279687116344908421447628582889452063689072802132038640299850958290246e155

14:

1.392384205135911829653010690852635801703869154480737525059593977465637061881980119490668136937814909e185

15:

4.628913886966125005170546673801289208904039876976311399573238161761151269706245138473989609461987434e217

16:

1.526549133806749576242700122729987290978053669936332550580477678121489841368115583423819134071462296e253

17:

5.648528568991848638080565508131124827092260452789984838634024512511431453358881410078603314927355558e291

18:

2.633482382128996286613410741384115003923131950538023830713461433580227516601493627091956743682438753e333

19:

1.726278356818082041110189135183097836098319516654088963175347043920869503706217256453275004840291761e378

20:

1.765289070201648524213044602682282514950007404934124577954195621814847192580007988134554755547553622e426

21:

3.108522291420057558595373408540546064728641893543985014679475108995422122907625627660280081238343640e477

22:

1.035650625770138175164395218886741602531189018931781574969039053805863781067800807442923131838923734e532

23:

7.142379190106962819221047988008608684317197121059105368241956051009173299212618098761848355414448536e589

24:

1.111249631233064706915884920323862196430250903326313206208481857646334850898750528205464244391927851e651

25:

4.235907237092471111744934048673849439310852132637553912944827420941822180081525623585467622932973227e715

26:

4.282079948927449783985575642040505147910055130779398136521807706534455493673084891325495580384195439e783

27:

1.238889955105761892899753611460217777996978735046152561204186484203214614158336251048386208023751979e855

28:

1.103980515225761077395331453663629370711059532329478064165251571664745637318886931617506589617759309e930

29:

3.252317486840765684873245061801777444556442780903275754678367962939520200804143752815895461709389123e1008

30:

3.391780368241998249307272403512681891353444314881808843722652297086847817284201773087418028479171222e1090

31:

1.337777953571114093498270998692028058378391131648356126672414640846643250163764146788333299817668052e1176

32:

2.127457389371498358081018577701882668080042768816262282995697863601576649607403055827364937912769369e1265

33:

1.451426333957363983446915802364142233712346236282285925352893373790287759336047547934094180162786275e1358

34:

4.511415651182662683696285341256276457948692015317175278974433765183461361702188748615005500065157251e1454

35:

6.772949166523256467864614581288222722368021791486729512584646004800260761800240978989521737829797450e1554

36:

5.197968405465908335373390310123345209403301378135982754136514203532613199502615220153190489334416467e1658

37:

2.154983923566185792401020102642768816882074212829035879550507231622288717973975085637688376482579516e1766

38:

5.092479501008246806094005290267583855280338464143061361669432524198742486287933900839973447805946744e1877

39:

7.227695909498539778876907166531592266978463088116282347087260931809599215484242773279951785755940281e1992

40:

6.483171844029679204375634383799004491113227730787981666708771078090678055421904798810653959309918556e2111

41:

3.862549565825646597825881848781297191312812269134264671906772994225536553728684110934461454698315825e2234

42:

1.604419664409348661733045960270604305725063586218997095745389589620640272810038155987083653042371333e2361

43:

4.871685724307336955998928991914523429717446433579479214258345925873207291896829680405913151747232150e2491

44:

1.132511123517566041955491680627824063231539941129100100414258853475888976211747919665326045612028261e2626

45:

2.108813654033022687785162517679175510878584100825162999155418886319116978301775794922849061748462758e2764

46:

3.287480254856642498382054195895957474700783666037479822662022106769506492056022791856475603420710615e2906

47:

4.480215491588642583047820658359183243732687988939156185956935796035395719142959721830385513440583646e3052

48:

5.568388041002818626287565129796378574846730484417716489720059095510340619364288362964210267365520004e3202

49:

6.578969768080848737251095854946555999673680231881702913708558842124410978060125275022683770628582697e3356

50:

7.695227857990952984051544793468567123325737828678173569997999720785772496755554498201968923341089273e3514

Definition

The Mehta integral [3] is $Z_{\beta,n}=\int_{\mathbb R^n}e^{-\frac12\sum_{i=1}^n x_i^2}\prod_{1\leq i<j\leq n}|x_i-x_j|^\beta\,dx_1\cdots dx_n$, with labelled coordinates and ordinary Lebesgue measure.

Parameters

$\beta$

— Dyson exponent ($\beta\in\{1,2,4\}$)

$n$

— number of eigenvalues ($n\geq1$)

Formulas

(1)

$Z_{\beta,n}=(2\pi)^{n/2}\prod_{j=1}^{n}\dfrac{\Gamma(1+jc)}{\Gamma(1+c)}$, where $c=\beta/2$ and $\Gamma$ is the Gamma function [3].

(2)

$Z_{\beta,n}=\sqrt{2\pi}\,\dfrac{\Gamma(1+nc)}{\Gamma(1+c)}Z_{\beta,n-1}$ for $n\geq2$, where $c=\beta/2$.

(3)

$Z_{\beta,1}=\sqrt{2\pi}$ for $\beta=1,2,4$.

(4)

$Z_{1,2}=4\sqrt{\pi}$, $Z_{2,2}=4\pi$ and $Z_{4,2}=24\pi$.

Comments

(5)

The exponents $\beta=1,2,4$ give the Gaussian orthogonal, unitary and symplectic ensembles, respectively, with real, complex and quaternion matrix symmetry [1]. The eigenvalues are scaled to the common weight $e^{-x^2/2}$. In the quaternion case, $n$ counts each eigenvalue once rather than twice in its complex representation.

(6)

The joint density of the labelled eigenvalues is the integrand divided by $Z_{\beta,n}$. The integral over one ordered chamber is $Z_{\beta,n}/n!$. Forrester and Warnaar use $F_n(\beta/2)=Z_{\beta,n}/(2\pi)^{n/2}$ [1]; this table stores the unscaled Lebesgue integral $Z_{\beta,n}$.

(7)

Integer and half-integer Gamma reduction writes each value as $q\,2^{s/2}\pi^{p/2}$, where $q$ is a positive rational number, $s$ is the parity of $n$, and $p=\lceil n/2\rceil$ for $\beta=1$ or $p=n$ for $\beta=2,4$. Thus $p>0$ throughout the positive-dimensional family, and none of these integrals is rational.

(8)

The one-variable weight $e^{-x^2/2}$ is the orthogonality weight of the probabilists' Hermite polynomials.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField


def mehta(beta, n, digits=100):
    if beta not in (1, 2, 4) or n < 1:
        raise ValueError('Require beta in {1,2,4} and n >= 1')
    R = RealBallField(numberdb.bits(digits, losing=96))
    c = QQ(beta) / QQ(2)
    z = (R(2) * R.pi()).sqrt() ** n
    for j in range(1, n + 1):
        z *= R(1 + j*c).gamma() / R(1 + c).gamma()
    return z


if __name__ == '__main__':
    for beta in (1, 2, 4):
        for n in range(1, 51):
            print(beta, n, mehta(beta, n))

References

[1]

Peter J. Forrester and S. Ole Warnaar, The importance of the Selberg integral, Bulletin of the American Mathematical Society 45 (2008), 489-534, Section 1 and equations (1.4)-(1.6). (arXiv) (doi)

[2]

Peter 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

[3]

DLMF 5.14.6: Mehta integral

Similar tables

Selberg integrals for classical Jacobi ensembles — normalization integrals for the corresponding classical ensembles on the unit interval

Laguerre–Selberg integrals for classical Wishart ensembles — normalization integrals with an exponential weight on the positive half-line

Morris integrals for circular ensembles with a coalescing charge — normalization integrals for the corresponding classical ensembles on the circle

Hermite polynomials in the probabilist convention — their orthogonality weight is $e^{-x^2/2}$

Values of the Gamma function — integer and half-integer Gamma values occur in (1)

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every $1\leq n\leq50$ for all three classical Dyson exponents $\beta=1,2,4$. The noncanonical dimension bound gives contiguous coverage of small and moderate finite matrices, including both parities, across all three symmetry classes before extending dimension further. The 150 values have 100 significant digits and occupy at most 106 characters each. The attached argument-selection.md explains the parameter families and the finite coverage choice)

How they were obtained:

The generator reduces the Gamma product exactly to a rational coefficient times powers of $\sqrt{2}$ and $\sqrt{\pi}$, then uses 429-bit real ball arithmetic. Every value retains at least 421 relative bits, and every serialized 100-significant-digit interval contains the resulting enclosure.

All 150 integrals are independently evaluated through exact rational Gaussian moments: a Hankel determinant for $\beta=2$, a confluent-Vandermonde Pfaffian for $\beta=4$, and a sign-kernel skew-moment Pfaffian with an odd-dimensional border for $\beta=1$. The integration identities are described in [2]. The resulting rational coefficients and radical exponents agree exactly with the generator. Independent 640-bit enclosures lie within every serialized interval. Known integral controls and the dimension recurrence are checked. The attached checks.py and independent-checks.md reproduce and explain the checks; numerical-evidence.json records the measured precision and sizes.