Values of the Tracy-Widom distribution functions $F_\beta(s)$

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Numbers

$\beta$

$s$

$F_\beta(s)$

GOE

-6:

2.707319325304773051449435716862581683957957840252003532432663956134740552224397868002964919528702023e-6

GOE

-23/4:

9.889863721678499380976415620306923132733617172329244621585586007431365128730245419723374644050678483e-6

GOE

-17/3:

1.491260126890795388801436151187762978778146925095081883198268625851332733016362254082505653906155895e-5

GOE

-11/2:

3.287508834018235604845965188023904911139458106506773281023163996670358398716952213853819767987708062e-5

GOE

-16/3:

6.961542203041748134701249709030756921985175170155622452833659613582519787722197369541053614839033957e-5

GOE

-21/4:

9.982163915922721305252441351015334153121216962033240879230434047776268215473189233752695405066991884e-5

GOE

-5:

2.779177549167903697987531983234842177698112743051314444115171102393151611029830927315889095169258191e-4

GOE

-19/4:

7.121827063817855366107102750361355553466108222592056797103356536518183086112333512552396011193719588e-4

GOE

-14/3:

9.574089257371530815651949966620564915426864090710453834072886207603841001860813101128329809392407111e-4

GOE

-9/2:

1.686151036139058862357567436700586919861362932360083802021649546576877787274575362384112741110341921e-3

GOE

-13/3:

2.871772518912001656463687673328613750544187772019804863132483822640254277716841889900807889480323754e-3

GOE

-17/4:

3.702321149260033731003481692897669917999485291317151464411406447144348039200308184060715638667351978e-3

GOE

-4:

7.567678598796400521936720529830926905217137795138816216050562148909213013944680457872769070276561540e-3

GOE

-15/4:

1.445425915053369167738597558103876299988394682536684880555620223042478347881347763683524949664147603e-2

GOE

-11/3:

1.767703699378237896794308728697866627149268267424082830633369414069447446005737588207637806550592819e-2

GOE

-7/2:

2.589414699301809795936698519781740747730253310024298529834170299449269269564248761777848657442675197e-2

GOE

-10/3:

3.692724405550717936067343038521723987680714587215095573057797050857372181317086081003951778402779979e-2

GOE

-13/4:

4.367185314659098600136396445687468780427393526019648780898490622263853727123084955209753754256865477e-2

GOE

-3:

6.960011886736988843622020111634477759228863522714858364133055203404848610380447980865729905432514124e-2

GOE

-11/4:

1.052039165691765418482755639465132772041533245087911370203908180447962602468646452064837124394954790e-1

GOE

-8/3:

1.194007553007836133858361714122991787011355025174252164856390485934718058655983795550235641224672973e-1

GOE

-5/2:

1.513777291781468037912019288384170055008556011935809521545235573925539893997226655968227404673080593e-1

GOE

-7/3:

1.880670534031887685822611030417500976415173256352896063720744092412016870397886539375374372754229013e-1

GOE

-9/4:

2.081047248174981702323722140266253341258089985118700019749687161619875179363082131691116466459721310e-1

GOE

-2:

2.743201979092178576716310476134368991287376315217689848988990531876725651788891210529907701899301278e-1

GOE

-7/4:

3.479654869953909492687261813798024763583104279285740044249931459723878390372837356377660065086282261e-1

GOE

-5/3:

3.736936528369147635053272580886156786656517467537199236089293347015166778383595020732973566841681260e-1

GOE

-3/2:

4.262258019375783973214940777291329026783851085232777619343328157388764794176287858910599222135842491e-1

GOE

-4/3:

4.793840848857157004666672558434698857715718745899960013052323778246050505155012285522709771350239251e-1

GOE

-5/4:

5.058960418367686110232115310887238288038411635166324608240566960981006659609415601511323931101725280e-1

GOE

-1:

5.837898955197322834588884463779700544033181994487244855028476109743115459570927846620792992699727322e-1

GOE

-3/4:

6.571064472482305342367670262324041185667041908056603379737493808994372902180667674345180255155380121e-1

GOE

-2/3:

6.801339330405905961905836794703772768685629061155547221459360827569481585840545362241430638777433900e-1

GOE

-1/2:

7.236910515483814718276021013676455784733881581216204656416148606543978818788910463011983314581726431e-1

GOE

-1/3:

7.636259680545654580490796293082610771072462917654476706253760206498790423757761827604774971295505159e-1

GOE

-1/4:

7.821619353991811567907830585591787031115726334616979444626575597671829874111467601555507887993261660e-1

GOE

8.319080662029519274622193818646768069313276435170821585562849746628248999380788480503853215621905935e-1

GOE

1/4:

8.729881052199939597539114612068769625466099215090775011969315766134992960582643826117633698903601047e-1

GOE

1/3:

8.848427458473229301843615221584581747742487944453293803089813246738560807931159005572345127680758118e-1

GOE

1/2:

9.059711835032338293324056764803246797144837411613223896274387965500262893373476517808881475875686167e-1

GOE

2/3:

9.238984408069079120461276266406397433449400081113743893752045348696438104255085501003820515034076451e-1

GOE

3/4:

9.317591707370399502715976651845551720620210247559309414055541897909269398216528297128399127955185030e-1

GOE

9.514212369115507347956186295073144506817551548015570249475249643105039934533743064848516364248811055e-1

GOE

5/4:

9.660595517028148370777456384905653846065871143897081514027922503325694955943551362307762086627626170e-1

GOE

4/3:

9.700053752982183318699761797602698096618176614304214001965585401301743432343575413617329492268525367e-1

GOE

3/2:

9.767136486166604462194412619958546844999173819556992588054987304509537079353276266246245738142760118e-1

GOE

5/3:

9.820626412987260107603707829717724519888187242744332259780320606057168038211769079624983960500229482e-1

GOE

7/4:

9.843024226914116341725534054135449264439823321382404366300051495732706053456692346247590556480692270e-1

GOE

9.895975710848269920731970333538613391783318762069888507735110707563887001248536019976665569989155812e-1

GOE

9/4:

9.932201562899101538592600176212543739161599714366496034091399338913002515509631733837928431079138343e-1

GOE

7/3:

9.941429483195429837848510350982820169806829310331126519652378794936692042190220378977736972783042978e-1

GOE

5/2:

9.956520227045179128234326563854759163646437960865393103105395030318680410976678349259691054268705040e-1

GOE

8/3:

9.967947605840982827407717208587310828941058344514196287271290371812032342149223134790927369169809930e-1

GOE

11/4:

9.972551060691690646897786831616478057680526678472077926825823760634569601108664869872386823443846092e-1

GOE

9.982934803498805951655028956206699745634293293416640177884045577459100350646027024921550361009364950e-1

GUE

-6:

1.062254674124451068774189210427600096597364535180305390746953229529986027505479307902181557277546096e-8

GUE

-23/4:

9.237738904321291746112679570098444343369870378600558234973119438741162222827716207446955238967003374e-8

GUE

-17/3:

1.824701449075159346752010699889436086286055630723565451903159256045781906419082654542175746916784632e-7

GUE

-11/2:

6.713838789865410167674493916412417817737075455005658174976848233127922634513016497128519800710851266e-7

GUE

-16/3:

2.288896553286656947388668525188167494943822864005512587698515540348506274123952522453992522135030067e-6

GUE

-21/4:

4.110054507736757339849374053921897982121833699889335367995843304356860413867639862906758815500340344e-6

GUE

-5:

2.135996984741115769872299325529972272179406768833766397917334137933444027612896575419454493285225740e-5

GUE

-19/4:

9.498107872155818299416144937615361673520181087033991048102342809362202875930535900082167857856387006e-5

GUE

-14/3:

1.510742187491057670429128530545641136843227339662795633934321750569264833556982137806509169441075094e-4

GUE

-9/2:

3.642223896787062398689015800150317322943599732241511478055935792067708134882174119967782556574777188e-4

GUE

-13/3:

8.250473837316331135362759508688492256831557302400394180558870408082542082115456260086533170251676255e-4

GUE

-17/4:

1.213954232299596718521016327555112449782342871559920436997464277764197255494411551895006593829477714e-3

GUE

-4:

3.544553595509200296340396504060891991200866923264779150648096840293425481832624727013368248555345297e-3

GUE

-15/4:

9.138343239873257906516985711554948087385570776619467273090540419477298105287075617930423835265216307e-3

GUE

-11/3:

1.220535097858596849653068900480398015482027873836025616000365434174942768728830932985031146893497194e-2

GUE

-7/2:

2.096769149276654325127495407445036960104321742776897373360640794160049644568091035325859035980251487e-2

GUE

-10/3:

3.432263767539289819778453892800561191785303569789644409660969244475781137385146877499800568756608416e-2

GUE

-13/4:

4.315698907872655444705479321220686464996327677271386382327904391647404303806448294470205288317212641e-2

GUE

-3:

8.031955293933454808137189279880225687583155576355300454260101116103537483120291507629512675541119881e-2

GUE

-11/4:

1.362473449792453178325756755761787878126240770063963067071825178880798733332986606609214436908614547e-1

GUE

-8/3:

1.593986041907293355157640798915415647668371891512520555577202188527970268019724411471442916307493727e-1

GUE

-5/2:

2.123514258195901062911634358606339567305458781991209084719018017154721909092563081330079088036135414e-1

GUE

-7/3:

2.734228967040098720124124518886113569235105127423954384423064629664882241583059389951314148926783647e-1

GUE

-9/4:

3.065513740959214297153060926289352209184692197655970462118504891382915455014720398638550946813258077e-1

GUE

-2:

4.132241425051225546880807613717181961899606376580800676674004880612281481487773868849523245320794261e-1

GUE

-7/4:

5.243451346147656733983908868896848200373085864818347510216959489633440828168374418657682768152159612e-1

GUE

-5/3:

5.608748517751349299376838262689267340785296832573763746231927683144652813997854433370269322723333832e-1

GUE

-3/2:

6.313808764207260466233171117108128061255311226927827737981206162533512837095623047417552944489836063e-1

GUE

-4/3:

6.968368081054779071999228256186260548720898279546931203827285014250056608220877323531418133415081577e-1

GUE

-5/4:

7.271806641670718180237423159040392124154405027901758339120381861882639422140209058043728896223286460e-1

GUE

-1:

8.072142419992852924789582879966889285812843902213863971175212510544940117741191065353594017014944068e-1

GUE

-3/4:

8.698862677504402838563167088346774455130410372296403402855702996227644405162681271086904336034344729e-1

GUE

-2/3:

8.870141411661208895942650442894634268015074426373076743863839558025672901805768639179684187448840452e-1

GUE

-1/2:

9.160651890092871965012354554438064979106252360535170588663395855325296121212078069659021280439248901e-1

GUE

-1/3:

9.388595253613166092530726962585442085257535680461143245056834014283562144341108756829811508574582748e-1

GUE

-1/4:

9.481953621617547605460593890552587795775651805257090535654214665234718467653186202434034002930305171e-1

GUE

9.693728283552626683498782494508137827306020107664514853125418085926383608838626593913161183187484530e-1

GUE

1/4:

9.826342074851229649052918244646654626983096639657488907672955354535064868035770037050727809267258613e-1

GUE

1/3:

9.857564432201757800130259108806402128629088499098546042753900985174661420038301118208837492894213401e-1

GUE

1/2:

9.905446073837164221162418432591923570577929786768908752461280322550695738849304273715693542021142085e-1

GUE

2/3:

9.938317989325114511659216999199479816845897997030338977760079631916927732617219265130467517345810532e-1

GUE

3/4:

9.950500145402366994468508403845407977834109038199366133454587402834737181663718393400415915049337147e-1

GUE

9.975054381493892493791531188651485010146251062622661558556224108980399833820920624346315156220454011e-1

GUE

5/4:

9.987884096721135958611399145531313598687962965092396747258530177361014426632659393656116510283081801e-1

GUE

4/3:

9.990552182637653326251650466010415518118256327584250310995419505264140158327965244946913253397657500e-1

GUE

3/2:

9.994322343111556701096197865896634495197404204339079409774857967700311338534074969146831613806850459e-1

GUE

5/3:

9.996640309950743811891082236806776676459016550476732297423780045479025684918278192458115319816247309e-1

GUE

7/4:

9.997430259757032668778241050988643365019126767361334251564186204461692336035542634745887918547227517e-1

GUE

9.998875536983091729250301355236344800715584271183514306970303348133784328549955599450541448905566837e-1

GUE

9/4:

9.999523852996880837695778375415552453896491217808632464253869011679891141044686685634321455875539772e-1

GUE

7/3:

9.999644992816280586300230040435154348365555914331478455688662222817910982843647515048247569084686587e-1

GUE

5/2:

9.999804720350555626267688982241826531765733489217920361616937349352908325687562436019766229976634578e-1

GUE

8/3:

9.999894058208518312933900962566612905427337158658679623945088202873935123478297549193292507827805948e-1

GUE

11/4:

9.999922366257760937865373438669161835656191775715033433290951405473086190903950783758573767924649663e-1

GUE

9.999970059566076483077760218866718349160270750592038228183225684283664978334650151185273643641933165e-1

GSE

-6:

1.451774540059169155423889819981252882164734957744257794650687559374561798293029560211527582974899098e-9

GSE

-23/4:

2.146616316375743324971757548249709728474073670979117604169730473746098186911787797491862177360621139e-8

GSE

-17/3:

4.985416445267526843564809801195036976216528917945019334883609345396885440513335318733654438770497202e-8

GSE

-11/2:

2.481261946380455144752892360090282881334052465448006349525686285220907624164087149488269067699780604e-7

GSE

-16/3:

1.112463808097253440783599973427826603415871236730815345071746771860785232085568746019586143793541832e-6

GSE

-21/4:

2.267440619827923058221511227065361651612844218769634545357720212721651324469454697591962943163075147e-6

GSE

-5:

1.656666130644060659963550044277662940070816719490813062289849337401023493235136233383107804239942769e-5

GSE

-19/4:

9.787494355097973591332334676395512905893731456274955195615629575805738973764095740644002845319545417e-5

GSE

-14/3:

1.691077504807961264923832533016620403440595105040286318793875229282130570951467402067097070394068321e-4

GSE

-9/2:

4.728922519136360382457489872373517662028506384020777614512109675995714238848128014174648355916810253e-4

GSE

-13/3:

1.215442477082758257582356476633631101028919523342940188358265975585586295919152362881733667727652235e-3

GSE

-17/4:

1.889925432476433609957464828555732190952668464804436628412644136991447788831091037947265565133608580e-3

GSE

-4:

6.319555940356641332663593738994786343297052350970576317524728532824209339086452848020759683183777992e-3

GSE

-15/4:

1.788540180655447443762886084911197544056417600706019063849812492097300910266445389899826129550839758e-2

GSE

-11/3:

2.442765581162364501367999613940346876717824575391598829637533212263566660954946830681526251482770337e-2

GSE

-7/2:

4.334648610424299449436722989035508281905304460627454993811622682540703414698181991628096140100914504e-2

GSE

-10/3:

7.217396614954808405372459820201992306561777380775192890621439220539208551114085310408108100800694167e-2

GSE

-13/4:

9.103583503782253994909757333295529191547229709804968496859585677755370755140064543373719139894173934e-2

GSE

-3:

1.677080381980468483385360683688484458756339232677074499880961008900712060149049105627790382397266783e-1

GSE

-11/4:

2.744187797215071237359678276697115645198476189839491670527586139343701065874324086177140463842008822e-1

GSE

-8/3:

3.156649259352803269170461737938082099817732179212877362969679122495238709251303407328798050971950569e-1

GSE

-5/2:

4.040326662225079113728487104388992061167584944521947562953056675135971442298692931026137892488423321e-1

GSE

-7/3:

4.963776527515610988263593139993782093498187212991449275189593954650566896821401685334652059646459070e-1

GSE

-9/4:

5.424883120072203493087525124630307249500404522936610014879229353422713629045512685396003200341736024e-1

GSE

-2:

6.735086715586111954664783239698542243450574888125471474399130869695665405127347247798990016995146102e-1

GSE

-7/4:

7.840923695345718464900325700188935841354731292830666486553526193155324888000065430771969333011636230e-1

GSE

-5/3:

8.151024109686439471527030725318110846335487705100290275328644353194688734718272538697309920750895658e-1

GSE

-3/2:

8.678684920408649303506810427266422026482231154166642695461769178203622501075817280013363628480122465e-1

GSE

-4/3:

9.087765975182180543534674508812291744200449192773374018801033797627243991661757221740053910829161543e-1

GSE

-5/4:

9.251713056250543587951289745141841782346541587873551338289820401075270444958122479135892481560062822e-1

GSE

-1:

9.607515364557128674649898936065849023425411495126681045933027297321738934618419609647130859201939056e-1

GSE

-3/4:

9.809062859400812952377904814162481494321146366182333124466755459030127922513642157536332776845796490e-1

GSE

-2/3:

9.852277541405131954509469617713405682757376990883935391627220641447268075479094310410391389748431643e-1

GSE

-1/2:

9.913693457709921312816414549855008388903550764308870052989519754030548127691618081239873079063965287e-1

GSE

-1/3:

9.951153239587871375342546541490078250100728792343794213719677738651632993068402681163515670054016700e-1

GSE

-1/4:

9.963680741286732215041442100088730734322343994914903293905382223421019921788508013486313042265242924e-1

GSE

9.985741973581685402167048089588001920045534684113751159403404896516268942355433412683067653814745767e-1

GSE

1/4:

9.994767564945506283414697987675131878007317766651169164395003441369545126229801966896125452445051696e-1

GSE

1/3:

9.996308258196390601302653967338233363505640293472747207235700172866261259509188373674492127668466420e-1

GSE

1/2:

9.998201349867107468867505644995219815408124820207106804707517654877283985774375852023730934001581881e-1

GSE

2/3:

9.999148044933374994359252210162747019104980989715469735806785032068282119531007038334368413355456778e-1

GSE

3/4:

9.999419722443951720144763235064714762237327327418319377814869022787445774516973808093044389122887345e-1

GSE

9.999823971365632391541633288746011916797126120311396232861620730935687244556029713034729836140191540e-1

GSE

5/4:

9.999949700633055773418102638272510802810416720086295900548107411163049324062776228110689470189749629e-1

GSE

4/3:

9.999967297620680815497417391889312702537371100625819707949561036076563455855770785810981515027351875e-1

GSE

3/2:

9.999986438543666200789380794193518496857770776916412029739123516236130864147138425056310351811289990e-1

GSE

5/3:

9.999994515634530729534519122635017316875537887296238059510781513130711302007233686994394161662958105e-1

GSE

7/4:

9.999996544524854764982718978366734310850132530355681202749997998944741904212773975420956192599516512e-1

GSE

9.999999166662101327208571195761056338215695967564509892323062456346809213111760589688831116451016138e-1

GSE

9/4:

9.999999809513726633299428760087501510159343704151031886748821722723403941651374292333052474076064628e-1

GSE

7/3:

9.999999884882454474894868231165477982098995316290329949553548813509127566860453182085040204347919500e-1

GSE

5/2:

9.999999958674538336152698414244339915990170517918400243925341690466465701650437632738495417797486556e-1

GSE

8/3:

9.999999985497097118603331108606860037567653418757407848845614694079610940934463927756731146574854258e-1

GSE

11/4:

9.999999991480136378110678578230755351657719186162575853241180562992483052017196018941790582197912150e-1

GSE

9.999999998328804780003568407875292745376769407884345820823704710984784225072746219733942991292294488e-1

Definition

For $\beta\in\{1,2,4\}$, let $F_\beta$ be the Tracy-Widom distribution function [3] in the soft-edge scaling used by [1]. This table gives values of $F_\beta(s)$ at rational $s$.

Parameters

$\beta$

— Dyson index

$s$

— argument ($s\in\mathbb{Q}$)

Formulas

(1)

The Airy kernel is $K_{\operatorname{Ai}}(x,y)=\frac{\operatorname{Ai}(x)\operatorname{Ai}'(y)-\operatorname{Ai}'(x)\operatorname{Ai}(y)}{x-y}$.

(2)

$F_2(s)=\det(I-K_{\operatorname{Ai}}\!\upharpoonright_{L^2(s,\infty)})$ [1].

(3)

Let $T_t$ be the operator on $L^2(0,\infty)$ with kernel $T_t(x,y)=\operatorname{Ai}(x+y+t)$.

(4)

$F_1(s)=\det(I-T_s)$, $F_2(s)=\det(I-T_s)\det(I+T_s)$, and $F_4(s)=\frac12(\det(I-T_{\sqrt2s})+\det(I+T_{\sqrt2s}))$ [1].

(5)

Equivalently, if $q$ is the Hastings-McLeod solution of $q''(s)=sq(s)+2q(s)^3$ with $q(s)\sim\operatorname{Ai}(s)$ as $s\to+\infty$, then $F_2(s)=\exp\left(-\int_s^\infty (x-s)q(x)^2\,dx\right)$ [1].

(6)

With the same $q$, $F_1(s)=F_2(s)^{1/2}\exp(-\frac12\int_s^\infty q(x)\,dx)$ and $F_4(s)=F_2(\sqrt2s)^{1/2}\cosh(\frac12\int_{\sqrt2s}^\infty q(x)\,dx)$ [1].

Comments

(7)

For $\beta=4$ the table uses Bornemann's soft-edge Tracy-Widom scaling, the same convention used by the table of Tracy-Widom cumulants. If $G_4(u)$ denotes the convention without the $\sqrt2$ in the GSE argument, then $F_4(s)=G_4(\sqrt2s)$ and $G_4(u)=F_4(u/\sqrt2)$.

(8)

The selected arguments are integers, halves, thirds and quarters in $[-6,3]$, shared with the density table. This is a bounded-denominator selection of simple arguments, not a decimal interpolation mesh. The website table and its attached generator record the current selection.

(9)

The Airy function $\operatorname{Ai}$ appears both in the Fredholm determinant kernel and in the boundary condition for the Hastings-McLeod solution. It is tabulated in the table of Airy function values.

Programs

(P1)

Python

from generate import TracyWidomDistributionFunctions

generator = TracyWidomDistributionFunctions()
print(generator.value({"beta": "1", "s": "0"}, digits=100))

References

[1]

Folkmar Bornemann, On the numerical evaluation of distributions in random matrix theory: a review, Markov Processes and Related Fields 16 (2010), 803-866. (arXiv)

[2]

Craig A. Tracy and Harold Widom, Level-spacing distributions and the Airy kernel, Communications in Mathematical Physics 159 (1994), 151-174. (doi)

Links

[3]

Wikipedia: Tracy-Widom distribution

[4]

Wikipedia: Random matrix

[5]

DLMF 32.11: Painleve transcendents

Similar tables

Values of the Hastings-McLeod solution $q(s)$ of Painleve II — stores the Painleve-II function used in (5)

Cumulants $\kappa_n$ of the Tracy-Widom distributions — stores cumulants of the same soft-edge distribution functions

Skewness and excess kurtosis of the Tracy-Widom distributions — stores standardised cumulants of the same soft-edge distribution functions

Values of the Airy function of the first kind $\operatorname{Ai}(x)$ — gives the Airy function in the Fredholm determinant kernel and in the Hastings-McLeod boundary condition

Random-matrix factors $g_G(k)$ in the moments of characteristic polynomials of unitary, orthogonal and symplectic matrices — is another random-matrix table involving the unitary, orthogonal and symplectic symmetry types

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds $\beta=1,2,4$ at every rational $s=a/b$ in lowest terms with $-6\leq s\leq3$ and $1\leq b\leq4$ (165 entries). The selection includes integers, halves, thirds and quarters in the central region and adjoining tails, with the same arguments as the density table. It is not complete over all real or rational arguments and does not cover the far tails.)

How they were obtained:

All 165 entries are given to 100 significant decimal digits using the Painleve-II formulas (5) and (6). The attached generate.py reproduces the values using the numberdb package and python-flint. Two Taylor integrations vary working precision, order, maximum step and right boundary: (230 digits, 180 terms, 1/8, 40) and (270 digits, 230 terms, 1/10, 46). Every entry is agreement-checked; the worst observed relative difference was 1.37e-144. Airy tail integrals are included at the right boundary.

A separately written Fredholm-determinant computation checked 12 distinct ordinary, endpoint and disputed values under two quadrature settings, including the beta=4 sqrt(2) scaling. All checks round to the stored 100-digit values. Arb is used internally, but Taylor step midpoints are retained. Taylor truncation and nonlinear boundary errors are tested by convergence and independent determinants, not rigorously enclosed.

On 2026-09-26, F2(-11/2), F2(-13/3) and F4(-11/2) were explicitly corrected after independent checking and user approval: the old low-precision values lay outside their claimed rounding intervals. The other 162 entries were precision refinements. No arguments or mathematical definitions were changed.