Superbosonization meets Free Probability

Transcription

1 Suprbosoato mts Fr Probablty M Zrbaur jot wor wth S Madt Eulr Symposum St Ptrsburg Ju Itroducto From momts to cumulats Larg- charactrstc fucto by fr probablty Suprbosoato Applcato to dsordrd scattrg

2 Itroducto

3 Itroducto: oto of fr probablty 'Fr probablty' troducd by D Voculscu 986 th study of vo uma algbras Gvs calculatoal schm by whch to hadl varat smbls of radom matrcs Larg- lmt of dsty of stats codd Voculscu R-trasform : Fr cumulats c R 0 + ar addtv udr addto of dpdt radom matrcs for c

4 Itroducto: fr probablty II Voculscu' s aalytcal approach : df R-trasform by vrtg a g avrag trac of rsolvt Combatoral dscrpto of fr cumulats trms of o-crossg parttos gv by R Spchr 994 Fr probablty thory has ot yt producd rsults for spctral corrlato fuctos th mcroscopc lmt

5 Itroducto: suprsymmtry mthod Mthod of commutg ad at-commutg varabls Wgr Eftov : rsults for corrlato fuctos g lvl statstcs of small mtallc gras localato thc dsordrd wrs scalg xpots at th Adrso trasto tc Tradtoal varat lmtd to Gaussa ubbard-stratoovch trasformato radom varabls Rct varat calld ' suprbosoato' much wdr class of dstrbutos allows to trat

6 Itroducto: suprsymmtry mthod frst stps Ω + Ω + Tr xp : Dt Dt Dt : Im Dt : sg d w w - Br w w μ μ fucto Charactrstc varabls commutg at ovr tgral Gaussa varabls commutg ovr Gaussa tgral spac vctor o rmta oprator lar

7 From momts to cumulats

8 From momts to cumulats I 0 0!! : Ω Ω d d c c m x d x d x m x grats th cumulats l Th logarthm charactrstc fucto ar gratdby th Momts Commutatv cas ω ω μ μ

9 From momts to cumulats II: combatoral dscrpto Momts ar xprssd trms of cumulats m d d Ω by summg ovr parttos p Π : whr ν p l 0 d d ω 0 p Π s th umbr of blocs of l l c ν l p l lν l p lgth l Exampl ν p 8 : ν p p { 36} { 8} { 45} { 7} ν p 3 ν p 0 4

10 From momts to cumulats III: R-trasform + + Ω Ω lm : Tr Tr R- c R R m g m m m d m d d trasform whr s vrtd by By dfto Lt dffrtato : ar gratd by Momts Charactrstc fucto : matrcs for Probablty masur a a μ μ μ

11 R-trasform xampls Luc Sommrs MZ: J Math Phys /3 / g R g R g R Exampl : smcrcl Wgr GUE Exampl :

12 From momts to cumulats IV: frss Rcall R c + R Spchr 994 : Combatoral dfto of fr cumulats Exampl c by 8 : p m p C { 58} { 34} { 67} whr th sum rus ovr o-crossg parttos p C l c ν l l p Fr cumulats add udr covoluto of masurs or addto of dpdt radom matrcs

13 Larg- charactrstc fucto by fr probablty

14 Cumulats o-commutatv cas ot : udr ω lm l Ω s addtv addto of dpdt radom matrcs Assum for g GL ad cumulats must b of th gral form [whr γ π s costat dμ Tr V o cojugacy classs] ω 0 d Th ω ω g π S γ π Graphcal mthods suggst larg- hypothss : j γ π c j j f π rrducbl cycl ad γ π l δ j l π l 0 ls g

15 urstcs from plaar graphs I Rcall Ω Tr V + Tr Prturbato thory for l Ω coctd graphs lads to topol xpaso : l Ω d χ 0 Ladg cotrbuto coms from summg all plaar graphs Eulr charactrstc χ χ ω χ Exampl : graph cotrbutg to Tr t ooft Wtt Gross 4 9 vrtcs: dgs: 4 facs: 9 4

16 urstcs from plaar graphs II χ χ 0 Tr 4 Tr 3 Tr

17 Larg- charactrstc fucto from fr probablty I Rcall th larg- scaro from plaar graphs : j j l Ω 0 c π [rr] l δ l j π l By Spchr' s combatoral dscrpto th umbrs c ar dtfd as th fr cumulats I fact tag th formula for th momt w hav m l Ω 0 p C l c ν l p l

18 Larg- charactrstc fucto from fr probablty II Summary ω : lm For ot : d d ω Π smbls w hav ω 0 ra- projctor c wth + l Ω 0 GL c + Π s th Tr + th drvatv of ω R -symmtry + for Π R-trasform :

19 o-prturbatv argumt: rducto to D tgral Lt Rcall : Π Ω Π ra-o projctor Π Tr Π TrV d Dagoal g Do tgral ovr Ω Π c whr dgr for ot : p p + x orthogoal polyomal of dag λ g U V x λ x V x g usg CIZ formula : x dx has ros [ ab] p

20 Asymptotcs for larg Saddl pot of x-tgral ls outsd of [ a b] Larg- asymptotcs of p x b a p l xy dν y wh x [ a b] : G x Itgral for scald logarthm of char fucto ω Π c l + lm l has good saddl pot : 0 x V x + G x V x + g x dx Fal rsult Dyso Coulomb gas : ω Π 0 R t dt

21 Asymptotcs for small o good saddl pot for prvous tgral saddl wats to b oscllatory rgo Us ow asymptotcs of to swtch to w rprstato cotour tgral : ω Π + l + lm wth saddl pot quato g p l G d Fal rsult aga : ω Π Guot & Mada R t dt

22 Suprbosoato acbroch Wdmüllr 95 Lhma Sahr Soolov Sommrs 95 Barruto Browr Svttsy 0 Eftov Schwt Taahash 04 Guhr 06; Basl Ama 07 Budr Eftov ravtsov Yvtusho MZ 07 Lttlma Sommrs MZ 08

23 Rmdr: suprsymmtry mthod ; ; xp ; Im ; ; sg Tr + Ω + Ω g g g g f f gg G d w f f f D d b b b b a a a a a j a aj j b q b b p a j a a j μ μ th actg by cojugato by som group varat If ad s alog whr th tgral fuctos : corrlato spctral for fucto Gratg s valuatd o fucto Charactrstc a

24 Suprbosoato spcal cas: commutg varabls oly Lt p q f : 0 ad cosdr GL -varat f f g g holom fct g GL Fact varat thory: thr xsts a holomorphc fucto F : such that F f By push forward of th tgral o has f d + c F r r dr f tgral xsts gralato to p > : s Fyodorov ucl Phys B

25 Suprbosoato spcal cas: Grassma varabls oly p 0 q Lt F : b holomorphc Atcommutg varabls Br tgral F d d : F + + F! 0 th U F th drvatv at d / π th org q > : awamoto ad Smt ucl Phys B

26 Th da of suprbosoato Rcall for g f ; G Lt G GL f g g or G O ; g g or G Sp Suprbosoato xplots to ma a stp of rducto : ths symmtry Th tgral ovr of th G-varat fucto f s covrtd to a tgral ovr a Rmaa symmtrc suprspac Th larg umbr of varabls th bcoms a paramtr of th tgral

27 Suprbosoato: utary symmtry U U / GL SDt MZ Sommrs Lttlma t t DQ - M Q F Q DQ f f p q p p M form Br tgrato varat ad doma wth tgrato th holomorphc ad Schwart alog ad If Thorm ; : ; GL F f Q F f G to Lft Lt

28 Applcato to dsordrd scattrg

29 Th sttg tral stats radom matrx amltoa coupld to M scattrg chals dlbrg approxmato to scattrg matrx : S E Id M W E + WW W To comput corrlatos C ab cd E E Z X Y S Dt E Dt E ab E δ us gratg fucto : ab S cd E δ + WXW Dt E WW + WW Dt E + WYW cd

30 Avragg trc Problm : ca' t us suprbosoato drctly sc prsc of WW bras U -symmtry Us trc of avragg tgrad ovr forc! U -symmtry U to For larg w hav formula lm f ra of both A ad B s pt fxd Etrs of g U U Tr AgBg dg Dt Id A B bcom Gaussa radom varabls

31 Uvrsalty of corrlato fucto Corrlato fucto by suprbosoato: C ab cd E E lm M DQ STr l Q + ω Q STr Q / E E STr ΛQ F ab cd Q Tag Q + R Q Soluto saddl-pot symmtry ad hc uvrsalup to scal factor Cocluso: gvs saddl-pot quato for Q : E Corrlatos of ar uvrsal dpdt mafold s dtrmd by S-matrx lmts radom matrx smbl th larg- + E of th choc of lmt

32 Coclusos Fr probablty thory provds th propr framwor whch to ta th larg- lmt of th dsty of stats for smbls whch ar varat but o-gaussa Fr cumulats ar th Taylor coffcts of th logarthm of th charactrstc fucto whch s coutrd wh usg suprbosoato Th group of suprsymmtrs dtrms crtcal tgrato mafold saddl pots th Our formalsm stablshs radom matrx uvrsalty of spctral corrlatos as wll as trasport obsrvabls