Y 1 Z 1,2 Z 1,3 Z 1,n. Z 1,3 Z 2,3 Y 3 Z 3,n. L n = 1 n

Transcription

1 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI Ditio. Lt {Z i,j } i<j ad {Y i } i b collctios of i.i.d. ma zro radom variabls. Suppos morvor that Zi,j ad that both Y ad Z hav it momts of vry ordr. For ay ow, w ca mak a radom symmtric matrix by puttig th Z's abov th diagoal, ad th Y 's o th diagoal blow th diagoal is th sam as abov, sic th matrix is symtric This is: Y Z, Z,3 Z, Z, Y Z,3 Z, Z,3 Z,3 Y 3 Z 3, X.... Z, Z, Z 3, Y This is calld a Wigr Matrix. If Z ad Y hav a Gaussia distributio, w call it a Gaussia Wigr Matrix. Rmark. W will b itrstd i th igvalus of th Wigr matrix, as thr ar som itrstig covrgc rsults to b had as. Ditio 3. Lt X b a Wigr matrix ad lt λ λ λ b its igvalus. Thy ar ral bcaus th matrix is symmtric. W d th mpirical distributio of th igvalus as th masur hr δ is th uit mass: L δ λ i So that L masurs th umbr of igvalus i ay st. i..: L a, b {i : λ i a, b} This is somthig lik th dsity of th igvalus i a, b sic is th total umbr of igvalus. Ditio 4. Th smicircl distributio is th probability distributio σxdx which is giv by a dsity with rspct to th Lbsgu masur o R: σx π 4 x x If you wr to graph this, you would s it is a smicircl! Thorm 5. [Wigr] For a Wigr matrix X th mpirical masur L covrgs wakly, i probability, to th smicircl distributio. I.. for ay cotiuous boudd fuctio f : R R w hav: Whr L, f fdl. P L, f σ, f > ɛ 0 Rmark 6. Th abov trmiology ca b a bit cofusig bcaus covrgs wakly i a a probabilisitc sttig ordiarily mas that for probability masurs P ad P mas that P, f P, f for vry boudd cotiuous fuctios f Thr ar lots of othr quivalt ways to phras this,.g, P a, b Pa, b for cotiuity sts a, b of P. Howvr for us th objct L is a radom probability masur i.. a probability masur valud radom variabl, so w hav to spcify i what ss.g. i probability, almost surly tc. w ma L, f σ, f wh w say L gos to σ wakly. Wigr's thorm tlls us that th covrgc is i th ss of covrgc i probability. xampl 7. Hr is a xampl to mak sur our hads ar scrwd o right for this covrgc i distributio. Lt A i b a squc of i.i.d. radom variabls ad lt δ A i b th mpirical distributio of th rst radom variabls. This is a radom masur! If P is th law of th A i 's, o ca chck that gos to P wakly, almost surly. This is tru sic a, b {i : A i a, b a.s. {A i a,b} {A a,b Pa, b. Th almost sur covrgc hr is from th strog law of larg umbrs sic th idicator radom variabls {Ai a,b ar all i.i.d.

2 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI Rmark 8. W will ow st out to prov Wigr's thorm. Th proof w giv hr will b basd o combiatorial argumts. W rst prov a fw facts about th smi-circl law. Lmma 9. [Momts of th Smi-Circl Law] Lt m k σ, x k b th k th momt of th smi-circl law. Th odd momts vaish, ad th v momts ar qual to th Catala umbrs. That is: m k+ 0 Whr C k is th k th Catala umbr C k k+ k k. m k C k Proof. m k+ is clar sic σxx k+ is a odd fuctio. To s m k C k look at σxx k dx ad do a chag of variabl to polar coordiats, ad th itgrat by parts i a clvr way to gt a rcurrc rlatio. Hav: m k k π k π k π k π x k σxdx π/ π/ π/ π/ π/ π/ π/ π/ si k θ cos θdθ si k θdθ k π si k θdθ k π π/ π/ π/ π/ si k θdθ k + m k si k+ θdθ [ si k+ θ ] [siθ] dθ Th last quality coms from doig itgratio by parts, dirtiatig si k+ ad itgratig si, ad th rcogizig m k si k cos. From hr w gt: m k k k + π π/ 4k k + m k π/ si k θdθ Whr w us th sam itgratio by parts trick agai to s that m k si k si k cos. From this rcurrc rlatioship w ca asily prov by iductio for istac that m k C k k+ k k Rmark 0. [About th Catala umbrs] Th Catala umbrs aris i all sort of combiatorial umratio problms. O is th umbr of -S paths th typ you ormally cosidr for radom walks. Th umbr of paths which hav k total stps, k stps, k S stps ad which vr go udr zro is C k. Ths ar calld Dyck paths. You ca driv this with th rctio pricipl. O of th most importat proprtis of th Catala umbr is th rcurrac: k C k C k j C j O ca also show that th gratig fuctio βz k0 C kz k is th covtio is C 0 : j βz 4z z This is do by writig C k k j C k jc j bcaus th Dyck paths of lgth k ca b dividd by th rst tim thy touch zro. If this rst tim is j th thr ar C j paths possibl o th lft of th rst hittig tim ad C k j paths possibl o th right. Summig givs th formula abov, ad th som gratig fuctio maipulatio yilds a quadratic quatio βz + zβz which w ca th solv for βz. This rcurrc proprty of th Catala umbr also lts us s th Catala umbrs apparig i othr placs, for xampl th umbr of rootd plaar trs with k dgs ad k + vrtics, bcaus it is ot hard to xhibit th sam rcurrc for ths objcts. A rootd plaar tr is a tr with a root ad a choic of ordrig o th childr of vry od For th rootd plaar trs, look at th root ad its rst child. Lt j b th umbr of dgs i th subtr comig from th child so thr ar k j i th subtr comig from th root which dos ot cotai th child. Ths subtrs ar xactly rootd plaar trs, so summig ovr j ow givs th sam rcurrc

3 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI 3 rlatio as abov. Aothr way to s this is a dirct bijctio btw roothd plaar trs ad Dyck paths. To do this you xplor aroud th outsid of th tr, ad crat a Dyck path by takig a stp up vry tim you mov dow a gratio, ad a stp dow vry tim you mov up a gratio. O ca show that o-crossig partitios of {, k}hav th sam rcurrc ad so ar coutd by C k as wll. This is bcaus if j is th largst lmt coctd to i th partitio, th th o-crossig proprty mas that w will iduc a o-crossig partio o { j }ad {j +,, k}too. Summig ovr j givs th sam rcurrc.. First Proof OF Wigr's Thorm Ditio. Rcall th ditio of L th mpirical distributio of th igvalus for th Wigr matrix. Lt L L b th o-radom masur o R giv by L a, b L a, b or quivaltly L, f L, f. Rcall that m k was th k th momt of th smi-circl lawσ. Lt m k L, x k b th k-th momt. Rmark. To prov that L P σ wakly, w will show that as gts larg, L is vry clos to L ad that L is vry clos to f. To mak this prcis, w will prov th followig two lmmas: Lmma 3. For vry k : lim Lmma 4. For vry k ad vry ɛ > 0: lim m k m k L, x k σ, x k lim P L, x k L, x k > ɛ 0 W will ow prov Wigr's thorm assumig ths two lmmas hav b prov, ad th w will com back to th proof of ths lmmas aftrward. Thorm 5. [Wigr] For a Wigr matrix X th mpirical masur L covrgs wakly, i probability, to th smicircl distributio. I.. for ay cotiuous boudd fuctio f : R R w hav: lim P L, f σ, f > ɛ 0 Proof. Assumig th two lmmas W will rst prov that: By Chbyshv iquality, w hav: lim sup P L, x k x >5 > ɛ 0 P L, x k x >B > ɛ Hav th, sic lim m k m k, that: ɛ L, x k x >B ɛ L, x k B k x >B ɛb k L, x k m k ɛb k lim sup P L, x k m x >B > ɛ lim sup k ɛb k m k ɛb k C k ɛb k If w choos B 5 ad th us th simpl iquality C k 4 k C k is th umbr of Dyck paths, whil 4 is th total umbr of -S paths of lgth th w s th right had sid is goig to zro as k. Sic x k x >B is icrasig i k wh B 5, w hav th that: So w coclud that: lim sup P L, x k x >5 > ɛ lim sup P L, x k+l x >5 > ɛ C k+l ɛ5 k+l 0 as l lim sup P L, x k x >5 > ɛ 0

4 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI 4 W will ow show that this rducs th problm to xamiig oly fuctios f which ar supportd i [ 5, 5]. Choosig k 0 givs us that lim P L, K x >5 > ɛ 0, so for boudd cotiuos fuctios f w hav: P L, f σ, f > ɛ P L, f x 5 + f x >5 σ, f x 5 + f x >5 > ɛ P L, f x 5 σ, f x 5 + L, f x >5 σ, f x >5 > ɛ P L, f x 5 σ, f x 5 > ɛ + P L, f x >5 σ, f x >5 > ɛ Hc: P L, f σ, f > ɛ P L, f x 5 σ, f x 5 > ɛ P L, f x >5 σ, f x >5 > ɛ P L, f x >5 0 ɛ > P L, sup f x >5 ɛ > 0 So showig that P L, f x 5 σ, f x 5 > ɛ 0 is sucit to show P L, f σ, f > ɛ 0. This mas w ca rstrict our atttio to fuctios f supportd o [ 5, 5]. Fix such a fuctio f ad ay δ > 0. By th Sto-Wirstrass thorm, w ca d a polyomial Q δ x L i0 c ix i dpdig o δ that approximatd f i th sup orm to withi δ/8 so that th dirc Q δ f has: sup x sup Q δ x fx δ x 5 x 5 8 otic that sic f is supportd i [ 5, 5] w may writ f Q δ Q δ x >5 + x <5. Hav th: P L, f σ, f > δ P L, Q δ Q δ x >5 + x <5 σ, Qδ Q δ x >5 + x <5 > δ P L, Q δ σ, Q δ + L, Q δ x >5 + L, x <5 + σ, Qδ x >5 + σ, x <5 > δ P L, Q δ σ, Q δ + L, Q δ x >5 + δ δ 8 > δ P L, Q δ L, Q δ + L, Q δ σ, Qδ + L, Q δ x >5 > 3δ 4 L P, Q δ L, Q δ δ > + P L, Q δ σ, Qδ δ > + P L, Q δ x >5 δ > : P + P + P 3 Sic Q δ is a polyomial, th rsult of th scod lmma 4 o th prcdig pag tlls us that P 0 ad th rsult of th rst lmma 3 o th prvious pag tlls us that P 0. W kow that P 3 0 by P L, x k x >5 > ɛ 0 which w provd at th bgiig of th proof, ad agai sic Qδ is a polyomial. Hc P L, f σ, f > δ 0 too... Proof of th Lmmas. Th startig poit of th proof of lmma 3 is to otic th followig idity : L, x k λ k i Tr X k : i,...i k i,...i k i,...i k X i, i X i, i 3 X i k, i k X i k, i Ti,i,...,i k : Ti i,...i k i,...i k T i T i,i,...,i k

5 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI 5 Whr i i, i,, i k ad Ti ad T i ar dd by th abov. Th proof of th lmma ow coms from combiatorial argumts ovr which i cotribut to th abov sum. Idd sic X i a, i b Z a,b or Y a,b ar idp ad ma zro, Ti 0 for may choics of i, for xampl if thr is a pair i a i b that oly appars oc i i. With som work, w will s that thr ar ordr k /+ o-zro trms. W will also s that thr ar ordr k / trms ivolvig momts of Z a.b or Y a,b highr tha or qual to 4. Sic all ths momts ar it, ad thy rprst a fractio of th sum, ths will ot cotribut i th limit that. W will ow crat som combiatorial objcts to ivstigat this i ail. Ditio 6. Giv a st L, a L-lttr is simply a lmt s L. A L-word w is a o-mpty it squc of L-lttr, s s s. A L-word is calld closd if its rst ad last lttrs ar th sam. Two L words ar calld quivalt if thr is a bijctio o L that maps o ito th othr. W also lt lw b th lgth of th word ad wtw th wight as th umbr of distict lttrs of L, ad suppw, th support of th word, is th st of distict lttrs which appar. If L {,,, } w oft us th trmiology -word, or if th st L is clar, w just say word. Ditio 7. Giv ay word w s s s, w d th graph associatd with th word w by G w V w, w b th graph with V w suppw th st of lttrs apparig i w ad with dgs w {{s i, s i+ } : i }. Th dg st ca b dividd ito slf-dgs s w {{u, u} : u V w } ad coctig dgs c w w s w. otic that two word ar quivalt if ad oly if th corrspodig graphs ar isomorphic. Ditio 8. Th graph G w is coctd bcaus th word w, wh rad i ordr, givs a spaig path. For w w lt dot th umbr of tim this path crosss th dg i ay dirctio. Rmark 9. Th tupl i i,, i k that appars i th valuatio of T i ds w i i i i k i a closd word of lgth k + o L {,,, }. If w lt wt i b th wight of this word, th th idpdc of th tris of th matrix X ad th fact thy ar all iically distributd lts us writ Rcall that X is scald by ad that it is symtric: T i k/ c i Sic th Z's ad Y 's ar ma zro, this product is o-zro oly if for all i. This forcs that wt i k +. W also s from this that quivalt words i th ss of isomorphisms o L {,,, } hav th sam valu for T i. Ditio 0. Lt S k,t dot th st of all closd words of lgth k + o L {,,, t} ad wight qual to t i.. vry lttr i {,,, t} is usd at last oc ad which hav th proprty that for vry dg w. Lt W k,t b th st of rprstativs of quivalc classs of S k,t udr th quivalc of words which w rcall corrspods to isomorphisms of L Rmark. For vry vry rprstativ w W k,t thr ar C,t t + words o th st L {,,, } that ar quivalt to w. All ths words i will hav th sam valu for T i. Propositio. From ths ditios w hav that: L, x k k/ + t C.t k/+ Z, w W k,t w c Rmark 3. otic that W k,t t k is lss tha th umbr of closd words of lgth k + from L {,,, t}. For xd k, this mas that i our sum W k,t t k k/ + k C cos t. Sic th momts of Z ad Y ar it, this mas that w hav th boud: T i s i k/ + C.t C k/+ t Sic C.t t + O t this logic us that for k odd, T i 0 sic k/ < k/, so 0 for vry t i our sum ad that for k v, oly th trm whr t k/ + survivs i th limit C,t k/+ Z, Y, ad th cocit i frot has C.k/+ W hav th th formulas: k/+ lim L, x k 0 for k odd L, x k for k v lim w W k,k/+ w c Z, c w c w Y Y

6 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI 6 This is th motivatio for th ditio of a Wigr word: Ditio 4. A closd word w of lgth k + is calld a Wigr word if ithr k 0 or k is v ad w is quivalt to a lmt of W k,k/+. Ths ar th oly words that appar i th valuatio of th momts of lim L, x k. Propositio 5. For a Wigr word w, th graph G w is a tr with o loops. Morovr, vry dg w has. Proof. G w is coctd, ad has V w k/ +, so it sucs to prov that w k/ to s that G w is a tr. Idd, w V w k/ or ls G w caot b coctd, ad w k/ by th coditio that for ach w sic k is th lgth of th path. Hc w k/ ad it is a tr with o slf loops. Morovr, sic w k, it must b th cas that for vry dg. Corollary 6. For k v:lim L, x k W k,k/+ Proof. This follows from th abov formula for lim L, x k, th propositio, ad sic Z,. Fially to s th lmma, w stablish a bijctio btw W k,k/+ Wigr words of lgth k + ad rootd plaar trs to stablish th lmma. Lmma 7. For vry k : lim L, x k σ, x k Proof. From our abov work, it sucs to crat a bijctio btw W k,k/+ ad rootd plaar trs with k + vrtics. Idd, for vry w W k,k/+ th graph G w is a rootd plaar tr whr th ordrig of th childr of vry od is th ordr i which thy ar visitd i th radig of w. This is a bijctio bcaus if G w G w th thy must b quilvalt words, so thy ar th sam lmt from W k,k/+. Th proof ca also b s by bijctio to Dyck paths by usig a xploratio procss btw Dyck paths ad plaar rootd trs. W ow work o th xt lmma. Ditio 8. A pair partitio of {,, k} is a partitio of {,, k} whr vry partitio is a pair i.. vry partitio cosists of xactly lmts A o-crossig pair partitio is a pair partitio that is also a o-crossig partitio. Fact 9. Th umbr of pair partitios of {,, k} is C k, th k th Catala umbr. Compar this to th fact that thr ar C k o-crossig partitios of {,, k} Propositio 30. Giv a Wigr word w i i i k+ of lgth k +, lt Π w b th partitio of {,, k} gratd by th fuctio f : {,, k} w by fj {i j, i j+ }. A partitio of {,, k} mas that th blocks ar dd by f {a} as a rags i th possibl targt spac. Th th followig hold: Proof. Π w is a o-crossig pair partitio. vry o-crossig pair partitio of {,, k} is of th form Π w for som Wigr word w of lgth k + 3 If two Wigr words w ad w of lgth k + satisfy Π w Π w, th w ad w ar quivalt. Bcaus vry Wigr word w ca b viwd as a walk o th graph G w ad ach dg is crossd xactly twic i th graph, soπ w is a pair partitio. Bcaus th graph G w is a tr, th partitio Π w is o-crossig. Say {a, b}ad {x, y} ar partitios, so that fa fb ad fx fy d. WOLOG a < x. Suppos by cotradictio a < x < b < y. Th i th walk o G w, w cotour rst, th d, ad th agai. But thr is o way to gt back to d agai sic w would hav to cross agai. Th argumt is th sam as showig that th xploratio procss of a tr givs a dyck path. vry o-crossig pair partitio ca b turd ito a rootd plaar tr by a xploratio. vry pair i th partitio corrspods to crossig a dg o th tr twic. This rul ca b usd to rcursivly build th tr up. As w'v s bfor, ths trs ar i bijctio with Wigr words. 3 I this cas th trs thy d would b th sam, ad so th words would b isomorphic. Rmark 3. To prov 4 o pag 3, by Chbyshv's iquality it is ough to prov that th variac gos to zro: L lim, x k L, x k lim Var L, x k 0

7 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI 7 Procdig as i th abov computatio for th momts of L goig through th trac of th -th powr of th matrix, sic L, x k i T i, w hav that: L, x k L, x k Var L, x k With: T i,i T i T i T i Cov Ti, Ti i,...i k i...i k T i To study this w d to furthr our stup to accout for pairs of words. W do this blow. Ditio 3. Giv a st L, a L-stc a is a it squc of L-words w w w at last o word log. Two Lstcs ar calld quivalt if thr is a bijctio o L that maps o to aothr. Ditio 33. Th graph associatd with a L-stc is obtaid by picig togthr takig th uio of th graphs associatd with th idividual words. otic that this may b discoctd! ow to valuat T i,i, w otic that th pair i, i ds a two word stc a w i w i, ad w hav: T i,i k Z a, Y a c a i,i c w i Z w i, s a i,i s w i Y w i c w i T i,i Z w i, Y w i w s i Sic ths radom variabls ar ma zro, w otic that T i,i is zro ulss a for all ai,i. It hlps hr to otic that a w for w w i or w w i. Also, if wi wi, th graphs G wi ad G wi ar disjoit ad so T i,i 0. This ca also b s dirctly from th formula abov with a bit of car. Cov Ti, Ti As bfor, to valuat this w look oly at rprstativs from a quivalc class of th rlvat stcs. D W k,t to b th st of rprstativs for quivalc class of stcs a of wight t rcall wight is th umbr of distict lttrs usd that cosist of two closd t-words w w ach of lgth k + ad with a for ach a ad with w w. With this ditio w hav a formula for T i,i aki to th o for T i w had arlir. Hr C,t is agai th umbr of stcs i ach quivalc class. Hav: Var L, x k k C,t k+ Z a, Y a t c w aw,w W k,t Z w, s w c a Y w s a c w Z w, s w Y w To show that this gos to zro as, sic th products i squar brackts ar boudd for xd k, it sucs to show that W k,t is mpty for t k +, bcaus i this cas th cocit C,t 0 for t < k + will tak th k+ whol sum to zro i th limit. W will actually prov a strogr claim, that W k,t is mpty for t k +, as this rsult will b usful latr. Propositio 34. W k,t is mpty for t k +. Proof. Lt a w w W k,t. Th G a is a coctd graph sic w w with t vrtics ad at most k dgs sic a for a, which is impossibl for t k + Sic a tr has th miimal umbr of dgs for a xd umbr of vrtics, ad a tr would hav t k + Wh t k +, it must b that G a is a tr ad that a for vry dg. But th path gratd by w starts ad ds at th sam vrtx, so it must visit ach dg o its rout at last twic. Th path gratd by w also visits ach dg o its rout at last twic. Sic ach dg is visitd xactly twic by ths two routs a for all dgs, it must b that th paths from w

8 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI 8 ad w ar disjoit. But th a is discoctd! This cotradicts th xistc of such a lmt a W k,t wh t k +. Lmma 35. For vry k ad vry ɛ > 0: lim P L, x k L, x k > ɛ 0 Proof. By Chbyshv, P L, x k L, x k > ɛ ɛ L, x k L, x k ɛ Var L, x k so it suf- cs to show that this gos to zro as. W hav that: Var L, x k k C,t k+ Z a, Y a t c w aw,w W k,t Z w, s w c a Y w s a c w Z w, s w Y w Wh t k i th sum, th cocit C,t 0 ad th trms i squar brackts ar boudd by somthig k+ lik k k, so ths trms vaish i th limit. Wh t k +, W k,t is mpty, so ths trms ar always zro. Hc th whol sum vaishs as... GO ad th GU. β. W us th ld F R, H U Ditios of dirt lttrs Mat R is th spac of ral symmtric matrics A A. is th st of orthogoal matrics A A Id. β. W us th ld F C, H Mat C is th spac of complx Hrmitia matrics A A. U is th st of uitary matrics A A Id. D is th st of diagoal matrics with ral tris. Ditio 36. GO Lt {ξ i,j } b a i.i.d. family of 0, radom variabls. D P to b th law of th radom ral symmtric matrix with X ii ξ ii o th diagoal, ad X ji X ij ξ i,j abov th diagoal. This is a probability masur o H For xampl: ξ ξ ξ 3 P D 3 ξ ξ ξ 3 H 3 ξ 3 ξ 3 ξ 33 A radom matrix X H with law P is said to blog to th Gaussia Orthogoal smbl GO Ditio 37. GU Lt {ξ i,j, η i,j } b a i.i.d. family of 0, radom variabls. D P to b th law of th radom complx Hrmitia matrix with X ii ξ ii o th diagoal, ad X ji X ij ξ i,j + ıη i,j abov th diagoal Hr ı is th imagiary uit. This is a probability masur o H A radom matrix X H P 3 D with law P ξ ξ +ıη ξ 3+ıη 3 ξ ıη ξ ξ 3+ıη 3 ξ 3 ıη 3 ξ 3 ıη 3 ξ 33 H 3 For xampl: is said to blog to th Gaussia Uitary smbl GU Rmark 38. Both th GU ad GO giv xampls of Wigr matrics. I particular Wigr's thorm applis ad lts us coclud that th mpirical distributio of th igvalus for th radom matrix X H β covrgs to th smicircl law w foud arlir. Rmark 39. What maks th GO ad th GU spcial? What do thy hav to do with orthogoal or uitary matrics? Rcall orthogoal matrics hav X X Id whil uitary matrics hav X X Id. Th aswr is that th built i symmtry of th ditio maks it so that th GO ad GU ar ivariat udr orthogoal matrics ad uitary matrics rspctivly. O way you ca s this is to calculat th probability dsity of th masur P β with rspct to th Lbsgu masur. W do this blow.

9 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI 9 Ditio 40. l β whβ Lt l Lbsgu masur o R +/ through th th o-to-o ad oto map H th o-or-abov th diagoal tris of a matrix i H b th Lbsgu masur o H which is dd by th pullback of th R+/ dd by takig as coordiats i R+/ Ditio 4. l β whβ Lt l b th Lbsgu masur o H which is dd by th pullback of th Lbsgu masur o R through th th o-to-o ad oto map H R C / R dd by takig th o-or-abov th diagoal tris of a matrix i H as coordiats i R C / Ditio 4. Th probability dsity i th ss of Rado-ikodym drivativ of P β Lbsgu masur l β is giv by: dp H dl / π +/4 xp Tr H /4 dp H dl / π / xp Tr H /4 I particular, w otic that th disty dpds oly o th trac of th matrix squard. with rspct to th Proof. This is a straightforward computatio usig th idpdc of th tris ad th fact that thy ar Gaussia. β otic that Tr H Tr H H H i,i + i<j H i,j. ow, sic ach ξ i,j is Gaussia ad idp, w hav that: dp H ρ ξi,i H dl i,i ρ ξ i,j H i,j i<j ρ ξ i,i H i,i ρ ξ i,j H i,j i<j +/ xp π Which givs th dsird rsult wh w us th xprssio for trac. β Firsty, otic that Tr H Tr H H Hi,i + dp H dl ρ ξ i,i H i,i Hi,i + i<j i<j RH i,j + ow, sic ach ξ i,j, η i,j is Gaussia ad idp, w hav that: ξi,j RH i,j ρ ξ i,i H i,i i<j i<j +/ xp H π i,i + ρ H i,i + H i,j i<j i<j ρ ξ i,j RH i,j i<j Which givs th dsird rsult wh w us th iity for trac. RH i,j + i<j ImH i,j H i,j ηi,j ρ ImH i,j i<j i<j ρ η i,j ImH i,j ImH i,j Ditio 43. For x, x,, x C w d th Vadrmod rmiat associatd with x by: x {x j i } i,j x j x i i<j

10 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI 0 Thorm 44. [Joit Distributio of th igvalus of th GO/GU] Lt X H β b a radom matrix with law P β, with β,. Th joit distributio of th givalus λ X λ X has dsity with rspct to th Lbsgu masur which quals: Hr C β! β C {x... x } x β is a ormalizig costat. This costat is: β! C! π / β HrΓs 0 x s x dx is ulr's Gamma fuctio. x β βx i /4 β /4+/ βx i /4 dx i j Γβ/ Γjβ/ Rmark 45. From this w ca asily s th dsity for th uordrd igvalus has dsity P β o R with dsity: dp β dlb β C x β βx i /4 Rmark 46. A cosquc of 44 is that th probability of a rpatd igvalu is zro sic vaishs if x i x j for ay i, j. This mas that vry igspac is o dimsioal. Lt v, v,, v b th basis of igvctors for th matrix X that is ormalizd so that th rst coordiat of v i is ral ad positiv ad so that v i. Th ivariac of X udr arbitrary orthogoal wh β or uitary wh β trasformatios mas that matrix costructd from th igvctors [v,, v ] is distributd lik th Haar masur o th st of orthogoal wh β or uitary wh β matrics. Rcall th Haar masur is th uiqu masur that is ivariat udr th actio of th matrics, i.. dµ U 0 U dµu. I particular, ay sigl vctor, say v, is distributd uiformly o th st of orm o vctors whos rst coordiat is ral ad positiv. Corollary 47. Lt S+ {x x,, x : x i R, x, x > 0}. Th v is uiformly distibutd i S+ for th GO or i S C,+ {x x,, x : x R, x i C, i, x, x > 0}for th GU. Furthrmor, v,, v is distributd lik a sampl of th Haar masur o orthogoal or uitary matrics, with ach colum multiplid by a orm o scalar so ach colum blogs to S+ for th GO ad S C,+ for th GU. Proof. Writ X UDU. Lt T b a orthogoal for GO or uitary for GU matrix distributd lik th Haar masur. Th cosidr T XT. This has th sam igvalus as X! Sic th law of X dpds oly o th igvalus of X, w kow T XT D D X. Morovr, sic T U T by ditio of th Harr masur, w hav th that X D T DT, i.. th orthogoal for GO or uitary for GU matrix that diagoalizs X is distributd lik th Haar masur. Th colums of this matrix mak a basis of igvctors, ad multiplyig ach by a orm o costat ormalizs thm th way w wat. Rmark 48. W will ow writ a outli of th proof for 44 of th distributio of th igvalus for th GO/GU. For ay X H β i.. X is symmtric/hrmitia, writ X UDU with U U β i.. U is orthogoal/uitary ad D D β i.. D diagoal with ral tris. Suppos th map Hβ U β D was a bijctio. It turs out it is ot a bijctio bcaus of th possibility of rpatd igvalus; w will mak a argumt to gt aroud this latr. Th o could paramatriz U β usig β / paramtrs i a smooth way + β / ral paramtrs to paramtriz H β ad subtract th dgrs of frdom comig from D. A asy computatio shows that th Jacobia of th trasformatio would th b a polyomial of dgr β / i th igvalus of X with cocits that ar fuctios of th paramtrizatio for U β. Sic th Jacobia must vaish o th st whr thr ar rpatd igvalus, w kow th roots of this polyomial! Symmtry ad dgr cosidratios th show that th Jacobia must b proportioal to th factor x β. Itgratig out th paramtrizatio for U β th givs 44. To gt aroud th fact that th map H β U β D is ot actually a bijctio, w hav to igor som masur zro sts. W do this blow.

11 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI { } Ditio 49. Lt U β,g U U β : vry diagoal try of U is a strictly positiv ral ad vry try of U is o-zro W call ths good uitary matrics or just good Lt D d {D D : vry try of D is distict} W call ths distict diagoal matrics or just distict. Lt D do {D D : vry try of D is distict ad thy ar dcrasig as w go dow th diagoal} W call ths distict ordrd { diagoal matrics or just distict ordrd. } Lt H β,dg H H β : H UDU whr D D d β,g ad U U. Lmma 50. Th st H β \Hβ,dg ca b thought of as a subst of R +/ or R as dscribd bfor. This is a ull st with rpsct to th Lbsgu masur. Furthrmor, th map D do U β,g H β,dg giv by D, U UDU is o-to-o ad oto, ad th map D do U β,g H β,dg giv by th sam map!-to-o. Proof. Firstly, otic that for a o-trivial polyomial p : R k R, th st {X : px 0} p {0} is a closd st ad is masur 0 with rspct to Lbsgu masur o R k sic it has at most k roots. Hc to prov th claim, it is ough to d a p : H β R which is a polyomial i th tris ad so that ph 0 for vry H H β \Hβ,dg. Lt us us th otatio that H i,j is th submatrix obtaid from H by dltig th ith colum ad jth row of H. Claim: Say X UDU for D D d so that X has distict igvalus ad suppos that X ad Xk,k do ot hav ay igvalus i commo for k,,,. Th all th tris of U ar ozro. Corollary: For vry H H β \Hβ,dg, ithr has som rpatd igvalus OR thr is som k so that th submatrix H k,k shars a igvalu with th matrix H. Pf of Corr: If H has rpatd igvalus w ar do. Othrwis, H has o rpatd igvalus. Suppos by cotradictio that X ad X k,k do ot hav ay igvalus i commo for k,,,. Th by th claim H UXU will hav U with all ozro tris. But th H has distic igvalus, ad a good uitary matrix, so H H β,dg which is a cotradictio. Pf of Claim: Lt λ b a igvalu of X, ad lt A X λi. D A adj as th matrix with A adj i,j i+j A i,j.this is th adjuct matrix from liar algbra, which ca b asily vrid to hav th proprty that AA adj AI. Sic λ is a igvalu of X, A 0 so w hav AA adj 0. ow, th dimsio of th ullspac of A is th dimsio of th λ igspac for X which is o sic th igvalus for X ar distict. ow, sic AA adj 0, w kow that ach colum of th adjugat is prpdicular to th row-spac of A. Sic th row-spac of A is dimsioal, by rak-ulltiy thorm, vry colum of A adj is i th -dimsioal spac orthogoal to th row-spac of A. Hav th that vry colum of A adj is a scalar multipl of som vctor v λ. Sic Av λ 0, w kow that v λ is a igvctor of X of igvalu λ. Sic X ad X k,k hav o igvalus i commo, w kow that X k,k λi 0. otic that by ditio of A, w hav A adj i,i X k,k λi 0, so th diagoal tris of A adj ar all o-zro. Sic th colums of A adj ar scalar multipls of v λ it must b that v λ has all o-zro compots! Or ls if v λ i 0, th tir i-th row of A adj would b zro, cotradictio th abov. Fially, sic ach colum of th diagoalizig matrix U is a scalar mulitpl of som v λ, w kow vry try of U is o-zro. Fact: For ay polyomials p, q : R R, thr is a fuctio f p, q which is polyomial i th cocits of p, q so that fp, q 0 if ad oly if p ad q shar a root. f is calld th rsultat of p, q. A corollary is that fp, p is a polyomial i th cocits of p which is zro if ad oly if p has a rpatd root. f is calld th discrimiat of p, q W ar ow rady to prov th lmma. Lt p b th charactristic polyomial of H ad lt p k b th charactristic polyomial of H k,k. Lt P k b th rsultat of p ad p k This is a polyomial i th cocits of p, p k. Sic roots of p, p k corrspod to igvalus of H, H k,k, P k is 0 if ad oly if H ad H k,k shar a igvalu. Lt P 0 b th discrimiat of p agai a polyomial i th cocits of p which is 0 if ad oly if H has a rpatd root. Sic th cocits of p, p k ar polyomial i th tris of th matrix H, P k ad P 0 ar also polyomial i th tris of H. Lt Q P P k P 0. This is a polyomial i th tris of H. Morovr by th corollary to th claim, for H H β \Hβ,dg, H ithr has a rpatd root i which cas P 0 0 or thr is som k so that H k,k shars a igvalu with H i which cas P k 0. I ithr cas Q 0. So Q works as th polyomial w ar lookig for! Th o-to-o or!-to-o atur of th map is clar bcaus ach igspac is of dimsio, ad th choic of ormalizatio i th ditio of a good uitary matrix forcs it. { } Ditio 5. lt U β,vg U U β,g : all miors of U hav ovaishig rmiat b a subst of th good matrics. W call ths vry good uitary matrics. Ths matrics hav a ic paramtrizatio.

12 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI Lmma 5. Th map T : U β,vg R β / whr w iify C R for β dd by: U, T U,, U,, U,3,, U, U,,,, U, U, U, U, U, is o-to-o with a smooth ivrs. Furthrmor, th st... I scoutd th rst of th proof but didt writ it up.. Itro To Frdholm Dtrmiats Ditio 53. A Polish spac is a complt mtric spac which is sparabl. T U β,vg is closd ad has zro Lbsgu masur. Ditio 54. Lt X b a Polish spac ad lt B X b its Borl sigma algbra. For a complx valud masur ν o X, B X d: ν x dνx W will cosidr oly masurs ν with ν <. X Ditio 55. A kral is a Borl masurabl, complx-valus fuctio Kx, y dd o X X with orm dd by: K sup x,y X X Kx, y Th trac of a kral Kx, y with rspct to som masur ν is: TrK Kx, xdνx Giv two krals Kx, y ad Lx, y thir compositio K Lx, y is aothr kral which is dd with rpct som masur ν as: K L x, y Kx, zlz, ydνz Ths will b wll dd i.. th itgrals will b it as log as K ad ν ar it Propositio 56. By Fubii Tr K L Tr L K ad K L M K L M ot 57. WARIG sic K is ot cotiuous it might b that K K almost vrywhr but Tr K Tr K thy could dir o th diagoal which is a masur zro st Rmark 58. If X is th spac {,,, } ad ν is th coutig masur, this fls a lot lik a matrix, with Tr big th trac ad big matrix multiplicatio. I fact, if w choos poits x,, x ad y, y th [Kx i, y j ] i,j is a matrix. It might b itrstig to tak th rmiat of th matrix. Lmma 59. Fix > 0 for ay two krals F x, y ad Gx, y w hav: F x i, y j Gx i, x j +/ F G max F, G Ad: Proof. D: F x i, y j / F H k i x, y Gx, y F x, y Gx, y F x, y if i < k if i k if i > k Sic rmiats ar liar i th rows, w ca do som maipulatio: F x i, y j Gx i, x j Hk i x i, y j This works by rwtritig th vctor that appars i th top row of F x i, y j as F k F G + G rcursivly doig this with th top F that appars o th lft had sid givs us xactly th rstult abov. Applyig Hadamards

13 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI 3 Thorm ow Hadamard: v,, v ar colum vctors of lgth with complx tris, th [v,, v ] v T i v i / v i. W gt: i x i, y j / F G max F, G Hk Which givs th dsird iquality. I th cas G 0 th abov rmiat is 0 ad w gt th dsird iquality o F alo. Ditio 60. For a giv kral K ad a masur ν d 0 as a covtio ad for > 0: K, ν K ξ i, ξ j dν ξ dν ξ By th iquality w just provd, ν K / so th itgral is wll dd.\ Ditio 6. Th Frdholm rmiat associatd with th Kral K is dd as: K K, ν K, ν! Rmark 6. Hr is som motivatio for this big calld a rmiat. Lt f x,, f x, g x,, g x b giv. Put: Kx, y f i xg i y This is a Kral, ad it will tur out that: K 0 δ ij f i xg j xdνx For this raso somtims popl us th otatio that K I K. I will ow skip to Lmma 3..4 from th book whr this xprssio appars. 3. Hrmit Polyomials ad th GU W ar goig to prov th followig thorm, which tlls us that th igvalus of th GU hav a probability dsity giv by a Frdholm rmiat. Prcisly, w ar workig towards: Thorm 63. Gaudi-Mhta For ay compact st A R: lim P λ, λ / A + K si k + k whr: K si x, y { π k π six y x y A x y x y A k K six i, x j k dx j To bgi rcall that w calculatd th joit distributio of th igvalus for th GU to b this masur is dotd by P : C x x i / C Whr is a ormalizig costat ad x is th Vadrmod rmiat. For p lt us dot by P p, th distributio of p uordrd igvalus of th GU; that is to say th law so that for fuctios f Pp, fλ,, λ p P fλ,, λ p. Bcaus th law P is symtric w hav that: p! Pp, fλ,, λ p f λσ,, λ σp! P σ S p, Whr S p, is th st of ijctiv maps from {,, p} to {,, }. W ow d th Hrmit polyomials H x. Ths com up i quatum mchaics as th ig-solutios to th quatum harmoic oscilllator, so thy might b somwhat familiar. j

14 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI 4 Ditio 64. Th -th Hrmit polymial H x is dd by: H x : x / d dx x / O ca vrify th followig proprtis of th polyomials H x, you might rmmbr doig this i your quatum mchaics class. I this sctio G x / dx is th Gaussia mausur ad, G is th L ir product with rspct to this masur, that is: f, g G fxgx x / dx π fzgz H 0 x, H x x ad H + x xh x H x H x is a moic polyomial of dgr 3 H x is v wh is v ad odd wh is odd. 4 x, H G 0 5 H k, H l G πk!δ kl 6 f, H G 0 for all polyomials fx of dgr < 7 xh x H + x + H x for 8 H x H x 9 H x xh x + H x 0 0 For x y: k0 H k xh k y k! HxH y H xhy!x y Ditio 65. Th -th ormalizd osciallator wav fuctio is th fuctio: This is ormalizd so that ψ k xψ l xdx δ kl. ψ x x /4 H x π! Lmma 66. For ay p, th law P p, is absolutly cotiuous with rspct to th Lbsgu masur ad it has dsity: ρ p, θ p! p,, θ p! k,l K θ k, θ l Whr: K x, y k0 ψ k xψ k y Proof. By th xplicit dsity calculatd for th GU, : ρ p, θ,, θ p C p, θ,, θ p, ζ p+,, ζ x i / ow, th fudmtal rmark of this sctio is th obsrvatio that th Vadrmod rmiat ca b writt i trms of th Hrmit polyomials: x i<j x j x i xj i H j x i This works bcaus vry H j x i is a moic polyomial with ladig trm x j i. Th rst of th polyomial dos ot cotribut to th rmiat ca b prov by iductio bcaus of 6. f, H G 0 for all polyomials fx of dgr < proprty 6 abov. Usig this w hav th: ρ ρ, θ,, θ p C, H j θ i x i / C, ψ j θ i C, ψ k θ i ψ k θ j k0 C, K θ i, θ j ip+ dζ i

15 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI 5 I th last li w usd th fact that AB A B with A B ψ j θ i so AB givs th trm whr K appars. Hr C, k0 πk! C, coms from th ormalizatio costats i th ditio of ψ i. Lmma 67. For ay squar-itgrabl fuctios f,, f ad g,, g o th ral li, w hav:! f k x i g k x j dxi! k f ix j f i xg j xdx g ix j dx i Proof. Us th iity AB A B applid to th matrix A [f k x i ] ik ad B [g k x j ] kj so that AB [ k f kx i g k x j ] ij. This iity givs: f k x i g k x j dx i k f ix j g ix j dx i ow usig th prmutatio xpasio for th rmiat, w hav: f ix j g ix j dx i sgσsgτ f σi x i g τi x i dx i σ,τ S sgσsgτ f σi xg τi xdx σ,τ S! sgσ f σi xg i xdx σ S! f i xg j xdx Which is th dsird rsult. Rmark 68. If w plug i f i g i ψ i ad ito th abov Lmma, w gt that from th orthogoality of th ψ i s that: K θ i, θ j dθ i! k0 ψ k θ i ψ k θ j dθ i ψ i θψ j θdθ! I! Corollary 69. Th ormalizig costat C, which appars i th joit distributio for th igvalus is: C,! k0 πk!! π / k! Proof. Itgrat th dsity ρ, to s that C, K θ i, θ j dθ i C,!. Th rsult follows from th rlatioship C, k0 πk! C,. Corollary 70. Th ormalizig costat C p, p!!

16 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI 6 Proof. Followig th abov stratgy, w will hav For covic, lt x i θ i if i p ad x i ζ i for i > p. ρ p, θ,, θ p C p, ψ j x i dζ i C p, sgσsgτ σ,τ S C p, C p, σ,τ S sgσsgτ σ,τ S sgσsgτ ip+ ψ σj x j ψ τj x j j p ψ σj θ j ψ τj θ j j p ψ σj θ j ψ τj θ j j ip+ jp+ jp+ dζ i ψ σj ζ j ψ τj ζ j δ σjτj This is ozro oly wh σ ad τ dir oly o {,, p} ad i this cas th factor at th d is. W ow divid th sum up basd o which lmts {,, p} map to. For v < < v p lt Lp, v b th bijctios from {,, p} to {v,, v p }. W hav: ρ p, θ,, θ p C p, C p, v <...<v σ,τ Lp,v v <...<v sgσsgτ p ψ v j θ j p ψ σj θ j ψ τj θ j From hr w ca itgrat both sids ad us th lmma to s that C p, p!!. This is ow rip for us to apply th Cauchy-Bit thorm. I our cas w us th followig vrsio of th thorm: Lt A b a p matrix ad lt C AA this is a p p matrix, th C K K p, A K A K whr K p, is th st of all p lmt substs of {,, } ad A K is th p p matrix which is obtaid from A by kpig oly th colums i K. This is xactly th st up w hav hr with A i,j ψ j θ i. Applyig this ad oticig that [Kθ i, θ j ] i,j [ψ i θ j ] i,j [ψ i θ j ] i,j from its ditio ally givs: j ρ p, θ,, θ p C p p, K θ i, θ j Fially, w gt to th mai rsult about th GU as a Frdholm rmiat. Thorm 7. For ay masurabl subst A of R: P k {λ i A} + k! A c Proof. From our prvious lmmas: {λ i A} P k A A C, C, [ A A A c ρ θ,, θ p A k K x i, x j dθ k k K θ i, θ j A [ k0 ψ i xψ j ydx A [ δ ij ] ] ψ i xψ j ydx A c dθ k k k dx i ] ψ k θ i ψ k θ j dθ k Th last li follows by th orthogoality of th ψ's. ow doigsom kid of rmit iity maipulatio which I dot quit s, you gt DIT: I gurd it out! Rplac th that appars with a x. Th th rmiat k ip+ dζ i

17 OTS FROM ITRODUCTIO TO RADOM MATRICS BY ADRSO, GUIOT AD ZITOUI 7 is a polyomial i x. By takig drivativs w.r.t to x you gt th formula ala Taylor xpasio. Fially, put back i for x. S th ots from Lax's fuctioal aalysis for this i ail! : P {λ i A} + k k ψ i xψ j ydx A c k 0 v <...<v k ow w just us our lmmas i rvrs to rwrit this i trms of K,. Hav: P k k {λ i A} + k! ψ v i x dxi A c k k + k! A c k A c 0 v <...<v k A c 0 v <...<v k k K x i, x j dxi Lastly, w ca sum to istad of without chagig th rsult, as th rak of th matrix [ K x i, x j ] k is at most bcaus it ariss as a product AA whr A is a k matrix of ψ's.