On the Moments of the Traces of Unitary and Orthogonal Random Matrices

Transcription

1 Proceedngs of Insttute of Mathematcs of NAS of Ukrane 2004 Vol. 50 Part On the Moments of the Traces of Untary and Orthogonal Random Matrces Vladmr VASILCHU B. Verkn Insttute for Low Temperature Physcs and Engneerng 47 Lenn Ave harkv Ukrane E-mal: vaslchuk@lt.kharkov.ua The complcated moments of the traces of untary and orthogonal Haar dstrbuted random matrces are studed. The exact formulas for dfferent values of moment and matrx orders are obtaned. 1 Introducton Consder the probablty space whose objects are n n untary matrces and whose probablty measure s unt Haar measure on the group Un). Denote the expectaton wth respect to the measure. Gven a non-negatve nteger k consder the moment Tr U n ) a 1 Tr U 2 n) a2 Tr U k n) a k Tr U n ) b 1 Tr U k n ) b k 1) where a =a 1...a k )andb =b 1...b k )arek-tuples of non-negatve ntegers. If κa) = ja j κb) := jb j 2) and κa) κb) then t s easy to see that 1) s equal to zero. Hence wthout loss of generalty we can restrct ourselves to the moments for whch κa) =κb). In ths case we call κ := κa) =κb) 3) the order of the respectve moment and we wrte κ a; b) = Tr U n ) a 1 Tr Un) 2 a2 Tr Un) k a k Tr U n ) b 1 Tr U k n ) b k 4) In recent paper [3] Dacons and Evans proved that f κ n then κ a; b) =δ ab k j a j a j! δ ab = k δ aj b j 5) Analogous result was obtaned for the orthogonal group also see formula 17) below). The proofs of 5) n [3] were based on the representaton theory of the groups Un) andon). Another proof of 5) was gven n [6] Appendx). The proof s based on certan denttes for the Töpltz determnants [1] Hughes and Rudnck [5] proved relaton 17) for SOn) and for κ usng some combnatoral technques. Smlar questons were consdered n [ ]. We wll gve below a proof of these relatons for the groups Un) andon) by usng elementary means.

2 1208 V. Vaslchuk 2 Untary group We begn from 5). Our proof s based on the followng smple mplcaton of the left nvarance of Haar measure on Un). Proposton 1. Let F U n Un) be a contnuously dfferentable complex valued functon. Then for any n n Hermtan matrx H we have F 1 U n Un) HU n F 2U n Un) UnH =0 6) where F 1 U nu n) and F 2 U nu n) are dervatves of F wth respect to U n and U n correspondngly. The proposton follows from the fact that F e th U n Une th) s ndependent of a real parameter t because of the left nvarance of Haar measure on Un). Choosng n 6) H = zx xy) z X xy)) T where z s an arbtrary complex number and X xy) = {δ px δ qy } n pq=1 we conclude that formula 6) s vald also n the case where H s replaced by X xy) n fact by any real matrx). Denote a j the frst from the left non-zero ndex of the k-tuple a =0...0a j...a k ). Then we can wrte 4) as k a; b) = n x=1 Un) j xx Tr Un) j a j 1 Tr Un) k a k Tr Un) b1 Tr Un) k ) b k. 7) We apply the proposton wth H = X xy) to the functon F U n U n)=u j n) xy Tr U j n) a j 1 Tr U k n) a k Tr U n) b1 Tr U n) k ) b k. Takng nto account the relatons Un m ) xy ) X xy) U n = UnX xy) Un m ) xy = =0 =0 U n) xx U m n = δ xx Un m ) yy Un) xx Un m ) yy Tr U m n ) X xy) U n = mtr U m n X = mu m n ) yx Tr U n) m ) X xy) U n = mu n) m ) yx and smlar ones we obtan where j 1 δ xx U j n ) yy α β U n ) xx Un j ) yy α β a j 1)j Un) j xy Un) j yx Tr Un) j aj 2 α β a l l Un) j xy Un) l yx αl)β b l l Un) j xy Un) l ) yx αβl) =0 8) α =TrUn) j a j 1 Tr Un) k a k β =TrUn) b1 Tr Un) k ) b k α =TrUn j1 ) a j1 Tr Un) k a k αl) =TrUn) j a j 1 Tr Un) l a l 1 Tr Un) k a k βl) =TrUn) b1 Tr Un) l ) b l 1 Tr Un) k ) b k. ) yy

3 On the Moments of the Traces of Untary and Orthogonal Random Matrces 1209 Applyng to 8) the operaton n 1 where κ a; b) 1 n j 1 n xy=1 and regroupng terms we obtan n vew of 7) κ 0...a l =1 0...a j l =1 0...a j 1a j1...a k ; b) ja j 1) κ 0...0a j 2a 2j)a 2j 1a 2j); b) la l κ 0...0a j 1a l)a l 1 ã a lj 1a l j); b) = jb j κ j 0...a j 1a j1...a k ; b j)b j 1b j)) 1 n j 1 lb l κ l 0...a j l =1 0...a j 1a j1...a k ; b l)b l 1b l)) lb l κ j 0...a j 1a ; b l j)b l j 1 b b l 1b l)) 9) a l) =a j1...a l 1 ) a l) =a l1...a k ) ã =a l1...a lj 1 ) b l) =b 1...b l 1 ) b l) =b l1...b k ) b =bl j1...b l 1 ). It wll be mportant n what follows that the order of all moments n the l.h.s. of 9) s κ whle the orders of all moments n the l.h.s. are less then κ κ j and κ l). We present now these relatons n a more convenent form. Gven a non-negatve nteger denote P the set of the k-tuples a =a 1...a k ) a j 0 such that ja j. Consder the vector space L k-tuples such that of collectons of complex numbers ndexed by pars a; b) of a b P ja j = jb j 10) and call the nteger κ of 3) the order of a component va; b) ofv L f the ndces a and b of the component satsfy 3). We defne n L the -norm.e. f v L then v =max va; b) ab where the maxmum s taken over the pars a b) satsfyng 10). Furthermore we vew the expresson n the parentheses of the l.h.s. of 9) the frst term of the r.h.s. and the expresson n the parentheses of the r.h.s. of 9) as the results of acton of the lnear operators A B andc on the vector whose components are the moments 4) of the orders κ. In other words f a j s the frst from the left non-zero component of the k-tuple a then Av) κ a; b) = 1 n j 1 v κ 0...a l =1 0...a j l =1 0...a j 1a j1...a k ; b) 11)

4 1210 V. Vaslchuk ja j 1)v κ 0...0a j 2a 2j)a 2j 1a 2j); b) la l v κ 0...0a j 1a l)a l 1 ã a lj 1a l j); b) Bv) κ a; b) =jb j v κ j 0...a j 1a j1...a k ; b j)b j 1b j)) 12) Cv) κ a; b) = 1 n j 1 lb l v κ l 0...a j l =1 0...a j 1a j1...a k ; b l)b l 1b l)) lb l v κ j 0...a j 1a j1...a k ; b l j)b l j 1 b b l 1b l)). 13) Wth ths notaton we can rewrte 9) as I A) = Bmn) Cmn). 14) On other hand f we denote as µ κ a; b) the r.h.s. of 5) for κa) =κ then we can easly obtan that the values µ κ a; b) verfy the followng recursve relaton for any a j 0 µ κ a; b) =jb j µ κ j a 1...a j 1 a j 1a j); b j)b j 1b j)). 15) By usng 11) 15) t s easy to prove the followng Lemma 1. Let A B and C be the lnear operators defned by 11) 13). Then ) A 1)/n ) f µ s the vector of L whose components are µ κ a; b) κ then Bµ = µ Cµ = Aµ. Remark 1. Relaton 15) and hence ts soluton 5)) can be also obtaned form 14). Indeed we obtan 15) just passng n 14) to the lmt n formally. To justfy ths usng the frst asserton of the lemma and nducton n we can prove that the sequence { } n=1 s unformly n n bounded and hence compact. The frst asserton of the lemma mples that f n then the operator I A s nvertble. Hence for n 14) or 9) ) s equvalent to =I A) 1 B ) Cmn). Snce B and Cmn) nclude the moments whose orders are strctly smaller than ths relaton allows us to fnd the moments of the order provded that the moments of lower orders are known. Ths suggests the use of the nducton n to prove formula 5). Indeed t s easy to check that for =0 1 the formula 5) holds. Assume that κ a b) = µ κ a b) κ 1. The r.h.s. of 12) 13) contan the components of v whose orders do not exceed 1. Hence B = Bµ C = Cµ B C) =I A)µ. These facts and the second asserton of the lemma yeld the followng relaton =I A) 1 Bµ Cµ )=I A) 1 I A)µ = µ whch complete the proof of 5).

5 On the Moments of the Traces of Untary and Orthogonal Random Matrces Orthogonal group Consder now the probablty space whose objects are n n orthogonal matrces and whose probablty measure s the normalzed to unt Haar measure on the group On) and denote the expectaton wth respect to the measure. Gven a non-negatve nteger k consder the moment κ a) = Tr O n ) a 1 Tr On) 2 a2 Tr On) k a k 16) where a =a 1...a k )sk-tuple. In recent paper [3] Dacons and Evans proved that f κ n/2 then k ) κ a) =E aj jξj y j y j = 1 1) j) /2 17) and ξ j are..d. standard normal varables. Hughes and Rudnck [5] we proved 17) for SOn) andκ usng some combnatoral technc. As n the prevous secton we wll use the followng proposton Proposton 2. Let F O n ) be a contnuously dfferentable complex-valued functon. Then for any n n real antsymmetrc matrx X we have cf. 6)) F O n ) XO n =0 18) where F O n ) s the dervatve of F wth respect to O n. The proposton follows from the fact that F e tx O n ) s ndependent of a real parameter t. Denote a j the frst from the left non-zero ndex of the k-tuple a =0...0a j...a k ). Then we can wrte 16) as k a) = n x=1 On) j xx Tr On) j a j 1 Tr On) k a k. 19) We apply the proposton wth X xy) = {δ px δ qy δ py δ qx } n pq=1 to the functon F O n )=O j n) xy Tr O j n) a j 1 Tr O k n) a k. Takng nto account the relatons O m n ) xy ) X xy) O n = = =0 =0 O n X xy) On m ) xy O n ) xx On m ) yy On) xy On m ) ) xy = δ xx On m ) yy O n) xx O m n ) yy δ xy On m ) xy Tr O m n ) X xy) O n = m Tr O m n X xy) = mo m n ) yx mo m n ) xy = mo m n ) yx m O m n ) T ) yx O n) xy O m n ) T ) yx

6 1212 V. Vaslchuk and smlar ones we obtan where j 1 δ xx O j n ) yy α δ xy O j n ) xy α O n ) xx On j ) yy α j 1 O n ) xy On j ) T yxα aj 1)j On) j xy On) j yx Tr On) j aj 2 α O j n) xy O j n) T yxtr O j n) a j 2 α ) a l l On) j xy On) l yx αl) a l l On) j xy On) l T yxαl) =0 20) α =TrOn) j a j 1 Tr On) k a k α =TrOn j1 ) a j1 Tr On) k a k αl) =TrOn) j a j 1 Tr On) l a l 1 Tr On) k a k Applyng to 20) the operaton n 1 n xy=1 and regroupng terms we obtan n vew of 19) κ a) 1 j 1 κ 0...a l =1 0...a j l =1 0...a j 1a j1...a k ) ja j 1) κ 0...0a j 2a 2j)a 2j 1a 2j)) la l κ 0...0a j 1a l)a l 1 ã a lj 1a l j)) = n y j 1 ) κ j 0...a j 1a j1...a k )ja j 1) κ 2j 0...a j 2a j1...a k ) 2 κ a j 2 =1 0...a j 1a j1...a k ) <j/2 where y j s defned n 17) la l κ 2j 0...a j 1a l j)a l j 1 â a l 1a l)) 21) a l) =a j1...a l 1 ) a l) =a l1...a k ) ã =a l1...a lj 1 ) â =a l j1...a l 1 ). As n the prevous secton we consder the vector space L of collectons of complex numbers ndexed by k-tuples a such that k ja j. and call the nteger κ of 3) the order of a component va)ofv L f the ndex a of the component satsfy 3). We defne n L the -norm.e. f v L then v =max a P va). Furthermore we vew the expresson n the parentheses of the l.h.s. of 21) the frst term of the r.h.s. and the expresson n the parentheses of the r.h.s. of 21) as the results of acton of

7 On the Moments of the Traces of Untary and Orthogonal Random Matrces 1213 the lnear operators A B andc on the vector whose components are the moments 16) of the orders κ. In other words f a j s the frst from the left non-zero component of the k-tuple a then Av) κ a) = 1 j 1 ν κ 0...a l =1 0...a j l =1 0...a j 1a j1...a k ) ja j 1)v κ 0...0a j 2a 2j)a 2j 1a 2j)) la l v κ 0...0a j 1a l)a l 1 ã a lj 1a l j)) 22) Bv) κ a) =y j ν κ j 0...a j 1a j1...a k ) ja j 1)ν κ 2j 0...a j 2a j1...a k ) 23) Cv) κ a) = 1 2 ν κ a j 2 =1 0...a j 1a j1...a k ) <j/2 y j ν κ j 0...a j 1a j1...a k ) ja j 1)ν κ 2j 0...a j 2a j1...a k ) la l ν κ 2j 0...a j 1a l j)a l j 1 â a l 1a l)). 24) Wth ths notaton we can rewrte 21) as I A) = Bmn) Cmn). 25) On other hand f we denote as µ κ a) the r.h.s. of 17) then usng Novkov Furutzu formula for the average of ξ j we obtan for any a j 0 the followng recursve relaton µ κ a) =y j µ κ j a 1...a j 1 a j 1a j)) ja j 1)µ κ 2j a 1...a j 1 a j 2a j)). 26) As n the prevous secton usng 22) 26) t s easy to prove the Lemma 1 wth the only modfcaton of ts frst asserton A 1)/). Thus the rest of the proof concdes wth untary case. It must be noted that we can also obtan relaton 26) by passng to the lmt n n 25). [1] Baxter G. Polynomals defned by a dfference system J. Math. Anal. Appl 1961 V [2] Brouwer P. and Beenakker C. Dagrammatc method of ntegraton over the untary group wth applcatons to quantum transport n mesoscopc systems J. Math. Phys V [3] Dacons P. and Evans S. Lnear functonals of egenvalues of random matrces Trans. of AMS 2001 V [4] Gorn T. Integrals of monomals over the orthogonal group J. Math. Phys V [5] Hughes C.P. and Rudnck Z. Moc Gaussan behavour for lnear statstcs of classcal compact groups math.pr/ [6] Johansson. On random matrces from the compact groups Ann. of Math V [7] Mehta M.L. Random matrces Boston Academc Press [8] Polya G. and Szego G. Problems and theorems n analyss II. Theory of functons zeros polynomals determnants number theory geometry Classcs n Mathematcs Berln Sprnger [9] Samuel S. UN) ntegrals 1/N and De Wt t Hooft anomales J. Math. Phys V [10] Szego G. Orthogonal polynomals AMS Provdence R.I [11] Wengarten D. Asymptotc behavor of group ntegrals n the lmt of nfnte rank J. Math. Phys V