A LAW OF LARGE NUMBERS FOR FINITE-RANGE DEPENDENT RANDOM MATRICES

Transcription

1 A LAW OF LARGE NUMBERS FOR FINITE-RANGE DEPENDENT RANDOM MATRICES GREG ANDERSON AND OFER ZEITOUNI Abstract. We consder random hermtan matrces n whch dstant abovedagonal entres are ndependent but nearby entres may be correlated. We fnd the lmt of the emprcal dstrbuton of egenvalues by combnatoral methods. We also prove that the lmt has algebrac Steltjes transform by an argument based on dmenson theory of noetheran local rngs. 1. Introducton Study of the emprcal dstrbuton of egenvalues of random hermtan (or real symmetrc) matrces has a long hstory, startng wth the semnal work of Wgner [Wg55] and Wshart [Ws28]. Except n cases where the jont dstrbuton of egenvalues s explctly known, most results avalable are asymptotc n nature and based on one of the followng approaches: () the moment method,. e., evaluaton of expectatons of traces of powers of the matrx; () approprate recursons for the resolvant, as ntroduced n [PM67]; or () the free probablty method (especally the noton of asymptotc freeness) orgnatng wth Voculescu [Vo91]. A good revew of the frst two approaches can be found n [Ba99]. For the thrd, see [Vo00], and for a somewhat more combnatoral perspectve, [Sp98]. These approaches have been extended to stuatons n whch the matrx analyzed nether possesses..d. entres above the dagonal (as n the Wgner case) nor s t the product of matrces wth..d. entres (as n the Wshart case). We menton n partcular the papers [MPK92], [KKP96], [Sh96] and [Gu02] for results pertanng to the model of random band matrces, all wth ndependent above-dagonal entres. In our recent work [AZ06] we studed convergence of the emprcal dstrbuton of egenvalues of random band matrces, and developed a combnatoral approach, based on the moment method, to dentfy the lmt (and also to provde central lmt theorems for lnear statstcs). Here we develop the method further to handle a class of matrces wth local dependence among entres (we postpone the precse defnton of the class to Secton 2). To each random matrx of the class we assocate a random band matrx wth the same lmt of emprcal dstrbuton of egenvalues by a process of Fourer transformaton, thus makng t possble to descrbe the lmt n terms of our prevous work (see Theorem 2.5). We also prove that the Steltjes transform of the lmt s algebrac (see Theorem 2.6), dong so by a general soft (. e., nonconstructve) method based on the theory of noetheran local rngs (see Theorem 6.2) whch ought to be applcable to many more random matrx problems. Date: September 10, O.Z. was partally supported by NSF grant number DMS

2 2 GREG ANDERSON AND OFER ZEITOUNI To get the flavor of our results, the reader should magne a Wgner matrx (.e., an N-by-N real symmetrc random matrx wth..d. above-dagonal entres, each of mean 0 and varance 1/N), on whch a local flterng operaton s performed: each entry not near the dagonal or an edge s replaced by half the sum of ts four neghbors to northeast, southeast, southwest and northwest. At the end of Secton 3 (Theorem 2.5 taken for granted) we analyze the (NE+SE+SW+NW)- fltered Wgner matrx descrbed above. We fnd that the lmt measure s the free multplcatve convoluton of the semcrcle law (densty 1 x <2 4 x2 ) and the arcsne law (densty 1 0<x<2 / x(2 x)). The appearance n ths example of a free multplcatve convoluton has a smple explanaton (see Proposton 3.6). We also wrte down the quartc equaton satsfed by the Steltjes transform of the lmt measure. Recently other authors have consdered the emprcal dstrbuton of egenvalues for matrces wth dependent entres, see [GoT05],[Ch05],[SSB05]. Ther class of models does not overlap sgnfcantly wth ours. In partcular, n all these works and unlke n our model, the lmt of the emprcal measure s always the same as that of a semcrcle law multpled by a random or determnstc constant. Closest to our work s the recent paper by [HLN05], that bulds upon earler work by [BDM96] and [G01]. They consder Gram matrces of the form X N X N where X N s a sequence of (non-symmetrc) Gaussan matrces obtaned by applyng a flterng operaton to a matrx wth (complex) Gaussan ndependent entres. They also consder the case (X N +A N )(X N +A N ) wth A N determnstc and Toepltz. The Gaussan assumpton allows them to drectly approxmate the matrx X N by a untary transformaton of a Gaussan matrx wth ndependent (but not dentcally dstrbuted) entres, to whch the results of [G01] and [AZ06] apply. In partcular, they do not need an assumpton on the support of the flter. Unlke the present work, the approach n [HLN05] and [BDM96] s based on studyng resolvants, rather than moments. We menton now motvaton from electrcal engneerng. The analyss of the lmtng emprcal dstrbuton of egenvalues of random matrces has recently played an mportant role n the analyss of communcaton systems, see [TV04] for an extensve revew. In partcular, when studyng mult-antenna systems, one often makes the (unrealstc) assumpton that gans between dfferent pars of antennas are uncorrelated. The models studed n ths work would allow correlaton between neghborng antenna pars. We do not develop ths applcaton further here. The structure of the paper s as follows. In the next secton we descrbe the class of matrces we treat, and state our man results, namely Theorem 2.5 (assertng a law of large numbers) and Theorem 2.6 (assertng algebracty of a Steltjes transform). In Secton 3 we dscuss the lmt measure n detal, and n partcular wrte down algebro-ntegral equatons for ts Steltjes transform, whch we call color equatons. Secton 4 provdes a computaton of lmts of traces of powers of the matrces under consderaton. In Secton 5 we complete the proof of Theorem 2.5 by a varance computaton. In Secton 6 we set up the algebrac machnery needed to prove Theorem 2.6. We fnsh provng the theorem n Secton 7 by analyzng the color equatons.

3 A LLN FOR FINITE-RANGE DEPENDENT RANDOM MATRICES 3 2. Formulaton of the man result After defnng a class of kernels n 2.1, we defne n 2.2 the class of matrces dealt wth n ths paper and n 2.3 descrbe the subclass of fltered Wgner matrces. To each kernel we assocate a measure n 2.4. Fnally, we state our man results, Theorems 2.5 and Kernels Color space (motvaton). As n [AZ06], the spatal change n the varance structure of the entres of a random matrx s captured by an auxlary varable, whch we call color. The dfference between the dervaton n [AZ06] and the present paper s that we append to color space an addtonal varable (related to the local averagng mentoned n the ntroducton), that one should thnk of as a Fourer varable. The precse defnton follows Color space (formal defnton). Let C = [0, 1] S 1, where S 1 s the unt crcle n the complex plane. We call C color space. We declare C to be a probablty space under the product of unform probablty measures on [0, 1] and S 1, denoted P. We say that a C-valued random varable s unformly dstrbuted f ts law s P. In the sequel, all L p (C) (resp., L p (C C)) spaces, p 1, are taken wth respect to the measure P (resp., P P) The kernel s. We fx a kernel s : C C R whch wll govern the local covarance structure of our random matrx model. We mpose on s the followng condtons. Assumpton (I) s s a nonnegatve symmetrc functon,.e. s(c, c ) = s(c, c) 0. (II) s has a Fourer expanson s(c, c ) = s j (x, y)ξ η j (c = (x, ξ), c = (y, η)),j Z where all but fntely many of the coeffcents vansh dentcally. s j : [0, 1] [0, 1] C (III) There s a fnte partton I of [0, 1] nto subntervals of postve length such that every coeffcent functon s j s constant on every set of the form I J wth I, J I. (IV) s s nondegenerate: s L 1 (C C) > 0.

4 4 GREG ANDERSON AND OFER ZEITOUNI The sets Q (N) K. For each N N (here and below N denotes the set of postve ntegers) and K > 0, we defne } Q (N) K { = {1,...,N} mn Nx > K x I where I [0, 1] s the fnte set consstng of all endponts of all ntervals belongng to the famly I Remarks. We have (1) s j = s, j, s j (y, x) = s j (x, y) because s s real-valued and symmetrc. Assumptons (I,II,III) mply that (2) 0 s s L (C C) < holds everywhere (not just P P-a.e.) The model. For each N N, let X (N) = [X (N) j ] N,j=1 be an N-by-N random hermtan matrx. We mpose the followng condtons, where s satsfes Assumpton Assumpton (I) (a) N N, j {1,...,N} (b) k N sup N max N=1,j=1 E X(N) j k <. EX (N) j = 0. (II) There exsts K > 0 such that for all N N, the followng hold: (a), j Z max(, j ) > K s j 0. (b) For all nonempty subsets A, B {(, j) {1,..., N} 2 1 j N} such that mn the σ-felds mn (,j) A (k,l) B σ({x (N) j are ndependent. (c), j, k, l Q (N) K EX (N) j X (N) kl max( k, j l ) > K, (, j) A}), σ({x (N) kl s.t. mn(j, l k) > K = s k,l j (/N, j/n). (k, l) B})

5 A LLN FOR FINITE-RANGE DEPENDENT RANDOM MATRICES The emprcal dstrbuton of egenvalues. Let λ (N) 1 λ (N) 2... λ (N) N denote the egenvalues of the hermtan matrx X (N) / N, and let L (N) = N 1 N =1 δ λ (N) denote the correspondng emprcal dstrbuton of the egenvalues. We are concerned wth the lmtng behavor of L (N) as N Remarks. () The exstence of K satsfyng Assumpton 2.2.1(IIa) s assured by Assumpton 2.1.4(IIb). () Assumpton 2.2.1(IIb) says that the on-or-above dagonal entres of X (N) form a fnte-range dependent random feld, wth K a bound for the range of dependence. Ths explans the ttle of the paper. () Assumpton 2.2.1(IIc) fxes varances at each ste and also short-range local correlatons for stes n suffcently general poston. (v) None of our assumptons rule out the possblty that all matrces X (N) are real. In other words, we can handle hermtan and real symmetrc cases unformly under Assumpton (v) The relatons (15), whch are equvalent to the realty and symmetry of the kernel s, play a key role n our analyss of the lmtng behavor of L (N). New methods would be needed were the realty and symmetry condtons to be relaxed Kernels of pure spatal type. Let s : [0, 1] [0, 1] R be a kernel. If the kernel s : C C R defned by the formula s((x, ξ), (y, η)) = s(x, y) satsfes Assumpton 2.1.4, then by abuse of termnology we say that s s a kernel of pure spatal type satsfyng Assumpton 2.1.4, and wth the evdent modfcaton of Assumpton 2.2.1(IIc) we can use s to govern the covarance structure of our model. In the specal case of kernels of pure spatal type our model essentally contans the model of [AZ06] n the specal case n whch color space s a fnte set Kernels of pure Fourer type. Let s : S 1 S 1 R be a kernel. If the kernel s : C C R defned by the formula s((x, ξ), (y, η)) = s(ξ, η) satsfes Assumpton wth I = {[0, 1]}, then by abuse of termnology we say that s s a kernel of pure Fourer type satsfyng Assumpton 2.1.4, and wth the evdent modfcaton of Assumpton 2.2.1(IIc) we can use s to govern the covarance structure of our model. We suggest that the reader focus on the pure Fourer case when frst approachng ths paper because lttle would be lost n terms of graspng the man deas. The man reason for us to work at a hgher level of generalty s to make sure that our theory handles not only fltered Wgner matrces (for whch I = 1) but also fltered Wshart matrces (for whch I = 2). (Here and below S denotes the cardnalty of a set S.) We then mght as well allow I > 2 as a possblty because there s no gan n smplcty by excludng t Fltered Wgner matrces. We descrbe now a natural class of random matrces fttng nto the framework of Assumptons and Ths class should be consdered the man motvaton for the paper. Members of the class arse by flterng Wgner matrces. The correspondng kernels are of pure Fourer type and depend n a smple way on the flter.

6 6 GREG ANDERSON AND OFER ZEITOUNI Wgner matrces. Let {Y j } <<j< be an..d. famly of real random varables. Assume that Y 01 has absolute moments of all orders. Assume that EY 01 = 0 and EY01 2 = 1. Put Then Y = 0 for < <, Y j = Y j for < j < <. (3) Y j = Y j, EY j = 0, EY j Y kl = (δ k δ jl + δ l δ jk )(1 δ j )(1 δ kl ) for all, j, k, l Z. Let Y (N) = [Y (N) j ] N,j=1 be the N-by-N matrx wth entres Y j. Then Y (N) / N (n the termnology of [AZ06]) s a Wgner matrx, and hence the emprcal dstrbuton of ts egenvalues for N tends to the semcrcle law. In partcular, f the Y j are standard normal random varables and one makes a sutable adjustment to the dagonal of Y (N) / N, the result s a Wgner matrx n the standard sense,.e., a member of the Gaussan orthogonal ensemble The flter, ts Fourer transform, and assocated kernel. Let a flter h : Z Z R be gven, wth H denotng ts Fourer transform, that s H(ξ, η) = h(, j)ξ η j (ξ, η S 1 ).,j Z We assume that h does not vansh dentcally. We assume that there exsts K > 0 such that (4) max(, j ) > K/2 h(, j) = 0, and hence H s well-defned. We assume that h satsfes the symmetry condton (5) h(, j) = h(j, ), whch mples the symmetry condton Put H(ξ, η) = H(η, ξ). s(ξ, η) = H(ξ, η) 2 = Z s j ξ η j j Z (s j R). Then s : S 1 S 1 R s a kernel of pure Fourer type satsfyng Assumpton In partcular, s L1 (C C) = h 2 L 2 (Z Z) > Fltered Wgner matrces (defnton). For, j {1,..., N} set X (N) j = N k=1 l=1 N Y kl h( k, l j), thus defnng an N-by-N random matrx X (N) whch n vew of the symmetry (5) s real symmetrc. We call X (N) / N a fltered Wgner matrx, wth flter h. We thnk of X (N) / N as the result of flterng the Wgner matrx Y (N) / N by h.

7 A LLN FOR FINITE-RANGE DEPENDENT RANDOM MATRICES Local covarance structure of X (N). From (3) and (4) we deduce that (6), j, k, l Q (N) K & mn(j, l k) > K EX(N) j X (N) kl = s k,l j, after a straghtforward (extremely tedous) calculaton. Ths s the man pont n verfyng that the real symmetrc random matrces X (N) satsfy Assumpton wth respect to the kernel s. The remanng detals needed to check Assumpton are easy to supply. Thus we can put fltered Wgner matrces nto the framework of our model The (NE+SE+SW+NW)-fltered Wgner matrx. Takng h = (1/2)1 {(1, 1),(1,1),( 1,1),( 1, 1)}, and hence s(e θ1, e θ2 ) = 4 cos 2 θ 1 cos 2 θ 2, we get a precsely defned verson of the model mentoned n the ntroducton, whch presently we wll analyze n detal. Note that wth θ unformly dstrbuted n [0, 2π), the law of 2 cos 2 θ has densty 1 0<x<2 / x(2 x) Remark. In [AZ06] we handled Wshart matrces Z Z (and more general matrces formed from matrces Z[ wth ndependent ] but perhaps not..d. entres) 0 Z n terms of band-type matrces Z. A smlar trck n the present settng 0 puts fltered Wshart matrces nto the framework of our model. For the kernels arsng n that connecton, the assocated partton I conssts of two ntervals. We do not dscuss the Wshart case further here The measure µ s. We make the last preparaton to state our man results. Let s be a kernel satsfyng Assumpton For each postve nteger N, let C (N) = {c (N) 1,..., c (N) N 2 } C be the set of pars (x, ξ) C where x [0, 1) 1 N Z and ξn = 1. Then the emprcal dstrbuton 1 N c C δ 2 (N) c tends weakly as N to the unform probablty measure P. Let C ( ) be the unon of the sets C (N). Let {Ỹe} e C ( ) s.t. e =1,2 be an..d. famly of standard normal (mean 0 and varance 1) random varables. (Recall from that S denotes the cardnalty of S.) Let X (N) be the N 2 -by- N 2 real symmetrc random matrx wth entres = 2 δj/2 s(c (N), c (N) )Ỹ{c (N) λ (N) X (N) j j,c (N) j }. (N) Let 1 λ N be the egenvalues of X (N) /N and let L (N) = 1 N 2 2 N 2 =1 δ λ(n) be the emprcal dstrbuton of the egenvalues. By [AZ06, Thm. 3.2] the emprcal dstrbuton L (N) tends weakly n probablty as N to a lmt µ s wth bounded support. (The strange-lookng factor 2 δj/2 n the defnton of X (N) could be dropped wthout changng the lmt of L (N). More generally, wthn the theory of [AZ06], one has many ways to construct models wth lmtng measure µ s. We made our concrete choce to smplfy the proof of Proposton 3.6 below.) In Secton 3 we wll provde a combnatoral descrpton of the moments of µ s and wrte down algebro-ntegral equatons (whch we call color equatons) satsfed by the Steltjes transform of µ s and certan auxlary functons.

8 8 GREG ANDERSON AND OFER ZEITOUNI The followng are our man results. In these results we fx a kernel s satsfyng Assumpton and a famly [X (N) ] N=1 of random hermtan matrces satsfyng Assumpton wth respect to s. As defned n 2.2.2, let L (N) be the emprcal dstrbuton of the egenvalues of X (N) / N. Let µ = µ s be the measure assocated to s by the procedure of 2.4. Theorem 2.5. L (N) converges weakly n probablty to µ. Theorem 2.6. The Steltjes transform S(λ) = µ(dx) λ x s an algebrac functon of λ,. e., there exsts some not-dentcally-vanshng polynomal F(X, Y ) n two varables wth complex coeffcents such that F(λ, S(λ)) vanshes for all complex numbers λ not n the support of µ. To prove Theorem 2.5 we frst prove convergence n moments n Secton 4, and then fnsh the proof n Secton 5 by consderng varances. To prove Theorem 2.6 we frst set up a general method for provng algebracty n Secton 6, and then fnsh the proof n Secton 7 by analyzng the color equatons. The method of proof does not yeld an explct polynomal F(X, Y ). We remark that the general method of Secton 6 ought to be applcable to many more random matrx problems. For example, t apples to the Steltjes transforms of the lmtng measures arsng from the model of [AZ06] n the case of a fnte color space, yeldng algebracty n all those cases. 3. The moments and Steltjes transform of µ We fx a kernel s satsfyng Assumpton and put µ = µ s. We provde a detaled descrpton of the moments and Steltjes transform of µ. We also ntroduce combnatoral tools needed throughout the paper Quck revew of key combnatoral notons Graphs. For us a graph G = (V, E) conssts by defnton of a fnte set V of vertces and a set E of edges, where every element of E s a subset of V of cardnalty 1 or 2. In other words, we are dealng here wth graphs () whch have fntely many vertces, () whch have unorented edges, () n whch a vertex may be joned to tself by an edge, but (v) n whch no two vertces may be joned by more than one edge. A graph G s a tree f G s connected and V = E + 1. We emphasze that every edge of a tree jons two dstnct vertces t s not allowed for a vertex of a tree to be joned to tself by an edge Set parttons. We say that a set π 2 {1,...,k} s a set-partton of k f π s a dsjont famly of nonempty sets wth unon equal to {1,..., k}. The elements of π are called the parts of π. For each {1,..., k}, let π() be the part of π to whch belongs. For convenence we extend π() to a perodc functon on Z by the rule π() = π( + k) The graph assocated to a set partton. To each set partton π of k there s canoncally assocated a graph G π = (V π, E π ), where V π = π and E π = {{π(), π( + 1)} = 1,...,k}. By constructon G π comes canoncally equpped wth a walk, namely π(1),..., π(k), π(k + 1) = π(1), whence n partcular t follows that G π s connected.

9 A LLN FOR FINITE-RANGE DEPENDENT RANDOM MATRICES Wgner set parttons. We say that a set partton π of k s a Wgner set partton f the correspondng graph G π has k/2 + 1 vertces and k/2 edges, n whch case G π, snce connected, s a tree. We denote the set of such π by W k. For k odd the set W k s empty. For k = 2l the set W k s canoncally n bjecton wth the set of rooted planar trees wth l + 1 nodes and hence, as s well-known [St99], 1 the cardnalty of W k s the Catalan number l+1 (2l l ). Lemma Fx π W k. () For each {1,...,k} we have π() π( + 1). () For each e E π the equaton e = {π(), π( + 1)} has exactly two solutons {1,...,k}, say 1 and 2, and moreover π( 1 ) π( 2 ). () For each {1,...,k} we have {π(), π( + 1)} = {π(j 1), π(j)}, where j s the least of the ntegers l > such that π() = π(l). Proof. The lemma formalzes facts about the tree G π and the canoncal walk on t whch are clear from a graph-theoretc pont of vew. () No edge of G π connects a vertex to tself. () The canoncal walk on G π vsts each edge of G π exactly twce. More precsely, the canoncal walk traverses each edge of G π exactly once n each drecton. () The canoncal walk on G π extended by perodcty returns to a gven vertex on the same edge by whch t departed Tree ntegrals. Let {κ A } be an..d. famly of C-valued unformly dstrbuted random varables ndexed by fnte nonempty sets of postve ntegers. Expectatons wth respect to these varables are denoted by E. For each π W k we defne a bounded random varable by the formula M π = s(κ A, κ B ), {A,B} E π whch s well-defned on account of the symmetry s(c, c ) = s(c, c). We call the expectaton EM π a tree ntegral. Proposton 3.2 (Combnatoral descrpton of the moments of µ). We have (7) µ, x k = π W k EM π for every nteger k > 0. The proposton s essentally just a specal case of [AZ06, Thm. 3.2], but a far amount of explanaton s needed because the set up n ths paper s (superfcally) ncompatble wth that of [AZ06] here we emphasze set parttons, whereas n [AZ06] we emphaszed words and spellng. We can mmedately reduce the proposton to the followng techncal lemma. The lemma s slghtly more detaled than needed for the proof of the proposton part () wll be needed for the dervaton of algebro-ntegral equatons for the Steltjes transform of µ. Lemma Put A = 2 s 1/2 L (C C). () There exsts a unque system {Φ n, Ψ n } n N of functons n L (C) such that (8) Ψ n (c) = s(c, c )Φ n (c )P(dc ) holds P-a.e. for every n and there holds an dentty ( ) 1 (9) Φ n t n = t 1 t Ψ n t n n=1 n=1

10 10 GREG ANDERSON AND OFER ZEITOUNI of formal power seres n t wth coeffcents n L (C). () The bounds (10) 0 Φ n A n 1, 0 Ψ n A n+1 /4 hold P-a.e. for every n. () The formula (11) P, Φ k+1 = holds for every nteger k > 0. π W k EM π Proof of the proposton grantng the lemma. Accordng [AZ06, Lemma 3.2] (takng there σ = P, D = 0, and s (2) = s), there exsts a unque probablty measure on the real lne wth k th moment P, Φ k+1 for every k 0. That measure accordng to [AZ06, Thm. 3.2] s none other than µ. Plan for the proof of the lemma. Part () of the lemma s nothng but an nductve defnton of Φ n and Ψ n. So only parts (,) requre proof. In prncple, part () follows from [AZ06, Lemmas 6.3 and 6.4], but because of the ncompatblty of setups noted above, those lemmas cannot be drectly appled here some amplfcaton s needed. Also part () s most easly explaned from from the pont of vew of [AZ06, loc. ct.] So, after recallng n 3.3 the needed background from [AZ06], we lghtly sketch n 3.4 a proof of parts (,) of the lemma The verbal approach. We brefly recall the pont of vew of [AZ06] and compare t to the present one. The materal revewed here wll be used n a substantal way only n Secton 3, not n later sectons of the paper Words. We fx a set of letters and defne a word to be a fnte nonempty sequence w = α 1 α k of letters. We say that words w = α 1 α k and x = β 1 β l are equvalent f k = l (the words are the same length) and there exsts a one-to-onecorrespondence ϕ : {α } {β j } such that ϕ(α ) = β for = 1,...k (each word codes to the other under a smple substtuton cpher). Each word w = α 1 α k of length k gves rse naturally to a set partton of k, namely the set of equvalence classes for the equvalence relaton j α = α j. Two words are equvalent f and only f they have the same length and gve rse to the same set partton. The upshot s that speakng of equvalence classes of words s equvalent to speakng of set parttons Wgner words. Let w be a word of at least two letters wth same frst and last letter, and let w be the word obtaned by droppng the last letter of w. The word w s a Wgner word n the sense of [AZ06] f and only f the set partton assocated to w s a Wgner set partton n the sense of ths paper. In [AZ06] we also declared every one-letter word to be a Wgner word. The Wgner words have a smple nductve characterzaton [AZ06, Prop. 4.5 and 4.7] whch s not so convenent to state n the set partton language. To wt, a word w s a Wgner word f and only f the followng condtons hold: The frst and last letters of w are the same. No letter appears twce n a row n w. Let α be the frst letter of w. Wrte w = αw 1 α αw r α, where α does not appear n any of the words w. Then each word w s a Wgner word, and moreover for j the words w and w j have no letters n common.

11 A LLN FOR FINITE-RANGE DEPENDENT RANDOM MATRICES 11 (If r = 0 then w conssts of a sngle letter and s by defnton a Wgner word.) Verbal descrpton of tree ntegrals. Let {κ α } be a letter-ndexed..d. famly of unformly dstrbuted C-valued random varables. Gven a Wgner word w, we defne a random varable M w nductvely by the followng procedure: Wrtng w = αw 1 α αw r α as n the nductve characterzaton of Wgner words, and lettng β be the ntal letter of w for = 1,...,r, we set M w = r =1 s(κ α, κ β )M w. (When w s one letter long, we put M w = 1.) The formula (11) clamed n Lemma can be rewrtten (12) P, Φ k+1 = w W k+1 EM w where the sum s extended over a set of representatves W k+1 for equvalence classes of Wgner words of length k + 1. Further and crucally, notaton as above n the nductve characterzaton of the random varables M w, we have a relaton (13) E(M w κ α ) = among condtonal expectatons. r E(s(κ α, κ β )E(M w κ β ) κ α ) = Proof of Lemma 3.2.1(,). It s enough to construct an example of a system {Φ n, Ψ n } n L (C) satsfyng (8,9,10,11), and to do so we follow the path of the proofs of [AZ06, Lemmas 6.3 and 6.4]. Fx a letter α. We may suppose that every word belongng to the set of representatves W k+1 fgurng n (12) begns wth α. There exst for each nteger k 0 well-defned Φ k+1, Ψ k+1 L (C) such that Φ k+1 (κ α ) = E(M w κ α ), Ψ k+1 (κ α ) = E(s(κ α, κ α )M w κ α ), w W k+1 w W k+1 where α s a letter not appearng n any of the words belongng to the set W k+1. The system {Φ n, Ψ n } has property (8) by constructon, and has property (11) snce the latter s equvalent to (12). Snce W k = W k+1 2 k, 0 M w s k/2 L (C C) for w W k+1, for all ntegers k 0, the system {Φ n, Ψ n } has property (10). Fnally, from (13) and the nductve characterzaton of Wgner words, we obtan denttes Φ k+1 = r r=0 P(l 1,...,l r) N r =1 r =1 (l+1)=k n L (C) for all ntegers k 0 whch together mply that the system {Φ n, Ψ n } has property (9). The proofs of Lemma and Proposton 3.2 are now complete The color equatons. We contnue n the settng of Proposton 3.2. Ψ l

12 12 GREG ANDERSON AND OFER ZEITOUNI Nce functons. We say that a complex-valued functon f on color space s nce (wth respect to the kernel s and assocated partton I of [0, 1]) f f has a Fourer expanson f(c) = Z f (x)ξ (c = (x, ξ) C) where all but fntely many of the coeffcent functons f : [0, 1] C vansh dentcally, and every coeffcent functon f s constant on every nterval of the partton I. It s not dffcult to see that Φ n and Ψ n can be corrected on a set of P-measure zero n a unque way to become nce. So we may and we wll assume hereafter wthout any loss of generalty that the functons Φ n and Ψ n are nce and that all the relatons asserted n Proposton 3.2 to hold P-a.e. n fact hold wthout excepton An auxlary functon. For all complex numbers λ > A and c C put Ψ(c, λ) = Ψ n (c)λ n, defnng a functon dependng holomorphcally on λ and satsfyng estmates A 2 (14) Ψ(c, λ) 1 A2, λ > 2A Ψ(c, λ) < 4 λ A 2 λ A 4 unform n c. n= The equatons. From the system of equatons descrbed n Proposton 3.2 we now deduce by the substtuton t = 1/λ and applcaton of domnated convergence the relatons (15) s(c, c )P(dc ) λ Ψ(c, λ) = Ψ(c, λ), P(dc) µ(dx) λ Ψ(c, λ) = S(λ) = λ x whch hold for all c C and λ > 2A. We call the relatons (15) the color equatons. Equatons of ths sort have appeared already n other contexts, see [G01], [HLN05, eq. 2.2], [KKP96]. Proposton 3.6. In the settng of the color equatons, assume further that for some nce nonnegatve functon f on color space s(c, c ) = f(c)f(c ), f L (C) = A/2, f L 1 (C) = 1. Let µ f be the law of f vewed as a random varable on C under P. Let S f (λ) be the Steltjes transform of µ f. Then: () There exsts a functon w(λ) defned and holomorphc for λ 0 such that (16) lm w(λ)λ = 1, λ (17) λs(λ) = 1 + w(λ) 2 = λ ( ) λ w(λ) S f w(λ) for λ 0. (We emphasze that we do mean S on the LHS and S f on the RHS.) () The measure µ s the free multplcatve convoluton of µ f wth the semcrcle law of mean 0 and varance 1.

13 A LLN FOR FINITE-RANGE DEPENDENT RANDOM MATRICES 13 Proof. () Put (18) w(λ) = f(c)p(dc) λ Ψ(c, λ) = Ψ(c, λ)p(dc), thus defnng a holomorphc functon n the doman λ > 2A such that w(λ) = 1 ( ) 1 λ + O λ 2, Ψ(c, λ) = w(λ)f(c). By defnton P(dc) S f (λ) = λ f(c), whch s a functon holomorphc n the doman λ > A/2. For λ 1 we have by (15) and the frst equalty n (18) that λs(λ) = λs(λ) 1 = λp(dc) λ w(λ)f(c) = w(λ)f(c)p(dc) λ w(λ)f(c) = w(λ)2, λ ( ) λ w(λ) S f, w(λ) whch proves the result. () We return to the settng of 2.4. Let W (N) be the N 2 -by-n 2 matrx wth entres W (N) j = 2 δj/2 Ỹ {c (N),c (N) j } /N. Note that W (N) belongs to the GOE (the factor 2 δj/2 s needed for orthogonal nvarance). Let Λ (N) be the N 2 -by-n 2 determnstc dagonal matrx wth dagonal entres Λ (N) = f(c (N) ). Then we have X (N) /N = Λ (N) W (N) Λ (N). Furthermore, as N, the emprcal dstrbuton of egenvalues of W (N) (resp., (Λ (N) ) 2 ) tends to the semcrcle law (resp., µ f ). Fnally, snce W (N) and Λ (N) are asymptotcally freely ndependent, see [HP00, Corollary 4.3.6] and the dscusson on page 157 there concernng the extenson from the untary to orthogonal case, µ has the clamed form of free multplcatve convoluton Analyss of (NE+SE+SW+NW)-fltered Wgner matrx. We consder the setup of 2.3 n the specal case mentoned n We are thus consderng the model mentoned n the ntroducton. Proposton 3.6 apples, wth µ f equal to the law of 2 cos 2 θ = 1 + cos2θ wth θ dstrbuted unformly n [0, 2π). Integratng, we get S f (λ) = 1 2π 2π 0 dθ λ 1 cos2θ = 1. λ(λ 2) It follows from (17) that (1 + w 2 ) 2 λ = λ 2w, and hence (after takng out an rrelevant factor of w) 2w 4 λw 3 + 4w 2 2λw + 2 = 0.

14 14 GREG ANDERSON AND OFER ZEITOUNI In turn, after formng the resultant of lefthand sde above and 1 + w 2 λs wth respect to w and takng out rrelevant factors, we get the equaton (19) 4λ 2 S 4 λ 3 S 3 λ 2 S 2 + λs + 1 = 0. Snce the latter s quartc, t can prncple be solved explctly by root extractons. We omt the detals, whch are avalable from the authors. We can wthout great dffculty fnd the boundary of the support of µ, as follows. The dscrmnant (λ) of the left sde of (19) relatve to S s (λ) = 16λ 6 (8λ λ ). The zeroes of (λ) are the only obstructons to analytc contnuaton of S n the λ-plane. The only nonzero real roots of (λ) are ± (approxmately ± ), 4 and so these have to be the endponts of the nterval n whch µ s supported. We fnally remark that t can be shown that dµ/dx has a spke 1/ x at the orgn. 4. The man lmt calculaton To the end of provng Theorem 2.5, we frst prove the followng result. Proposton 4.1. Let Assumpton hold. For each postve nteger k, lm N k/2 1 E trace((x (N) ) k ) = EM π. N π W k The proof requres some preparaton and wll not be completed untl Notaton, termnology and strategy (N, k)-words. Let N and k be postve ntegers. An (N, k)-word s by defnton a functon : {1,...,k} {1,...,N}. To each (N, k)-word we attach a random varable where X (N) = k α=1 X (N) (α),(η k (α)) η k = (12 k) S k. (Here and below we employ cycle notaton for permutatons.) We have EX (N) = E trace((x (N) ) k ), where the sum on the left s extended over (N, k)-words.

15 A LLN FOR FINITE-RANGE DEPENDENT RANDOM MATRICES The set partton assocated to an (N, k)-word. Gven an (N, k)-word, consder the graph G K = (V K, E K ), where V K = {1,..., k} and E K = {{α, β} {1,..., k} (α) (β) K}. Here and below K s the constant fgurng n Assumpton 2.2.1(II). We defne π to be the set partton of k the parts of whch are the equvalence classes for the relaton α and β belong to the same connected component of G K. (Although the set partton π depends on K, we suppress reference to K n the notaton.) We have (α) (β) K π (α) = π (β), for all α, β {1,...,k}. (α) (β) > Kk π (α) π (β), Dstngushed (N, k)-words. Let be an (N, k)-word. Consder the followng condtons: (I) π W k. (II) For all A π we have (mn A) Q (N) (k+1)k. (III) For all dstnct A, B π we have (mn A) (mn B) > 5kK. (IV) max max (α) (β) Kk. A π α,β A k (V) mn (α) (η k(α)) > 3Kk. α=1 (VI) For α = 1,...,k we have (α) Q (N) K. (VII) For all A π and α, β A, the numbers (α) (β) N and N belong to the same nterval of the partton I. If condtons (I,II,III) hold we say that s dstngushed, n whch case automatcally also satsfes condtons (IV,V,VI,VII) Strategy. We wll show that the only nonneglgble contrbutons to (20) lm N k/2 1 EX (N) N = lm N k/2 1 E trace((x (N) ) k ) N : (N, k)-word come from dstngushed (N, k)-words. Then we wll evaluate EX (N) for dstngushed. Fnally, we wll calculate the lmt on the left wth the summaton restrcted to dstngushed Neglgblty of nondstngushed (N, k)-words. Lemma Let π be a set partton of k. There exsts C π > 0 such that for every postve nteger N the sum E X(N) extended over (N, k)-words such that π = π does not exceed C π N π. Proof. By Assumpton 2.2.1(Ib) and the Hölder nequalty, t suffces smply to estmate the number of (N, k)-words such that π = π. A crude estmate of the latter s (1 + 2Kk) k π N π. Lemma Let π be a set partton of k. Let be an (N, k)-word such that π = π. If π k/2 + 1 and EX (N) 0, then π W k (and hence π = k/2 + 1).

16 16 GREG ANDERSON AND OFER ZEITOUNI Proof. Let G π = (V π, E π ) be the graph assocated to π. For each α {1,..., k} put e(α) = {π(α), π(α + 1)} E π. Now fx e E π. It s enough to show that e = e(α) for at least two α {1,..., k}, for then, snce G π s connected, we have k/2 + 1 π = V π E π + 1 k/2 + 1, and hence π s a Wgner set partton. To derve a contradcton, suppose rather that e = e(α) for unque α {1,...,k}. For every γ {1,...,k} let (γ) j(γ) be the ntegers (γ) and (η k (γ)) rearranged. Then for every β {1,..., k} \ {α} we have max( (α) (β), j(α) j(β) ) > K, for otherwse e(α) = e(β). It follows by Assumpton 2.2.1(IIb) that the random varable X (N) (α),(η k (α)) s ndependent of the rest of the random varables appearng n the product X (N), and hence EX (N) = 0 by Assumpton 2.2.1(Ia). Ths contradcton proves the lemma. Lemma Let π be a Wgner set partton. There exsts C π > 0 such that for every postve nteger N the sum E X(N) extended over (N, k)-words such that π = π but s not dstngushed does not exceed C π Nk/2. Proof. The proof s smlar to that of Lemma We omt the detals Evaluaton of EX (N) for dstngushed Defntons of τ π and σ π. Let π be a set partton of k. Enumerate π and ts parts thus: π = {I 1,..., I π }, mn I 1 < < mn I π, I α = { α1 < < α, Iα } for α = 1,..., π. Put τ π = ( 11 1, I1 ) ( π,1 π, I π ) S k, σ π = η 1 k τ π S k. By constructon π(τ π ()) = π() for = 1,...,k. Lemma Let π W k be a Wgner set partton. Then the permutaton σ π s fxed-pont-free and squares to the dentty. Furthermore, for all dstnct, j {1,..., k}, we have {π(), π( + 1)} = {π(j), π(j + 1)} j = σ π (). Proof. Let G π = (V π, E π ) be the graph (tree) assocated to π. For each e E π, there are by Lemma 3.1.5() exactly two ndces such that e = {π(), π( + 1)}, and σ π swaps them by Lemma 3.1.5(). Lemma Let π W k be a Wgner set partton. Put σ = σ π and τ = τ π. Then we have EX (N) = α {1,...,k} s.t. α σ(α) for every (N, k)-word such that π = π. EX (N) (α),(τ(σ(α))) X(N) (σ(α)),(τ(α)) Proof. Put η = η k. By defnton we have ησ = τ. By the prevous lemma we have τ σ = η. Therefore after rearrangng the product X (N) = X (α),(η(α)), α {1,...,k}

17 A LLN FOR FINITE-RANGE DEPENDENT RANDOM MATRICES 17 we have X (N) = α {1,...,k} s.t. α σ(α) X (N) (α),(τ(σ(α))) X(N) (σ(α)),(τ(α)). It suffces to prove enough ndependence to justfy pushng the expectaton under the product. For every γ {1,...,k} let (γ) j(γ) be the ntegers (γ) and (η(γ)) rearranged. By defnton of π and the precedng lemma, for all dstnct α, β {1,...,k}, f max( (α) (β), j(α) j(β) ) K, then β = σ(α). By Assumpton 2.2.1(IIb) we deduce the desred ndependence The dfference operator assocated to a set partton π. Let π be a set partton of k. Let τ = τ π S k be the canoncally assocated permutaton. Let f : {1,...,k} Z be a functon. We defne π f : {1,..., k} Z by the formula ( π f)() = f() f(τ()) for = 1,...,k. Gven a functon g : {1,..., k} Z, the equaton π f = g has a soluton f : {1,..., k} Z f and only f α A g(α) = 0 for all parts A π, and f s unque up to the addton of a functon {1,...,k} Z constant on every part of π. Lemma Let π W k be a Wgner set partton. Put σ = σ π, τ = τ π and = π. Fx α {1,..., k} such that α σ(α). We have ( ) (21) EX (N) (α),(τ(σ(α))) X(N) (σ(α)),(τ(α)) = s (mn π(α)) ( )(α),( )(σ(α)) N, (mn(π(σ(α)))) N for every dstngushed (N, k)-word such that π = π. In partcular, the expectaton n queston vanshes unless K by Assumpton 2.2.1(IIa). Proof. Let E(α) be the left sde of (21). Assume at frst that (α) (τ(σ(α)). Then (α) + 3Kk < (τ(σ(α)) by property (V) of a dstngushed (N, k)-word, and hence (τ(α)) + Kk < (σ(α)) by property (IV) of a dstngushed (N, k)-word. So we have (22) E(α) = EX (N) (α),(τ(σ(α))) X(N) (τ(α)),(σ(α)) = s ( )(α),( )(σ(α)) ( ) (α) N, (τ(σ(α))) N by Assumpton 2.2.1(IIc) and property (VI) of a dstngushed (N, k)-word. Then we deduce that the desred formula holds by Assumpton 2.1.4(III) and property (VII) of a dstngushed (N, k)-word. Assume next and fnally that (α) (τ(σ(α)). Reasonng as above we have E(α) = EX (N) (σ(α)),(τ(α)) X(N) (τ(σ(α)),(α) = s ( )(σ(α)),( )(α) ( (σ(α)) N ), (τ(α)) N. Now we apply the symmetry (1), and then contnue to reason as above. We deduce the desred formula just as before Contrbuton of EX (N) to the lmt for dstngushed. Lemma Let π W k and an (N, k)-word be gven such that the followng hold: For all A π we have (mn A) Q (N) (k+1)k. For all dstnct A, B π we have (mn A) (mn B) > 5Kk. K. Then π = π and (hence) π s dstngushed.

18 18 GREG ANDERSON AND OFER ZEITOUNI No proof s needed, but ths pont bears emphass as an mportant step n the proof of Proposton 4.1. Lemma Fx a Wgner set partton π W k. Then we have (23) EM π = lm N N k/2 1 EX (N), where the sum s extended over dstngushed (N, k)-words such that π = π. Proof. Let = π, σ = σ π, and τ = τ π. Let {t A } (resp., {z A }) be an..d. famly of random varables unform n [0, 1] (resp., S 1 ), ndexed by fnte nonempty sets A of postve ntegers. We further suppose that the famles {t A } and {z A } are defned on a common probablty space and are ndependent. We denote expectatons wth respect to these varables by E. We have by Lemma and the defntons that EM π = E ( ) s mn (t π(), t π(σ()) )zπ() m zn π(σ()) = = {1,...,k} s.t. σ() f:{1,...,k} Z m Z n Z E {1,...,k} s.t. σ() f:{1,...,k} Z [ K,K] A π, P A f()=0 E s f(),f(σ()) (t π(), t π(σ()) ) {1,...,k} s.t. σ() A π s f(),f(σ()) (t π(), t π(σ()) ). P A z f() A At the last step we ntegrate out the z s and take nto account Assumpton 2.2.1(IIa). We then have EM π = E s ( f)(),( f)(σ()) (t π(), t π(σ()) ), f:{1,...,k} Z f K A π f(mn A)=f 0(A) {1,...,k} s.t. σ() where f 0 : π Z s any fxed functon defned on the parts of π. After some straghtforward bookkeepng whch we omt, t follows by the precedng two lemmas that (23) holds wth the summaton on restrcted to dstngushed (N, k)-words such that π = π and K. But then by Assumpton 2.2.1(IIa), the lmt does not change f we drop the restrcton K (the further terms all vansh), whence the result Completon of the proof of Proposton 4.1. The sute of lemmas proved n 4.3 shows that the lmt on the left sde of equaton (20) does not change f we restrct attenton to dstngushed (N, k)-words. The last lemma above evaluates the lmt on the left sde of (20) wth restrcted to dstngushed (N, k)-words, and gves the desred value. The proof of Proposton 4.1 s complete. 5. Completon of the proof of Theorem 2.5 Fx a postve nteger k. As n [AZ06, pf. of Thm. 3.2, Secton 6, p. 305], Theorem 2.5 wll follow as soon as we can prove that (24) Var( L (N), x k ) N 0.

19 A LLN FOR FINITE-RANGE DEPENDENT RANDOM MATRICES 19 We wll prove ths by lghtly modfyng the arguments of 4.3. Now (25) Var( L (N), x k ) = N ( ) k 2 E[X (N) X (N) j ] E[X (N) ]E[X (N) j ] (,j) where the sum s extended over pars (,j) of (N, k)-words. By Assumpton 2.2.1(Ib) and the Hölder nequalty, t s (more than) enough to show that the number of pars (,j) makng a nonzero contrbuton to the sum on the rght sde of (25) s O(N k+1 ). Now fx a par (,j) of (N, k)-words ndexng a nonzero term n the sum on the rght sde of (25). Let j be the (N, 2k)-word obtaned by concatenatng and j,. e., { (α) f α {1,...,k}, j(α) = j(α k) f α {k + 1,...,2k}. Consder the set partton π = π j defned n We need also to consder a graph assocated to π. The relevant graph s no longer G π, but rather a slghtly modfed verson G π = (Ṽπ, Ẽπ), where Ṽπ = π and Ẽ π = {{π(1), π(2)},..., {π(k), π(1)}} {{π(k+1), π(k+2)},..., {π(2k), π(k+1)}}. By constructon G π comes equpped wth two walks, namely π(1),..., π(k), π(1) and π(k + 1),..., π(2k), π(k + 1). Argung as n the proof of Lemma 4.3.2, we fnd that nonvanshng of the term on the rght sde of (25) ndexed by (,j) mples that the walks jontly vst every edge of Gπ at least twce, and moreover there must exst some edge of Gπ vsted by both walks. Thus Ẽπ k and G π s connected. It follows that π = Ṽπ k + 1. Fnally, argung as n the proof of Lemma 4.3.1, we fnd that the number of nonzero terms on the rght sde of (25) s ndeed O(N k+1 ). The proof of Theorem 2.5 s complete. 6. An algebracty crteron In ths secton, whch s completely ndependent of the precedng ones, we develop a soft method for provng that a holomorphc functon s algebrac under hypotheses commonly encountered n random matrx theory Formulaton of an algebracty crteron Notaton. Let {X } =1 be ndependent (algebrac) varables. Let C[X 1,...,X n ] denote the rng of polynomals n X 1,..., X n wth coeffcents n C. We vew C[X 1,..., X n ] as a subrng of C[X 1,...,X n+1 ]. Gven F C[X 1,...,X n ], let F(0) C be the result of settng X 1 = = X n = 0. Let C(X 1,..., X n ) be the feld of ratonal functons n the varables X 1,...,X n,. e., the rng consstng of fractons A/B where A, B C[X 1,...,X n ] and B does not vansh dentcally. We say that F C(X 1,...,X n ) s defned at the orgn f F = A/B wth A, B C[X 1,..., X n ] such that B(0) 0, n whch case we put F(0) = A(0)/B(0) DIRE famles. Let ϕ 1,...,ϕ N be a fnte famly of holomorphc functons each defned n a connected open neghborhood of the orgn n C n ; the domans need not be the same. We wll say that ϕ 1,..., ϕ N are defned mplctly by ratonal equatons (DIRE for short) f there exst Φ 1,...,Φ N C(X 1,..., X n+n ) such that (I) Φ s defned at the orgn and Φ (0) = 0 for = 1,...,N, (II) (det N Φ,j=1 X j+n )(0) 0, and

20 20 GREG ANDERSON AND OFER ZEITOUNI (III) Φ (z 1,...,z n, ϕ 1 (z) ϕ 1 (0),..., ϕ N (z) ϕ N (0)) = 0 for = 1,...,N and z = (z 1,..., z n ) C n suffcently near the orgn. The relatonshp between ths defnton and the mplct functon theorem for holomorphc functons [Ca63, Proposton 6.1] s close. Indeed, gven Φ 1,..., Φ N C(X 1,...,X n+n ) wth propertes (I,II) above, the mplct functon theorem for holomorphc functons says that there exst holomorphc functons ϕ 1,..., ϕ N each defned n a connected open neghborhood of the orgn such that (III) holds, and the theorem further asserts unqueness of these functons n the sense that f ψ 1,...,ψ N are holomorphc functons each defned n a connected open neghborhood of the orgn n C n and also satsfyng (III), then for = 1,...,N there exsts a neghborhood of the orgn n whch ψ and ϕ dffer by a constant. In the sequel, for brevty, when we say ϕ 1,..., ϕ N s an n-varable DIRE famly, ths s short for the asserton that ϕ 1,..., ϕ N are holomorphc functons each defned n some connected open neghborhood of the orgn n C n whch together have the property of beng defned mplctly by ratonal equatons Algebrac functons. A holomorphc functon ϕ defned n a nonempty open subset D C n s called an n-varable algebrac functon f there exsts a notdentcally-vanshng polynomal F = F(X 1,..., X n+1 ) C[X 1,...,X n+1 ] such that F(z 1,..., z n, ϕ(z)) = 0 for all z = (z 1,...,z n ) D. If D s connected and U D s any nonempty open subset, then algebracty of F n U mples algebracty of F n D by the prncple of analytc contnuaton. The man result of Secton 6 s the followng. Theorem 6.2. Let ϕ 1,...,ϕ N be an n-varable DIRE famly. Then each ϕ s an n-varable algebrac functon. The proof takes up the last several subsectons of Secton 6. Before turnng to the proof we gve key examples of DIRE famles, and descrbe technques for constructng new DIRE famles from old. Proposton 6.3. Fx a postve nteger l and put L = 2l+1. For z = (z 1,..., z L ) C L such that L =1 z < 1 and ntegers j = 1,...,L, consder the quantty (26) ϑ j (z) = δ j,l π 2π 0 exp( (j l 1)x)dx 1 L k=1 z k exp((k l 1)x), whch depends holomorphcally on z and vanshes for z = 0. Then the famly ϑ 1,..., ϑ L can be extended to an L-varable DIRE famly ϑ 1,...,ϑ N. Proof. Fx z = (z 1,..., z L ) C L such that L =1 z < 1. Let p = [p j ],j Z be the unque matrx of complex numbers wth rows and columns ndexed by Z wth the followng propertes: p j = 0 for all and j such that j > l. p j = z j +l+1 for all and j such that j l. Let p be the Z-by-Z matrx wth entres p j. We have the crude estmate ( L ) t (27) ( p t ) j z holdng for all postve ntegers t. =1

21 A LLN FOR FINITE-RANGE DEPENDENT RANDOM MATRICES 21 Now supposng p were a Markov matrx (whch of course t s not), we could vew each entry p j as a transton probablty P(S t+1 = j S t = ) = p j, for a random walk {S t } t=0 on Z wth step-length bounded by l and we would have, for any subset M Z t, an equalty P((S 1,..., S t ) M S 0 = 0 ) = p 0 1 p t 1 t. ( 1,..., t) M Because t s a valuable ad to ntuton, we wll make the lne above a defnton. Notce that by (27) the sum on the rght s absolutely convergent, and that the sum of the absolute values of the terms s ( L =1 z ) t. We wll be able to calculate usng the usual rules of probablty, provded we never nvoke postvty p j 0 or the Markov property j p j = 1. We are nterested n functons of z descrbable n the language of random walk because, as we explan presently, the functons ϑ j belong to ths class, and moreover wthn ths class we can easly fnd the extra functons needed to extend ϑ 1,...,ϑ L to a DIRE famly. Let U, V, A, B, C be l-by-l matrces wth complex entres defned as follows: U j = t=1 P(S t = j, mn t u=0 S u > 0 S 0 = ), V j = t=1 P(S t = j, max t u=0 S u < l + 1 S 0 = ), A j = P(S 1 = j + l S 0 = ), B j = P(S 1 = j S 0 = ), C j = P(S 1 = j l S 0 = ). Let W, D be L-by-L matrces wth complex entres defned as follows: W j = t=1 P(S t = j S 0 = ), D j = P(S 1 = j S 0 = ). By (27), all the seres n queston here converge absolutely and defne functons of z holomorphc n the doman L =1 z < 1. Notce that A, B, C, D depend lnearly on z, and that all matrces A, B, C, D, U, V, W vansh for z = 0. Consder the expanson of the ntegrand of (26) by geometrc seres and then ntegrate term by term. One fnds n ths way a seres expresson for ϑ j dentcal to the seres expresson defnng some entry of the matrx W. In short, every ϑ j appears n W. By breakng paths down accordng to vsts to the sets { l,..., 1}, {0} and {1,..., l}, we obtan n the usual way recursons (28) 0 = U + B + A(1 + U)C(1 + U), 0 = V + B + C(1 + V )A(1 + V ), C(1 + V )A = W + D (1 + W). 0 0 A(1 + U)C Extend the gven famly ϑ 1,..., ϑ n to an enumeraton ϑ 1,..., ϑ N of all entres of U, V and W. Rewrte the system of equatons (28) as a system of N polynomal equatons n z 1,..., z L, ϑ 1,..., ϑ N, n order to fnd polynomals such that Φ (X 1,..., X L+N ) C[X 1,..., X L+N ] for = 1,...,N Φ (0,...,0, X L+1,..., X L+N ) = X +L, Φ (z 1,..., z L, ϑ 1,..., ϑ N ) = 0.

22 22 GREG ANDERSON AND OFER ZEITOUNI Note that part (II) of the defnton of DIRE famly holds because X j+l (0) = δ j. Thus ϑ 1,..., ϑ N s ndeed a DIRE famly Natural operatons on DIRE famles. We wrte down some lemmas whch wll be helpful n applyng the noton of DIRE famly. The frst three are trval but deserve beng stated for the sake of emphass. The last s the trck decsve for the applcaton of Theorem 6.2 to the proof of Theorem 2.6. Lemma Let ϕ 1,..., ϕ N and ψ 1,..., ψ M be n-varable DIRE famles. Then the concatenaton ϕ 1,..., ϕ N, ψ 1,..., ψ M s an n-varable DIRE famly. Lemma Let ϕ 1,...,ϕ N be an n-varable DIRE famly. Let ϕ N+1 be a C- lnear combnaton of ϕ 1,..., ϕ N. Then the extended famly ϕ 1,...,ϕ N+1 s an n-varable DIRE famly. Lemma Let ϕ 1,...,ϕ N be an n-varable DIRE famly. Let A be an n by m matrx wth complex entres. Identfy C n and C m wth spaces of column vectors. Then there exsts an m-varable DIRE famly ψ 1,..., ψ N such that for = 1,...,N we have ψ (z) = ϕ (Az) for all z C m suffcently near the orgn. Lemma Let ψ 1,...,ψ n be a famly of holomorphc functons defned n an open dsk centered at the orgn n C. Let ϕ 1,...,ϕ N be an n 0 -varable DIRE famly, where N n n 0. Assume that ψ (z) = zϕ (zψ 1 (z),...,zψ n (z)) for = 1,...,n and z C suffcently near the orgn. Then ψ 1,..., ψ n can be extended to a 1-varable DIRE famly. Proof. By the precedng lemma we may assume wthout loss of generalty that n = n 0. For = n + 1,...,n + N the formula ψ (z) = ϕ n (zψ 1 (z),...,zψ n (z)) ϕ n (0) defnes a holomorphc functon ψ n some open neghborhood of the orgn n C. We wll prove that the extended famly ψ 1,..., ψ N+n s a 1-varable DIRE famly. Note that all the functons ψ vansh at the orgn. Let Φ 1,...,Φ n+n C(X 1,...,X N+n ) be wth respect to ϕ 1,..., ϕ N as called for by the defnton of an n-varable DIRE famly. For = 1,...,N + n defne Ψ C(X 1,..., X N+n+1 ) by the formula = Then: Ψ (X 1,..., X n+n+1 ) { X+1 X 1 (X +n+1 + ϕ (0)) f 1 n, Φ n (X 1 X 2,..., X 1 X n+1, X n+2,...,x n+n+1 ) f n + 1 n + N. (I) Ψ s defned at the orgn and Ψ (0) = 0 for = 1,...,N + n, (II) (det N+n Ψ,j=1 X j+1 )(0) 0, and (III) Ψ (z, ψ 1 (z),..., ψ N+n (z)) = 0 for = 1,...,N + n and z C suffcently near the orgn. In other words, Ψ 1,..., Ψ n+n are wth respect to ψ 1,..., ψ n+n as called for by the defnton of 1-varable DIRE famly. We next formulate a purely algebrac result and explan how to deduce Theorem 6.2 from t. Φ