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Semicircle law for a matrix ensemble with dependent entries 4 Winfried Hochsta¨ttler, Werner Kirsch 1 0 Fakulta¨t fu¨r Mathematik und Informatik 2 FernUniversita¨t in Hagen, Germany b e F Simone Warzel 4 Zentrum Mathematik 2 Technische Universita¨t Mu¨nchen, Germany ] h p - h Abstract t a m We study ensembles of random symmetric matrices whose entries ex- hibit certain correlations. Examples are distributions of Curie-Weiss-type. [ Weprovide acriterion on thecorrelations ensuring thevalidity of Wigner’s 3 semicircle law for the eigenvalue distribution measure. In case of Curie- v 6 Weissdistributions thiscriterion applies abovethecriticaltemperature (i.e. 3 β < 1). We also investigate the largest eigenvalue of certain ensembles of 6 Curie-Weisstypeandfindatransition initsbehavioratthecriticaltempera- 6 . ture. 1 0 4 1 : v 1 Introduction i X r Inthisarticleweconsider random matricesXN oftheform a X (1,1) X (1,2) ... X (1,N) N N N X (2,1) X (2,2) ... X (2,N) N N N XN =  .. .. ..  (1) . . .    X (N,1) X (N,2) ... X (N,N)  N N N     The entries X (i,j) are real valued random variables varying with N. N WewillalwaysassumethatthematrixX issymmetric, suchthat N 1 X (i,j) = X (j,i) foralli,j. Furthermore wesuppose that allmoments N N oftheX (i,j)existandthatE(X (i,j)) = 0andE(X (i,j)2) = 1. N N N Itisconvenienttoworkwiththenormalized versionA ofX ,namely N N with 1 A = X (2) N N √N As A is symmetric it has exactly N real eigenvalues (counting multi- N plicity). Wedenote themby λ λ ... λ 1 2 N ≤ ≤ ≤ anddefinethe(empirical) eigenvalue distribution measureby N 1 σ = δ N N λj j=1 X anditsexpectedvalueσ ,thedensityofstatesmeasureby N N 1 σ = E δ . N N λj j=1 X   If the random variables X (i,j) are independent and identically dis- N tributed (i.i.d.) (except for the symmetry condition X (i,j) = X (j,i)) N N then itis well known that the measures σ and σ converge weakly to the N N semicircledistribution σ (almostsurelyinthecaseofσ ). Thesemicircle sc N distributionisconcentratedontheinterval[ 2,2]andhasadensitygivenby − σ (x)= 1 √4 x2 forx [ 2,2]. ThisimportantresultisduetoEugen sc 2π − ∈ − Wigner[23]andwasprovedbyArnold[3]ingreatergenerality, seealsofor example[17],[18]or[2]. Recently, there was a number of papers considering random matrices with some kind of dependence structure among their entries, see for example [7], [13], [12] and [20]. In particular the papers [6], [10] and [11] consider symmetric random matrices whose entries X (i,j) andX (k,ℓ)are N N independent if they belong to different diagonals, i.e. if i j = k ℓ , | − | 6 | − | but may be dependent within the diagonals. It was in particular the work [11] which motivated the current paper. Among other models Friesen and Lo¨we [11] consider matrices with independent diagonals and (independent copies of) Curie-Weiss distributed random variables on the diagonals. (For adefinition oftheCurie-Weissmodelseebelow). The main example for the results in our paper is a symmetric random matrix whose entries X (i,j) are Curie-Weiss distributed for all i,j (with N i j). The models considered in this paper also include the Curie-Weiss ≤ 2 modelondiagonalsinvestigated byFriesenandLo¨we. Forthereader’scon- venience we define our Curie-Weiss ensemble here, but we’ll work with abstract assumptions inthefollowingtwochapters. In statistical physics the Curie-Weiss model serves as the easiest non- trivial model of magnetism. There are M sites with random variables X i attached to the sites i taking values +1 (”spin up”) or 1 (”spin down”). − Each spin X interacts with all the other spins prefering to be aligned with i theaveragespin 1 X .Moreprecisely: M j=i j 6 P Definition1 Random variables X withvalues in 1,+1 are j j=1,...,M { } {− } distributed according to a Curie-Weiss law P with parameters β 0 β,M ≥ (calledtheinverse temperature) andM N(calledthenumberofspins)if ∈ Pβ,M(X1 = ξ1,X2 = ξ2,...,XM = ξM) = Zβ−,1M 21M e2βM(Pξj)2 (3) whereξ 1,+1 andZ isanormalization constant. i β,M ∈ {− } For β < 1 Curie-Weiss distributed random variables are only weakly correlated, while for β > 1 they are strongly correlated. This is expressed for example by the fact that a law of large numbers holds for β < 1, but is wrong for β > 1. This sudden change of behavior is called a ”phase transition” in physics. In theoretical physics jargon the quantity T = 1 is β called the temperature and T = 1 is called the critical temperature. More information about the Curie-Weiss model and its physical meaning can be foundin[22]and[8]. Our Curie-Weiss matrix model, which we dub the full Curie-Weiss ensemble, is defined through M = N2 random variables Y (i,j) { N }1 i,j N whichareP -distributed. ToformasymmetricmatrixwesetX (≤i,j)≤= β,M N Y (i,j)fori j andX (i,j) = Y (j,i)fori > j anddefine N N N ≤ 1 A = X N N √N By the diagonal Curie-Weiss ensemble we mean a symmetric random matrixwiththerandomvariablesonthekth diagonal i,i+k beingP - β,N { } distributed (0 k N 1 and 1 i N k) and with entries on ≤ ≤ − ≤ ≤ − different diagonals being independent. This model was considered in [11]. Forβ < 1wewillprovethesemicircle lawforthesetwoensembles. In the following section we formulate our general abstract assumptions andstatethefirsttheoremofthispaperwhichestablishes thesemicirclelaw forourmodels. TheprooffollowsinSection3. In Section 4 we discuss our main example, the full Curie-Weiss model, infactwewillstudy various random matrixensembles associated toCurie- Weiss-like models. In this section we also discuss exchangeable random variables andtheirconnection withtheCurie-Weissmodel. 3 In Section 5 we investigate the largest eigenvalue (and thus the matrix norm)ofCurie-Weiss-typematrixensemblesbothbelowandabovethecrit- icalvalueβ = 1. AcknowlegmentIt is a pleasure to thank Matthias Lo¨we, Mu¨nster, and WolfgangSpitzer,Hagen,forvaluablediscussion. Twoofus(WKandSW) would like to thank the Institute for Advanced Study in Princeton, USA, wherepartofthisworkwasdone,forsupport andhospitality. 2 The semicircle law Definition2 Suppose I is a sequence of finite index sets I . A { N}N N N family X (ρ) of ran∈dom variables indexed by N N and (for givenN{)bNythe}sρe∈tIIN,Niscalledan I -schemeofrandomva∈riables. Ifthe N N { } sequence I isclearfromthecontext wesimplyspeakofascheme. N { } To define an ensemble of symmetric random matrices we start with a ‘quadratic’ scheme of random variables Y (i,j) with I = (i,j) 1 i,j N anddefinethematrixe{ntrNiesX}((ii,,jj)∈)IbNyX (i,Nj) = N N { | ≤ ≤ } Y (i,j)fori j andX (i,j) = Y (j,i)fori > j. N N N ≤ Remark3 Todefine thesymmetric matrix X itwould beenough tostart N witha‘triangular’ schemeofrandom variables, i.e.onewith I = (i,j) 1 i j N , thus with M = 1N(N + 1) random N { | ≤ ≤ ≤ } 2 variablesinsteadofM = N2 variables. Toreducenotationalinconvenience we decided to use the quadratic schemes. In a slight abuse of language we willnolonger distiguish innotation between the random variables Y (i,j) N and their symmetrized version X (i,j). We will always assume that the N random matriceswearedealingwitharesymmetric. In this paper we consider schemes X (i,j) of random vari- ableswithN = 1,2,...andI = (i,j){ 1N i,j}(i,Nj)∈INwiththefollowing N { | ≤ ≤ } property: Definition4 Ascheme X (i,j) iscalledapproximatelyuncorre- lated,if { N }(i,j)∈IN ℓ m C E X (i ,j ) X (u ,v ) ℓ,m (4) N ν ν N ρ ρ (cid:12)  (cid:12) ≤ Nℓ/2 (cid:12) ν=1 ρ=1 (cid:12) (cid:12) Y Y (cid:12) (cid:12)  ℓ (cid:12) (cid:12) (cid:12) (cid:12) E X (i ,j )2 1(cid:12) 0 (5) N ν ν (cid:12) ! − (cid:12) → (cid:12) νY=1 (cid:12) (cid:12) (cid:12) for all sequences (i1(cid:12),j1),(i2,j2),...,(iℓ,jℓ) whi(cid:12)ch are pairwise disjoint (cid:12) (cid:12) anddisjointtothesequence(u ,v ),...,(u ,v )withN-independentcon- 1 1 m m stantsC . ℓ,m 4 Notethatforanyapproximately correlatedschemethemeanasymptotically vanishes, E(X ((i,j))) C N 1/2 by (4), and the variance is asymp- N 1,0 − | | ≤ totically one, E(X (i,j)2) 1 by (5). Moreover, by (4) we also have N → sup E X (i,j)2k < forallk. N,i,j N ∞ The main examples we have in mind are schemes of Curie-Weiss- dis- (cid:0) (cid:1) tributed random variables (fullordiagonal) withinverse temperature β 1 ≤ (fordetailsseeSection4). Theorem5 If X (i,j) isanapproximatelyuncorrelatedscheme { N }1 i j N of random variables then t≤he≤e≤igenvalue distribution measures σ of the N corresponding symmetricmatrices A (asin(2))converge weaklyinprob- N abilitytothesemicirclelawσ ,i.e.forallbounded continuous functions f sc onRandallε > 0wehave P f(x)dσ (x) f(x)dσ (x) > ε 0. N sc − → (cid:18)(cid:12)Z Z (cid:12) (cid:19) (cid:12) (cid:12) Inparticular,w(cid:12)eprovetheweakconvergenceofthe(cid:12)densityofstatesmeasure (cid:12) (cid:12) σ to the semicircle law σ . In Section 4 wediscuss various examples of N sc approximately uncorrelated schemes. 3 Proof of the semicircle law The proof is a refinement of the classical moment method (see for example[2]). Wewillsketchtheproofemphasizingonlythenewingredients. As in[2],Theorem5followsfromthefollowingtwopropositions. Proposition 6 Forallk N: ∈ 1 C fork even E trAk k/2 (6) N → 0 fork odd (cid:26) (cid:16) (cid:17) whereC = 1 2k denotetheCatalannumbers. k k+1 k The right hand si(cid:0)de(cid:1)of (6) gives the moments of the semicircle distribution σ . Infact,thisproposition impliestheweakconvergence ofthedensity of sc statesmeasuresσ toσ . N sc Proposition 7 Forallk N: ∈ 1 2 C2 fork even E trAk k/2 . (7) N2 → 0 fork odd (cid:20) (cid:21) (cid:26) (cid:16) (cid:17) 5 ObservethatProposition 6andProposition 7together implythat 2 2 1 1 E trAk E trAk 0 (8) N − N → " # (cid:18) (cid:19) (cid:20)(cid:18) (cid:19)(cid:21) which allows us to conclude weak convergence in probability from weak convergence intheaverage(see[2]). Foraproof ofthe abovepropositions, whichcan befound in thesubse- quentsubsections, wewrite N 1 1 trAk = X (i ,i )X (i ,i ) ... X (i ,i ) N 1 2 N 2 3 N k 1 N N1+k/2 i1,i2X,...,ik=1 N 1 = X (i) (9) N N1+k/2 i1,i2X,...,ik=1 whereweusedtheshorthandnotation X (i) = X (i ,i )X (i ,i ) ... X (i ,i ) (10) N N 1 2 N 2 3 N k 1 for i= (i ,i ,...,i ). The associated k + 1-tupel (i ,i ,...,i ,i ) con- 1 2 k 1 2 k 1 stitutes a Eulerian circuit through the graph (undirected, not necessary i G simple) with vertex set V = i ,i ,...,i and an edge between the ver- i 1 2 k { } tices v and w whenever v,w = i ,i for some j = 1,...k with the j j+1 { } { } understanding that i = i , a convention we keep for the rest of this pa- k+1 1 per. More precisely, the number of edges ν(v,w) linking the vertex v and thevertexwisgivenby ν(v,w) = # m v,w = i ,i . (11) m m+1 { | { } { }} Let us call edges e = e parallel if they link the same vertices. An edge 1 2 6 whichdoes nothave aparallel edgeiscalled simple. So,ifelinks v andw, theneisasimpleedgeiffν(v,w) = 1. Thegraph maycontainloops,i.e. i G edges connecting avertexv withitself. Byaproper edge wemeananedge which is not a loop. We set ρ(i) = # i ,i ,...,i the cardinality of the 1 2 k { } vertexsetV ,i.e.,thenumberof(distinct)verticestheEuleriancircuitvisits. i We also denote by σ(i) the number of simple edges in the Eulerian circuit (i ,i ,...,i ,i ). Withthisnotation wecanwrite(9)as 1 2 k 1 k 1 1 trAk = X (i) N N N1+k/2 r=1i:ρ(i)=r X X k k 1 = X (i). (12) N N1+k/2 r=1s=0 ρ(i)=r XX X σ(i)=s 6 The sum extends over all Eulerian circuits with k edges and vertex set V 1,2,...,N .Tosimplifyfuturereferences weset i ⊂ { } S = E[X (i)] (13) r,s N | | ρ(i)=r X σ(i)=s and S = E[X (i)] (14) r N | | ρ(i)=r X Obviouslyρ(i)andσ(i)areintegerswith1 ρ(i) k and ≤ ≤ 0 σ(i) k. Forρ(i) = r <N thereare N Nr choicesforthevertex ≤ ≤ r ≤ setV . Moreover, i (cid:0) (cid:1) # i ρ(i)= r η Nr (15) k { | }≤ whereη isthenumberofequivalenceclassesofEuleriancircuitsoflengthk. k WecalltwoEulerian circuits (i ,i ,...,i ,i )and(j ,j ,...,j ,j )with 1 2 k 1 1 2 k 1 corresponding vertex sets V and V equivalent if there is a bijection ϕ : i j V V suchthatϕ(i ) = j forallm. i j m m → 3.1 Proof of Proposition 6 Weinvestigate theexpectation valueofthesum(12). Lemma8 Forallk NthereissomeD < suchthatforallN: k ∈ ∞ S = E[X (i)] D Nr s/2. (16) r,s N k − | | ≤ ρ(i)=r X σ(i)=s Proof. The assertion follows using (4) from the estimate E[X (i)] N | | ≤ D N s/2 together with(15). k − Evidently, in case r s/2 < 1+k/2 the term 1 S vanishes in − Nk/2+1 r,s tehe limit. This is in particular the case if r < k/2+1. If r > k/2+1 we usethefollowingproposition whichisoneofthekeyideasofourproof: Proposition 9 Let = (V,E) denote aEulerian graph withr = #V and G k = #E, and let t be a positive integer such that r > k +t then has at 2 G least2t+1simpleproperedges. WenotethefollowingCorollary toProposition 9. 7 Corollary 10 Foreachk-tupleiwehave ρ(i) σ(i)/2 k/2+1 . − ≤ Moreoverρ(i) σ(i)/2 = k/2+1iffρ(i)= k/2+1andσ(i)= 0. − Proof(Corollary 10). Setr = ρ(i)ands = σ(i). Ifr k/2+1theassertion isevident. ≤ Ifr > k/2+1thereissomet Nsuchthat ∈ k k +t < r +t+1. 2 ≤ 2 Proposition 9henceimplies s 1 k 1 k r r t + < +1. − 2 ≤ − − 2 ≤ 2 2 2 Postponing the proof of Proposition 9, we continue to prove Proposi- tion6. FromLemma8andCorollary10welearnthat 1 S 0 (17) Nk/2+1 r,s → unlessbothr = k/2+1ands = 0. Thusitremainstocomputethenumberofk-tuplesi,withρ(i)= 1+k/2 such that thecorresponding graph isa ‘doubled’ planar tree. ThereareC k 2 differentrootedplanartreeswith k (simple)edges,whereC = 1 2ℓ are 2 ℓ ℓ+1 ℓ theCatalannumbers(seee.g. [19],Exercise6.19e,p.219-220). Eachindex (cid:0) (cid:1) i ischosen fromtheset 1,...,N . Aswehave1+k/2different indices ν { } thereare N! suchchoices. Fromthisoneseesthat (N 1 k/2)! − − k 1 1 lim E trAk = lim E (X (i)) (18) N N N N N1+k/2 →∞ (cid:16) (cid:17) →∞ Xr=1 ρX(i)=r 1 = lim E (X (i)) N N N1+k/2 →∞ ρ(i)=1+k/2,σ(i)=0 X 1 N! = lim C = C . N N1+k/2 (N 1 k/2)! k2 k2 →∞ − − ThisendstheproofofLemma6modulotheproofoftheProposition 9. For futurepurpose,wenotethattheaboveproofalsoshowstheslightlystronger assertion. 8 Corollary 11 Forallk N: ∈ 1 C fork even lim E[X (i)] = k/2 (19) N N1+k/2 | N | 0 fork odd →∞ i (cid:26) X wheretheabovesum(in(19))extendsoverallEuleriancircuitsoflengthk. Foraproofwenotethattheleading contribution inthesum(18)isnon- negative. Thesubleading termswerealreadyshowntovanish. Proof (Proposition 9). If the graph = (V,E) contains loops, wedelete G all loops and call the new graph (V,E˜). This graph is still Eulerian and satisfies#V > #E˜/2+t.Thuswithoutlossofgenerality wemayassume that(V,E)containsnoloops. Weproceedbyinductiononthenumberofedgeswithmultiplicitygreater than one. If there is no such edge, then the number of simple edges is k. Since is Eulerian wehave k r and r > k +t implies k > 2t and thus G ≥ 2 theassertion. Hence, assume there exists an edge of multiplicity m 2. If the graph ≥ that arises from the deletion of2 copies ofthis edge is still connected, then the resulting graph is Eulerian and denoting its number of edges by k we ′ have k k ′ r > +t +t+1 2 ≥ 2 Thus, by inductive assumption, we find at least 2t + 3 edges without parallels inthereduced graphandhenceatleast2t+2in . G We are left with the case that the removal of the edges disconnects the graph into two Eulerian graphs = (V ,E ) and = (V ,E ). We use 1 1 1 2 2 2 G G theabbreviations r := #V andk := #E . Then: i i i i k k +1 k +1 1 2 r = r +r > +t = + +t 1 2 2 2 2 Hencewecanpartition tintointegerst ,t suchthat 1 2 k k 1 2 r > +t and r > +t 1 1 2 2 2 2 If, say t 0, then t t and the inductive assumption yields at least 1 2 ≤ ≥ 2t+1simpleedgesin andhencein . 2 G G Otherwisewefindatleast2t +1simpleedgesineachofthe andthus i i G intotal2t+2 > 2t+1suchedgesin . G 9 3.2 Proof of Proposition 7 Wewritetheexpectation value 1 2 1 E trAk = E X (i)X (j) (20) N2 N2+k N N (cid:20)(cid:16) (cid:17) (cid:21) Xi,j (cid:2) (cid:3) wherethesumextendsoverallpairsofEuleriancircuits (i ,...,i ,i )and 1 k 1 (j ,...,j ,j ) of length k with vertex sets V and V in 1,...,N . We 1 k 1 i j { } distinguish twocases. In case V V = the union of the corresponding Eulerian graphs i j ∩ 6 ∅ isconnected andeach vertexhasevendegree. Therefore thisunion i j G ∪G is itself aEulerian graph with 2k edges. The corresponding contribution to the sum (20) is then estimated by extending the summation to all Eulerian circuits ℓoflength2k: 1 1 C E X (i)X (j) E[X (ℓ)] k . N2+k N N ≤ N2+k | N | ≤ N Xi,j (cid:12) (cid:2) (cid:3)(cid:12) ℓ=(ℓX1,...,ℓ2k) e Vi∩Vj6=∅(cid:12) (cid:12) (21) ThelastestimateisduetoCorollary11. IncaseV V = weusethefollowinganalogue ofLemma8. i j ∩ ∅ Lemma12 Forallk NthereissomeD < suchthatforallN: k ∈ ∞ E X (i)X (j) D Nr1+r2 s1/2 s2/2, (22) N N k − − ≤ Vi∩XVj=∅ (cid:12) (cid:2) (cid:3)(cid:12) ρ(i)=r1,ρ(j)=r2 (cid:12) (cid:12) σ(i)=s1,σ(j)=s2 where the sum extends over non-intersecting pairs of Eulerian circuits of lengthk. TheproofmirrorsthatofLemma8. From Lemma 8 we know that r s /2 k/2+1 and likewise r 1 1 2 − ≤ − s /2 k/2+1. Sotheuniquepossibility that 2 ≤ s +s 1 2 r +r k+2 1 2 − 2 ≥ giving rise to a non-vanishing term in the limit, is that r = r = k/2 + 1 2 1 and s = s = 0. Similarly as in the proof of Lemma 6 we con- 1 2 clude that in this case i and j constitute disjoint ’doubled’ planar trees and E X (i)X (j) 1 by assumption (5). The proof of Lemma 7 is con- N N → cluded using the same arguments relating the number of planar trees to the (cid:2) (cid:3) Catalannumbers. 10