Random band matrices in the delocalized phase, II: Generalized resolvent estimates

Transcription

1 Random band matrces n the delocalzed phase, II: Generalzed resolvent estmates P. Bourgade F. Yang H.-T. Yau J. Yn Courant Insttute bourgade@cms.nyu.edu U. of Calforna, Los Angeles fyang75@math.ucla.edu Harvard Unversty htyau@math.harvard.edu U. of Calforna, Los Angeles jyn@math.ucla.edu Ths s the second part of a three part seres abut delocalzaton for band matrces. In ths paper, we consder a general class of N N random band matrces H = H j whose entres are centered random varables, ndependent up to a symmetry constrant. e assume that the varances E H j form a band matrx wth typcal band wdth N. e consder the generalzed resolvent of H defned as GZ := H Z, where Z s a determnstc dagonal matrx such that Z j = z + z > δj, wth two dstnct spectral parameters z C + := {z C : Im z > 0} and z C + R. In ths paper, we prove a sharp bound for the local law of the generalzed resolvent G for N 3/4. Ths bound s a ey nput for the proof of delocalzaton and bul unversalty of random band matrces n []. Our proof depends on a fluctuatons averagng bound on certan averages of polynomals n the resolvent entres, whch wll be proved n [0]. The model and the results... Tools for the proof of Theorem Proof of Theorem Propertes of M... 6 The model and the results.. The model. Our goal n ths paper s to establsh estmates on Green s functons whch were used n the proof of delocalzaton conjecture and bul unversalty for random band matrces. All results n ths paper apply to both real and complex band matrces. For smplcty of notatons, we consder only the real symmetrc case. Random band matrces are characterzed by the property that the matrx element H j becomes neglgble f dst, j exceeds the band wdth. e shall restrct ourselves to the conventon that, j Z N = Z N/, N/], and j s defned modular N. More precsely, we consder the followng matrx ensembles. Defnton. Band matrx H N wth bandwdth N. Let H N be an N N matrx wth real centered entres H j :, j Z N whch are ndependent up to the condton H j = H j. e say that H N s a random band matrx wth typcal bandwdth = N f s j := E H j = f j. The wor of P.B. s partally supported by the NSF grant DMS# The wor of H.-T. Y. s partally supported by NSF Grant DMS and a Smons Investgator award. The wor of J.Y. s partally supported by the NSF grant DMS#559.

2 for some non-negatve symmetrc functon f : Z N R + satsfyng x Z N fx =,. and there exst some small postve constant c s and large postve constant C s such that c s x fx C s x Cs,, j Z N..3 The method n ths paper also allows to treat cases wth exponentally small mass away from the band wdth e.g. fx C s e cs x /. e wor under the hypothess.3 manly for smplcty. e assume that the random varables H j have arbtrarly hgh moments, n the sense that for any fxed p N, there s a constant µ p > 0 such that max E H j p /p µ p Var H j /.4,j unformly n N. In ths paper, we wll not need the followng moment condton assumed n Part I of ths seres []: there s fxed ε m > 0 such that for j, mn j E ξ 4 j E ξ 3 j N εm, where ξ j := H j s j / s the normalzed random varable wth mean zero and varance one. All the results n ths paper wll depend on the parameters c s, C s n.3 and µ p n.4. But we wll not trac the dependence on c s, C s and µ p n the proof. Denote the egenvalues of H N by λ... λ N. It s well-nown that the emprcal spectral measure N N = δ λ converges almost surely to the gner semcrcle law wth densty ρ sc x = 4 x +. π The am of ths paper s to estmate the generalzed resolvent Gz, z of H N defned by Gz, z := zi H N 0, z, z C + R,.5 0 zi N N where C + denotes the upper half complex plane C + := {z C : Im z > 0}. The generalzed resolvent s an mportant quantty used n Part I of ths seres []. The ey pont of ths generalzaton, compared wth the usual resolvent, s the freedom to choose dfferent z and z. To the best of our nowledge, the local law for ths type of generalzed resolvent has only been studed n the precedng paper [], where t was assumed that cn for some constant c > 0. To understand the role of the generalzed resolvent, we bloc-decompose the band matrx H N and ts egenvectors as A B wj H N =, ψ B D j :=, p j where A s a gner matrx. From the egenvector equaton Hψ j = λ j ψ j, we get Q λj w j = λ j w j, Q e := A B D e B. Thus w j s an egenvector of Q e := A B D e B wth egenvalue λ j when e = λ j. A ey nput to the proof of unversalty and QUE for random band matrces s an estmate on the Green s functon of Q e. Snce some egenvalues of D can be very close to e, the matrx D e can be very sngular. It s thus very dffcult f possble to estmate the Green s functon of Q e drectly. On the other hand, the Green s functon of Q e s just the mnor of the generalzed resolvent Gz, e of H N, whch we fnd to be relatvely more doable. Due to the need n Part I, we wll consder generalzed resolvent for a general class of band matrces. More precsely, we ntroduce the followng Defnton.. Here and throughout the rest of ths paper, we wll use the notaton that for any a, b Z, a, b := [a, b] Z.

3 Defnton. Defnton of H g ζ. For any suffcently small ζ > 0 and any g = g, g,, g N R N, H ζ and H g ζ wll denote N N real symmetrc matrces satsfyng the followng propertes. The entres H ζ j are centered and ndependent up to the symmetry condton, satsfy.4, and have varances E H ζ j = s ζ j := s j ζ + δ j,j,, where s j,, j Z N, satsfy the condtons n Defnton.. Then the matrx H g ζ H g ζ j := H ζ j g δ j. s defned by e denote by S 0 and Σ the matrces wth entres S 0 j = s j and Σ j = +δj,j,, respectvely. Then the matrx of varances s S ζ := S 0 ζσ, S ζ j = s ζ j.. The results. The generalzed resolvent G g ζ z, z of Hg ζ s defned smlarly as n.5 by G g ζ z, z := H g ζ zi 0. 0 zi N N Defne M g ζ z, z N as the soluton vector to the system of self-consstent equatons = M g ζ z, z = z, z /, g s ζ j M g ζ jz, z,.6 for z, z C + R and Z N, wth the constrant that M 0 0 z, z = m sc z + 0 +, where m sc denotes the Steltjes transform of the semcrcle law j m sc z := z + z 4, z C +..7 The exstence, unqueness and contnuty of the soluton s gven by Lemma.3 below. For smplcty of notatons, we denote by M g ζ z, z the dagonal matrx wth entres M g ζ j := M g ζ δ j. e wll show that M g ζ z, z s the asymptotc lmt of the generalzed resolvent Gg ζ z, z. e now lst some propertes of M g ζ needed for the proof of local law stated n Theorem.4. Its proof s delayed to Secton 4. Lemma.3. Assume Re z κ and z κ for some small constant κ > 0. Then there exst constants c, C > 0 such that the followng statements hold. Exstence and Lpschtz contnuty If then there exst M g ζ z, z, Z N, whch satsfy.6 and max ζ + g + z z c,.8 M g ζ z, z m sc z C ζ + g + z z..9 If, n addton, we have ζ + g + z z c, then max M g ζ z, z M g ζ z, z C g g + z z + z z + ζ ζ..0 3

4 Unqueness The soluton vector M g ζ z, z N to.6 s unque under.8 and the constrant = max M g ζ z, z m sc z c. e now state our results on the generalzed resolvent of H g ζ. In ths paper, we wll always use τ to denote an arbtrarly small postve constant ndependent of N, and D to denote an arbtrarly large postve constant ndependent of N. Defne for any matrx X the max norm X max := max X j.,j The notatons η, η and r n next theorem were used n Assumptons.3 and.4 of Part I of ths seres []. Ther meanngs are not mportant for ths paper and the reader can smply vew them as some parameters. In ths paper, all the statements hold for suffcently large N and we wll not repeat t everywhere. Theorem.4 Local law. Defne a set of parameters wth some constants ε, ε > 0: η := N ε, η := N ε r := N ε +3ε, T := N ε +ε, 0 < ε ε /0.. Fx any e < κ for some constant κ > 0. Then for any determnstc z, ζ, g satsfyng and, ε, ε satsfyng Re z e r, η Im z η, 0 ζ T, g 3/4,. { 6 log N max 7 + ε, } 4 ε + ε,.3 we have that for any fxed τ > 0 and D > 0, P G gζ z, e M gζ z, e max N τ + N / N D..4 Im z In fact, the last estmate holds under the weaer assumpton { 3 log N max 4 + ε, + ε + ε }..5 e wll refer to the frst statement,.e.,.4 under the assumpton.3, as the wea form of ths theorem, and the statement.4 under assumpton.5 as the strong form. Ths paper gves a full and self-contaned proof for the wea form, whch helps the reader understand the basc strategy of our proof. On the other hand, the proof for the strong form s much more nvolved, and we nclude a substantal part nto a separate paper [0]. Only the strong form of Theorem.4 was used n part I of ths seres [], where we too log N > 3/4, ε < /4 and ε to be a suffcently small constant. The man purpose of ths part and part III [0] of ths seres s to prove the above Theorem.4. In fact, the bound.4 s almost optmal under our settng n the sense that t at least gves the correct sze of E G g ζ j for j up to an N τ factor. Ths sharp bound s very mportant for the proof of the complete delocalzaton of egenvectors and the bul unversalty of random band matrces n part I []. As explaned there, the bound must be of order o/n to allow the applcaton of the so-called mean feld reducton method, whch was ntroduced n [] and s the startng pont of ths seres. Compared wth the local law for regular resolvents, the man dffculty n provng the local law for the generalzed resolvents s due to the small and even vanshng magnary part of z. As a result, some ey nputs, such as ard s dentty see 3. for the regular resolvents estmates are mssng. In fact, as dscussed before, the case Gz, z max = could occur when z = e s real. Ths dffculty has already appeared n the case cn n [], where some uncertanty prncple was ntroduced to solve ths problem. Unfortunately, ths method seems dffcult to apply n the N case. Instead, n ths paper, we shall use a totally dfferent strategy,.e, the T -equaton method, whch was ntroduced n [4]. Moreover, we have to mprove the nducton bootstrap argument used n [4], as explaned below. e remar that the proofs of the wea form and strong form of Theorem.4 are completely parallel, except that we wll apply a stronger T -equaton estmate Lemma.4 than the one Lemma.8 used n the proof of the wea form. e shall gve a smple proof of the wea T -equaton estmate usng the standard fluctuaton averagng mechansm as n the prevous proof of local semcrcle law [5, 8]. The proof of the strong T -equaton estmate s based on an mproved and substantally more nvolved fluctuaton averagng result, whose proof s delayed to part III of ths seres [0]. 4

5 .3 Setch of proof. In the followng dscusson, for two random varables X and Y, we shall use the notaton X Y f for any fxed τ > 0, X N τ Y wth hgh probablty for large enough N. e defne the T matrx wth entres T j := S G j, G G g ζ, S S ζ,.6 th a standard self-consstent equaton estmate see Lemma., one can show that G M max T max, M M g ζ..7 Our proof of Theorem.4 s based on an nducton argument combned wth a self-consstent T -equaton estmate as explaned below. e ntroduce the followng notaton: G z, z := max G j z, z, Λz, z := G M j max z, z..8 N Fx z and Re z = e. e perform the nducton wth respect to the magnary part of z. Defne a sequence of z n such that Im z n = N nε Im z, Re z n = e, for small enough constant ε > 0. In the n = 0 case wth Im z 0 = Im z, usng the methods n [5, 8], we can obtan the local law.4 for Gz, z 0. Suppose we has proved the local law for Gz, z n : Λz, z n Φ goal, Φ goal := + N / Im z..9 Then wth Im z n = N ε Im z n and a smple but qute sharp up to an N ε factor L -estmate, we get a bound on the n-th level: G z, z n N Φ, Φ := N ε Φ goal,.0 whch gves a rough bound Φ 0 by the self-consstent equaton estmate.7: T max z, z n C s G z, z n Φ 0 Λz, z n Φ 0, Φ 0 := N Φ,. where C s s the constant from.3. Note that Φ s very close to Φ goal, whle Φ 0 s not. Now wth the strong T -equaton estmate see Lemma.4, one can get an mproved bound Φ on T as follows: N T max z, z n Φ Λz, z n Φ, Φ := Φ goal + Im z + N Φ + N / Φ 0,. where we used.7 to get a better bound Λz, z n Φ. th.5, one can verfy that Φ Φ goal + N ε Φ 0 for some constant ε > 0. After at most l := /ε many teratons wth. and.7,.e. Φ 0 Φ Φ l, we can obtan the local law.9 for Gz, z n, whch s used as the nput for the next nducton. The ey pont of ths nducton argument s that one has a good L -bound.0 nherted from the local law on the upper level, and ths L -bound can be used n the T -equaton estmate. to gve an mproved bound for Λz, z n on ths level. Fnally, after fntely many nductons n n, we can obtan the local law.4 for, say, Gz, e + N 0. Then wth a contnuty argument, we can prove the local law.4 for Gz, e. In Fg., we llustrate the flow of the nducton argument wth a dagram. e remar that the above nducton argument s not a contnuty argument, as used e.g. n the wors [5, 7, 8] on local semcrcle law of regular resolvents. The multplcatve steps Im z n N ε Im z n that we made are far too large for a contnuty argument to wor. The man reason for choosng ths multplcatve step s that the T -equaton estmate can only be appled for O number of tmes due to the degrade of the probablty set see Remar.9. The man dffculty of our proof les n establshng the T -equaton estmate.. The startng pont s a self-consstent equaton for the T matrx,.e. the T -equaton, see.4 below. In ths paper, we focus on provng the stablty of the T -equaton,.e. boundng S M S max n.4, where we abbrevate S S ζ. For regular resolvent of generalzed gner matrces.e. z = z, ζ = 0 and g = 0, we have 5

6 L bound Λ Φ goal z 0 G Φ Λ Φ 0 Λ Φ Λ Φ goal z S T+S T+S T+S L bound Λ Φ goal z n G Φ Λ Φ 0 Λ Φ Λ Φ goal z n S T+S T+S T+S Fgure : The dagram for the nducton argument wth respect to n. At each level n, we obtan the local law.9, whch gves the rough bound Φ 0 on level n through.0 and.. Applyng. and.7 teratvely, one can mprove the ntal bound Φ 0 to the sharp bound Φ goal. In the dagram, S stands for an applcaton of the self-consstent equaton estmate.7, and T+S stands for an applcaton of the T -equaton estmate. followed by a self-consstent equaton estmate.7. M c Im z for some constant c > 0. However, n our general settng and n partcular when Im z s small, we actually have M > and S M l l >. Therefore, the usual Taylor expanson approach cannot be used n fact, t s not even easy to see that s outsde the spectrum of M S. In ths paper, we wll establsh the followng bound S M S max = O Im z + N. One mportant component for the proof s the estmate M c Im z for some constant c > 0. To see ths bound s useful, we can ntutvely vew M S n as an n-step nhomogeneous random wal on Z N wth annhlaton, where the average annhlaton rate s Im z/n by the above bound. Ths shows that we can explore some decay propertes of M S n as n ncrease, whch may gve some useful bounds on the Taylor expanson of S M. However, our proof actually wll not follow ths heurstc argument, see Secton 4. Fnally, to fnsh the proof of the strong verson of the T -equaton estmate Lemma.4, we need a fluctuaton averagng results for a quantty of the form N E, where E s are some polynomals of the generalzed resolvent entres. The proof nvolves a new graphcal method and we nclude t n part III of ths seres [0]. Tools for the proof of Theorem.4 The basc strategy to prove Theorem.4 s to apply the self-consstent equaton estmate: Lemma., and the T -equaton estmate: Lemma.8 or.4, n turns. e collect these results n ths secton, and use them to prove Theorem.4 n next secton. For smplcty, we wll often drop the superscrpts ζ and g from our notatons. In partcular, G and M are always understood as G g ζ and M g ζ, whle H and S are understood as H ζ and S ζ n the rest of ths paper. In the proof, for quanttes A N and B N, we wll use the notatons A N = OB N and A N B N to mean that A N C B N and C B N A N C B N, respectvely, for some constant C > 0. 6

7 . The self-consstent equaton estmate. The self-consstent equaton estmate s the startng pont of almost every proof of the local law of the generalzed resolvents of random matrces. e now state the self-consstent equaton estmate for our model. Lemma. Self-consstent equaton estmate. Suppose that Re z κ for some constant κ > 0. Then there exsts constant c 0 > 0 such that f ζ + g + z z c 0, then the followng statement holds. If there exst some fxed δ > 0 and some determnstc parameter Φ / such that Gz, z Mz, z max N δ, T max Φ,. n a subset Ω of the sample space of the random matrces, then for any fxed τ > 0 and D > 0, P Ω Gz, z Mz, z max N τ Φ N D.. Note that by the defnton of T -matrx n.6, we have T max Gz, z Mz, z max + O. Hence we can always choose Φ = ON δ n.. The proof of Lemma. follows the standard dea of usng a vector-level self consstent equaton method [5,8]. In preparaton for the proof, we recall the followng defnton of mnors. Defnton. Mnors. For any N N matrx A and T {,..., N}, we defne the mnor of the frst nd A [T] as the N T N T matrx wth A [T] j := A j,, j / T. For any N N nvertble matrx B, we defne the mnor of the second nd B T as the N T N T matrx wth B T j = B [T],, j / T, whenever B [T] s nvertble. Note that we eep the names of ndces when defnng the mnors. By defnton, for any sets U, T {,..., N}, we have A [T] [U] = A [T U], B T U = B T U. For convenence, we shall also adopt the conventon that for T or j T, j A [T] j = 0, B T j = 0. For T = {a} or T = {a, b}, we shall abbrevate {a} a and {a, b} ab. Remar.3. In prevous wors, e.g. [6,8], we have used the notaton for both the mnor of the frst nd and the mnor of the second nd. Here we try to dstngush between and [ ] n order to be more rgorous. The followng denttes were proved n Lemma 4. of [8] and Lemma 6.0 of [6]. Lemma.4 Resolvent denttes. For an nvertble matrx B C N N and / {, j}, we have and B j Moreover, for j we have B j = B j B + B B j B, = B B,.3 B B B B B = B B B l B l..4,l j = B B B j = B jj B j B j..5 The above equaltes are understood to hold whenever the expressons n them mae sense. 7

8 Snce the N τ factor and the N D bound for small probablty event appear very often n our proof, we ntroduce the followng notatons. Defnton.5. For any non-negatve A, we denote O τ A := ON Oτ A. e shall say an event E N holds wth hgh probablty w.h.p. f for any fxed D > 0, PE N N D for suffcently large N. Moreover, we say E N holds wth hgh probablty n Ω f for any fxed D > 0, for suffcently large N. PΩ \ E N N D The followng lemma gves standard large devaton bounds that wll be used n the proof of Lemma.. Lemma.6 Lemma 3.5 of [9]. Let X be a famly of ndependent random varables and b, B j be determnstc famles of complex numbers, where, j =,..., N. Suppose the entres X satsfy EX = 0, E X = and the bound.4. Then for any fxed τ > 0, we have wth hgh probablty. b X N τ b /, X B j X j N,j τ B j,j The followng lemma provdes estmates on the entres of M S and S M S. It wll be used n the proof of Lemma. and Theorem.4, and ts proof s delayed untl Secton 4. Lemma.7. Suppose that the assumptons for the strong form of Theorem.4,.e.,.,. and.5, hold. If z satsfes Re z = e, 0 Im z Im z, then we have for M M g ζ z, z and S S ζ, and [ M S ] j = {δ j + O, f j log N ON c log N, f j > log N,.6 Now we can gve the proof of Lemma.. S M max S = O Im z + N..7 Proof of Lemma.. The followng proof s farly standard n random matrx theory and we wll omt some detals. For smplcty, we drop ζ and g n superscrpts. Usng.5, we have G j = G H G j for j. Snce the elements n {H } N = are ndependent of G, by the standard large devatons estmates n Lemma.6, we have that for any fxed τ > 0 and D > 0, P G j N τ G S G j / N D, j..8, Snce G n Ω,.8 mples that P Ω G j = O τ S G j N D, j. 8

9 By.3, the defnton of T n.6, and the bound for T n., we have S G j S G j + Therefore, we obtan. for the j case. For the dagonal case, we defne Z := Q H H l G l H. l S G G j G = OΦ n Ω. Usng.4,.3, the off-dagonal case for. we just proved, and the standard large devatons estmates n Lemma.6, we can get that for any fxed τ > 0, G = z, z /, g j S j G jj Z + O τ Φ, wth Z = O τ Φ, holds wth hgh probablty n Ω. th the defnton of M n.6, we have G M = j S j G jj M j + O τ Φ, w.h.p. n Ω, whch mples M G = j M S j G jj M j + O τ Φ + O max G M, w.h.p. n Ω. e rewrte the above estmate as M S j G jj M j = O τ Φ + O max G M. j Then wth.6 and the frst bound n., we can get. for the dagonal entres and complete the proof of Lemma.. The T -equaton estmate. A ey component for the proof of Theorem.4 s the self-consstent equaton for the T varables. It leads to a self-mproved bound on G M max. Ths nd of approach was also used n [4] to prove a wea type delocalzaton result for random band matrces. To help the reader understand the proof, we frst prove a wea T -equaton estmate,.e. Lemma.8, whch wll gve the wea form of Theorem.4. The stronger T -equaton estmate wll be stated n Lemma.4, and ts proof s put n the companon paper [0]. Lemma.8 ea T -equaton estmate. Under the assumptons of Theorem.4.e.,.,.,.5 and the assumpton on e, the followng statements hold provded ε > 0 s a suffcently small constant. Let z satsfy Re z = e, N 0 Im z Im z,.9 and Φ be any determnstc parameter satsfyng Φ N δ for some fxed δ > 0. Fx some z and z whch can depend on N. If for any constants τ > 0 and D > 0, P Gz, z Mz, z max N τ Φ N D,.0 then for any fxed small τ > 0 and large D > 0, we have P T z, z max N τ Φ w # N D, Φ w # := N Im z + N Φ 3 + N.. 9

10 Furthermore, f the parameter Φ satsfes { Φ mn N +ε +ε, },. N +ε then for any fxed τ > 0 and D > 0 we have Gz, z Mz, z max ΦN 3 ε + N τ + N / Im z wth probablty at least N D..3 Remar.9. The above statements should be understood as follows. For any small constant τ > 0 and large constant D > 0,. and.3 hold f.0 holds for some constants τ, D that depend on τ and D. In general, we need to tae τ < τ to be suffcently small and D > D to be suffcently large. Compared wth Lemma., we lose a much larger porton of the probablty set. Hence Lemma.8 can only be terated for O number of tmes, whle Lemma. can be appled for ON C tmes for any fxed C > 0. Proof of Lemma.8. From the defnng equaton.6 of T, we add and subtract S M T j so that T j = S M T j + S Gj M T j. Therefore, we have T j = [ S M S ] Gj M T j..4 Isolatng the dagonal terms, we can wrte the T -equaton as T j = Tj 0 + [ S M S ] Gj M T j, T 0 j := [ S M S ] Gjj M j j T jj. j.5 By the defnton of T, the assumpton.0 and the estmate.9 on M, we can get the smple bounds G jj = O and T jj = O τ Φ. Applyng these bounds to the defnton of Tj 0, we get [ Tj 0 = O S M S ],.6 j whch wll be shown to be the man term of T j up to an N τ factor. By.7 and the condton. on Im z, we have [ S M S ] = O j Im z + N..7 Defnton.0 E, P and Q. e defne E as the partal expectaton wth respect to the -th row and column of H,.e. E := E H []. For smplcty, we wll also use the notatons P := E, Q := E..8 Usng ths defnton and the bound.7, we rewrte the off-dagonal terms n.5 nto two parts: [ S M S ] Gj M T j j N = Im z + N c E G j M T j + c Q G j, j j.9 where c s a sequence of determnstc numbers satsfyng c := [ S M S ] N Im z + N = ON. The followng two lemmas provde estmates for the two parts n.9, where Lemma. s a standard fluctuaton averagng lemma. 0

11 Lemma.. Suppose that b, Z N, are determnstc coeffcents satsfyng max b = ON. Then under the assumptons of Lemma.8, we have that for any fxed small τ > 0, wth hgh probablty. b E G j M T j = Oτ Φ 3, j Z N,.0 j Proof. By.5 and.0, we have l Then we can obtan that for j, l H l G lj E G j = E M H l G lj wth hgh probablty. Usng.3, we have = G j /G = O τ Φ and G M = O τ Φ w.h.p.. + O τ Φ 3 = M G s l lj G lj = G lj + O τ G l G j = G lj + O τ Φ, l, j, l + O τ Φ 3. wth hgh probablty. Insertng t nto. and usng the defnton.6, we can obtan.0. Lemma.. Suppose that b, Z N are determnstc coeffcents satsfyng max b = ON. Then under the assumptons of Lemma.8, we have for any fxed large p N and small τ > 0, E b Q G j j p N τ Φ 3 p, j ZN.. Proof. Our proof follows the arguments n [5, Appendx B]. e consder the decomposton of the space of random varables usng P and Q defned n.8. It s evdent that P and Q are projectons, P + Q =, P Q = 0, and all of these projectons commute wth each other. For a set A Z N, we denote P A := A P and Q A := A Q. Now fx any j Z N, we set X := Q G j. Then for p N, we can wrte p p p p E b X = c E X s = c E P r + Q r X s j,,..., p s= s= r= = c A,...,A p [] E p s= PA c s Q A s X s, where :=,,..., p, [] := {,,..., p }, means summaton wth ndces not equal to j, and c are determnstc coeffcents satsfyng c = ON p. Then wth the same arguments as n [5] more specfcally, the ones between B.-B.4, we see that to conclude., t suffces to prove that for A Z N \ {j} and any fxed τ > 0, Q A X = O τ Φ A + w.h.p..3 e frst recall the followng smple bound for partal expectatons, whch s proved n Lemma B. of [5]. Gven a nonnegatve random varable X and a determnstc control parameter Ψ such that X Ψ wth hgh probablty. Suppose Ψ N C and X N C almost surely for some constant C > 0. Then for any fxed τ > 0, we have max P X = O τ Ψ w.h.p..4 In fact,.4 follows from Marov s nequalty, usng hgh-moments estmates combned wth the defnton of hgh probablty events n Defnton.5 and Jensen s nequalty for partal expectatons. In the applcaton to resolvent entres, the determnstc bound follows from G Im z N 0 by.9.

12 Now the bound.3 n the case A = follows from.4 drectly. For the case A = n, we assume wthout loss of generalty that j =, = and A = {,..., n + }. It suffces to prove that Usng the dentty.3, we can wrte Q 3 G = Q 3 G 3 + G 3G 3 G 3 G + G 3G 3 = Q 3 33 G 33 Q n+ Q 3 G = O τ Φ n+..5 G 3 G 3 G 3 + G 3 G 3 G 3 + G 33 G 33 G 3 G 3 G 33. G 3 Note that the leadng term Q 3 vanshes snce G 3 s ndependent of the 3rd row and column of H, and the rest of the three terms have at least three off-dagonal resolvent entres. e now act Q 4 on these terms, apply.3 wth = 4 to each resolvent entry, and multply everythng out. Ths gves a sum of fractons, where all the entres n the numerator are off-dagonal and all the entres n the denomnator are dagonal. Moreover, the leadng order terms vansh, Q 4 Q 3 G 34 G 4 3 G4 3 G G 34 G 4 3 G4 3 G 4 33 = 0, and each of the survvng term has at least four off-dagonal resolvent entres. e then contnue n ths manner, and at each step the number of off-dagonal resolvent entres n the numerator ncreases at least by one. Fnally, Q n+ Q 3 G j s a sum of fractons where each of them contans at least n + off-dagonal entres n the numerator. Together wth.4, ths gves the estmate.5, whch further proves.3. Remar.3. Lemma. asserts that the Q operaton yelds an mprovement by a factor Φ. In fact, for the regular resolvents of band matrces, a stronger verson of averagng fluctuaton results was proved n [3]. e beleve that followng the methods there, the bounds n Lemma. and Lemma. can be mproved to O τ Φ 4 + / Φ..6 In ths paper, however, we wll sp the dscusson on the strategy n [3], snce ts proof s rather nvolved, and more mportantly, we wll prove an even stronger bound,.e.,.30 below, n Part III of ths seres [0]. th.6, the Φ w # n. can be mproved to N Φ w # = Im z + N Φ 4 + / Φ + N, and the condton. becomes { Φ mn N +ε +ε, }..7 N +ε Usng ths estmate, the condtons.3 can be weaen to { 4 log N max 5 + ε, ε + ε }..8 Now we fnsh the proof of Lemma.8. Usng.9, Lemma., Lemma. and Marov s nequalty, we can get that [ S M S ] Gj M N T j = Oτ Im z + N Φ 3 j wth hgh probablty. Note that t only ncludes the off-dagonal terms,.e. j terms. Now pluggng t nto the T -equaton.5 and usng.6, we obtan.. Fnally, we need to prove.3. Clearly, f. holds, then Φ N δ and Φ w # N δ for some constant δ > 0. Thus. s satsfed, and then.3 follows from an applcaton of. and Lemma.. Ths completes the proof of Lemma.8.

13 The followng lemma gves a stronger form of Lemma.8. It wll be proved n the companon paper [0]. Here we recall the notaton n.8. Lemma.4 Strong T -equaton estmate. Suppose the assumptons of Theorem.4.e.,.,.,.5 and the assumpton on e and.9 hold. Let Φ and Φ be determnstc parameters satsfyng Φ Φ Φ N δ.9 for some constant δ > 0. Fx some z and z whch can depend on N. If for any constants τ > 0 and D > 0, P Gz, z Mz, z max N τ Φ + P G z, z N +τ Φ N D, then for any fxed small τ > 0 and large D > 0, we have P T z, z max N τ Φ # N D, Φ # := Furthermore, f the parameter Φ satsfes { Φ mn N Im z + N Φ Φ + Φ N / + N..30 N +ε +ε, },.3 N +ε then for any fxed τ > 0 and D > 0 we have Gz, z Mz, z max ΦN 3 ε + N τ + N / Im z.3 wth probablty at least N D. The Remar.9 also apples to ths lemma. Note that.3 or.3 gves a self-mproved bound on G M max, whch explans how we can mprove the estmate on G from Φ to Φ # va T equatons. As long as we have an ntal estmate such that. or.3 holds, we can then terate the proof and mprove the estmate on G to Φ goal = Im z + N / n.4. Proof of Lemma.4. See the proof of Theorem.7 n part III of ths seres [0]. 3 Proof of Theorem.4 Fx a parameter 0 < ε 0 < ε /5. e defne z n := Re z + N nε0 Im z, so that Im z n+ = N ε0 Im z n. The basc dea n provng Theorem.4 s to use mathematcal nducton on n N. The proofs of the wea form and strong form of Theorem.4 are completely parallel. In the followng proof, we wll only remar on the mnor dfferences between them. Step 0: The specal case wth z = z and ζ = 0, g = 0.e. GH, z s the ordnary resolvent of a generalzed gner matrx was proved n [5]. The proof gven there can be carred over to our case wthout changes under the assumptons of Theorem.4 when z = z and Im z +δ for some fxed δ > 0. Ths gves that P Gz, z Mz, z max N τ Im z N D, for any fxed τ > 0. Ths bound s clearly stronger than the one n.4. 3

14 Step : Consder the case n = 0,.e., Gz, z 0, where we have e clam that for any w, w C +, Re z 0 = Re z, Im z 0 = Im z. Gw, w L L mnim w, Im w. 3. To prove t, we frst assume that Im w = a + Im w wth a 0. e wrte where A s a symmetrc matrx. Then Gw, w = A aj Im w, J l =, δ l, A aj Im w A aj Im w = A aj A aj + aim wj + Im w Im w. Obvously, we have a smlar estmate wth Im w replaced by Im w when Im w Im w. Ths proves the clam 3.. Now by the defnton of T and.3, we now T j z, z 0 C s G j z, z 0 = C s Im G jj z, z 0, Im z where n the second step we used the so-called ard dentty that for any symmetrc matrx A and η > 0, Obvously, the same argument gves that R j A, η = Im R jja, η, RA, η := A η. 3. η T z, z 0 t max C s max j Im G jj z, z 0 t, z 0 t := tz + t z 0, t [0, ]. 3.3 Im z Now we clam that for any small enough τ > 0, sup P s [0,] Gz, z 0 t Mz, z 0 t max To prove 3.4, we frst note that for any w, w C, Ths mples that N τ Im z N D. 3.4 Gz, w = Gz, w + Gz, ww w JGz, w, Jl = /, δ l. 3.5 z Gz, z max N Gz, z L L Gz, z max In partcular, n ths step we have N mnim z, Im z G max. s Gz, z 0 t max CN /+ε z z 0 Gz, z 0 t max. 3.6 Ths provdes some contnuty estmate on Gz, z 0 t, whch shows that 3.4 can be obtaned from the followng estmate: Gz, max P z0 N 5 Mz, z 0 N 5 N τ 0,N 5 max N D. 3.7 Im z From Step 0, ths estmate holds for = 0. By nducton, we assume that 3.7 holds for = 0. Then usng 3.6 and.0, we now that the frst estmate of. holds for Gz, z 0 t wth t = 0 + N 5. 4

15 Then by 3.3 and applyng Lemma., we obtan 3.7 for = 0 +. Ths completes the proof of 3.7 and 3.4. Note that the estmate 3.4 appled to Gz, z 0 s the result we want for ths step Step : Suppose that for some n N wth Im z n N 0,.4 holds for Gz, z n and Mz, z n for any large D > 0. e frst prove the followng estmate for Gz, z n+ Mz, z n+, whch s weaer than.4: for any fxed τ > 0. For any w, w C + satsfyng N P Gz, z n+ Mz, z n+ max N τ /+ε 0 Im z + N +ε0 N D 3.8 3/ Re w = Re w, N ε0 Im w Im w Im w, 3.9 usng 3.9 and 3.5, we have G j z, w + w w Gz, w L L G j z, w sup T z, w max Re w=re z n, Im z n+ Im w Im z n + Im w Im w G j z, w 3N ε0 Gz, w, where we have used 3. to bound Gz, w L L. e apply ths nequalty wth w = z n and w satsfyng 3.9. Usng.4 and the defnton.8, we can bound Gz, z n as C N +ε Gz, 0 w = O τ Im z + N +ε sup Re w=re z n, Im z n+ Im w Im z n wth hgh probablty for any fxed τ > 0. e now consder nterpolaton between z n and z n+ : z n,m = z n Im z n Im z n+ mn 50, m 0, N 50. e would le to use Lemma. and nducton to prove that 3.8 holds for Gz, z n,m Mz, z n,m for all m. Frst, we now 3.8 holds for Gz, z n. Then suppose 3.8 holds for Gz, z n,j for all j m. e now verfy that. holds for Gz, z n,m wth Φ = N τ Φ 0 for any fxed τ > 0, where Φ 0 := N +ε0 Im z + N +ε0 3. To ths end, we note that 3.0 already verfes the bound on T z, z n,m max n. for all m 0, N 50. By usng z G max N G max whch follows from 3.5,.0, z n,m z n,m N 50, and 3.0 to bound G max by G, we note that for suffcently small constant δ > 0, Gz, z n,m Mz, z n,m max N δ = Gz, z n,m Mz, z n,m max N δ. Ths proves the frst bound n. for Gz, z n,m. Then Lemma. asserts that. holds for Gz, z n,m wth N τ Φ 0 for any fxed τ > 0. Ths proves 3.8.e. the m = N 50 case by nducton. Step 3: Suppose that for some n N wth Im z n N 0,.4 holds for Gz, z n and Mz, z n for any large D > 0. e have proved that 3.8 and 3.0 hold for Gz, z n+. e now apply Lemma.8 to prove the wea form of Theorem.4. Frst, the condton.0 holds wth Φ = N /+ε 0 + N +ε0. In order for the Im z 3/ condton. to hold, we need whch s satsfed f N /+ε0 Im z + N +ε0 3/ { mn N +ε +ε, max N ε0+ 7 ε, N ε + ε0+ ε. }, 3. N +ε 5

16 If we tae ε 0 < ε,.0 mples.3 under the condton.3. e then apply Lemma.8 agan, and after at most 3/ε teratons we obtan that Gz, z n+ Mz, z n+ max N τ + N /. 3. Im z By nducton on n wth the number of nductons 0/ε 0, the man estmate 3. for Gz, z n holds for all n as long as Im z n N 0. Smlarly, we can apply Lemma.4 to prove the strong form of Theorem.4. As n the prevous argument, 3.8 and 3.0 hold for Gz, z n+ assumng.4 for Gz, z n and Im z n N 0. Therefore, we can choose Φ and Φ as Φ = N /+ε0 Im z + N +ε0 3/, Φ = N ε0 Im z + N /+ε0, where the choce of Φ follows from usng 3.0. It s easy to see that.9 holds. In order to apply Lemma.4, we need.3,.e., N ε 0 + N { /+ε0 mn Im z N +ε +ε, }, N +ε whch s satsfed f max N ε0+ 4 ε, N +ε +ε0+ ε. Clearly, the assumpton.5 guarantees ths condton f we choose ε 0 < ε /. Agan, we can apply Lemma.4 teratvely untl we get 3. for Gz, z n+. The rest of the proof for the strong form of Theorem.4 s the same as the proof for the wea form. Step 4: e now prove.4 for Gz, z wth Im z = 0 by usng contnuty from the estmate for Gz, z wth Im z = N 0 establshed n Step 3. It s easy to see that z Gz, z max z Gz, z max N Gz, z max. 3.3 th 3.3 and usng 3. for Gz, Re z + N 0, we can obtan that sup Gz, Re z + η max = O, 0 η N 0 w.h.p. Then usng.0, 3.5 and 3. for Gz, Re z + N 0, we obtan that.4 holds for Gz, Re z. Remar 3.. If we use the bound n Remar.3 and the condton.7 nstead of., then the restrcton 3. becomes N /+ε 0 Im z + N { +ε0 mn 3/ N +ε +ε, } N +ε whch gves restrcton n.8. So we get a result n between the wea and strong forms of Theorem.4. 4 Propertes of M The man goal of ths secton s to derve some determnstc estmates related to M g ζ, Z N. In partcular, we wll fnsh the proof of Lemma.3 and Lemma.7. 6

17 4. The stablty. The system of self-consstent equatons.6 s a perturbaton of the standard selfconsstent equaton = z m sc m sc for m sc z. Thus our basc strategy s to use the standard perturbaton theory see 4.3 below combned wth a stablty estmate for the self-consstent equaton.e. the operator bound 4.4. e frst recall the followng elementary propertes of m sc, whch can be proved drectly usng.7. Lemma 4.. e have for all z = E + η wth η > 0 that m sc z = m sc z + z. Furthermore, there s a constant c > 0 such that for E [ 0, 0] and η 0, 0] we have c m sc z cη, 4. as well as z m sc z c κ + η /, 4. m scz κ + η, Im m sc z { κ + η f E η κ+η f E, where κ := E denotes the dstance of E to the spectral edges. The followng lemma wll be used n the proof of Lemma.3 and Lemma.7. Recall that S 0 s the matrx wth entres s j, whch s defned n Defnton.. Lemma 4.. Assume Re z κ for some constant κ > 0 and denote m = m sc z Then for any fxed τ > 0, there exst constants c, C > 0 such that m S 0 + τ < c τ L L Furthermore, m S 0 L L C. 4.4 Proof. For some small constant τ > 0 we wrte Assumng 4.3, we get that m S 0 = + τ m S 0 L L + + τ m S 0 + τ τ =0 m S 0 + τ + τ L L j=0 m S 0 + τ + τ whch proves 4.4. e now prove 4.3. Suppose that there s a vector v C N so that v = and [m S 0 + τ v] + τ = ε for some Z N and ε ε N 0 +. Hence j C, L L + τ + τ ε = m 4 b + τm a + τ v b + τ a + τ v + τ + τ, 4.6 where a := S 0 v, b := S 0v and we have used the bounds m, a and b snce S 0 L L =. It wll be clear that the m = case s most dffcult and we wll assume ths condton n the followng proof. Moreover, we assume wth loss of generalty that v > 0 by changng the global phase of v. Now m, 7

18 a and b are complex numbers, and the nequalty 4.6 mples that m 4 b, m a and v have almost the same phases. Snce v, b and a, 4.6 mples that for some constant C > 0 ndependent of ε, v Cε, b m 4 Cε, a m Cε. 4.7 Snce m s a unt modulus complex number wth magnary part of order, we have that δ := m m 4 s a number of order and a b > δ/. Fx the ndex and denote c j := S 0 j, d j := S 0 j. Then j c j = = j d j. Hence 4.7 mples Oε = Reaā = j c j Rev j ā, Oε = Reb b = j d j Rev j b, where Oε denote a postve number bounded by Cε for some constant C > 0 ndependent of ε. For any 0 < r <, denote by A r := {j : Rev j ā r} and let α r := j A r c j. Then we have α r j A r c j Rev j ā = Oε whch mples that j A r c j Rev j ā Oε α r r = α r + r α r r Oε, j A r c j = α r Oεr, Smlarly, f we defne B r := {j : Rev j b r}, then j / A r c j = Oεr. 4.8 j / B r d j = Oεr. 4.9 e clam that f r Cε for some large enough constant C > 0, then A r B r. To see ths, we defne U := {j : j }. By.3 and the defnton of c j, we have c j c s for j U. Clearly, we also have d j c s for j U. Then wth 4.8 and 4.9, we have #{j U \ A r } = Oεr c s, #{j U \ B r } = Oεr c s. If we choose r = Cε for some large enough constant C > 0, then the above two nequaltes mply A r B r, snce U =. Thus there s an ndex j such that Rev j ā r, Rev j b r. 4.0 Snce a, b, v j and a b > δ/, 4.0 s possble only f r δ, whch contradcts the fact that r 0 when ε 0. Ths proves Proof of Lemma.3. th Lemma 4., we can now gve the proof of Lemma.3. Proof of Lemma.3. e frst prove the exstence and contnuty of the solutons to.6. The proof s a standard applcaton of the contracton prncple. Denote by z := z,..., z N, x := x,..., x N and M := M g ζ,..., M g ζ N wth z = z, + z /,, and x x g ζ z, z := M g ζ z, z m, M = x + me, m := m sc z + 0 +, e =,,,. 4. Usng the above notatons and recallng Defnton., we can rewrte.6 nto the followng form m+x = M = z g S 0 M +ζσm = z g S 0 x ms 0 e +ζσx +ζmσe. 4. Subtractng m = z m from the last equaton and usng S 0 e = e, we get that m m + x = g + z z + S 0 x ζmσe ζσx. 8

19 Then 4. s equvalent to [ m S 0 x] = m g + z z + m m + x m + x m ζm 3 Σe ζm Σx. 4.3 Defne teratvely a sequence of vectors x C N such that x 0 = 0 C N and [ m S 0 x +] := m g + z z + m m + x m + x m ζm 3 Σe ζm Σx. 4.4 In other words, 4.4 defnes a mappng h : l Z N l Z N : x + = hx, h x := j m S 0 [ j m g j + z j z + qx j ζm 3 Σe j ζm Σx ] j, 4.5 where qx := m m + x + x m = x m m + x. Note by the assumptons of Lemma.3, c κ m for some constant c κ > 0 dependng only on κ. Then wth 4.4, t s easy to see that there exsts a suffcently small constant 0 < α < c κ /, such that h s a self-mappng h : B r l Z N B r l Z N, B r l Z N := {x l Z N : x r}, as long as r α and ζ + g + z z c r 4.6 for some constant c r > 0 dependng on r. Now t suffces to prove that h restrcted to B r l Z N s a contracton, whch then mples that x := lm x exsts and s a unque soluton to 4.3 subject to the condton x r. From the teraton relaton 4.5, we obtan that x + x = m S 0 [ qx qx ] ζm m S 0 Σx x, 4.7 where qx denotes a vector wth components qx. Usng q 0 = 0 and 4.4, we get from 4.7 that x + x C κ ζ + x + x x x for some constant C κ > 0 dependng only on κ. Thus we can frst choose a suffcently small constant 0 < r < α and then the constant c r > 0 such that C κ c r + r <, and h s a self-mappng on B r l Z N under the condton 4.6. In other words, h s ndeed a contracton, whch proves the exstence and unqueness of the soluton. Note that wth 4.4 and x 0 = 0, we get from 4.5 that th the contracton mappng, we have the bound x =0 x = O z z + ζ + g. x + x x C κ ζ + r = O z z + ζ + g. Ths gves the bound.9. e now prove.0. e have proved above that both M g ζ z, z and M g ζ z, z exst and satsfy.9. Denote by m := m sc z and x g := Mζ z, z m. By 4., we have m m = O z z

20 Then usng 4.3 we can obtan that x x C m S 0 L L { z z [ x + g + z z + x + ζ + x ] + [ g g + z z + z z + ζ ζ + x + ζ + x + x x x ]} C ζ + x + x x x + C g g + z z + z z + ζ ζ. Applyng.9 to both M g ζ z, z and M g ζ z, z, we see that for small enough c, x x C g g + z z + z z + ζ ζ. Together wth 4.8, we obtan.0 as desred. 4.3 Proof of Lemma.7. To prove Lemma.7, t suffces to prove the result for the case g = 0, and we wll descrbe how to relax to the condton g = O 3/4 by usng the Lpschtz contnuty estmate.0 at the end of the proof. In preparaton for the proof, we frst prove the followng lemma. Lemma 4.3. Suppose that g = 0 and the assumptons.,. and.5 hold. Then there exst constants c > 0 and C > 0 such that Mζ 0 n m n c C z z + ζ e, n ZN, 4.9 and where m := m sc z n Z N m M 0 ζ n cim z Im z ζ + O N 3 ε + N ε Im z, 4.0 Proof of Lemma 4.3. Frst wth 4.5 and the fact that S 0 j = 0 f j C s, we get that [ m S 0 ] j δ j = [m m S 0 S 0 ] j = O Therefore wth 4.3, we obtan mmedately that j Cs m S 0 + τ + τ L L [ m S 0 ] j δ j C j c e 4. for some constants c, C > 0. As n the proof of Lemma.3, wth x defned n 4.4, we now that x n = M n m = x n + x + n x n, M n := M 0 ζ n. 4. Recall that we have proved that x n = lm x n n the proof of Lemma.3 above. In partcular, accordng to 4.4, x s gven by [ m S 0 x ] = m z z ζm 3 Σe. 4.3 Then wth 4. and 4.3, one can show that By 4.7 and 4., we have x + x C j j c e x n c n Ce z z + ζ, n ZN. 4.4 [ + δj x j + x j ] x j x j + ζ j, max j, x j x j. By nducton, t s easy to prove that there are constants c, C > 0 such that x + n x n c n Ce z z + ζ

21 Together wth 4.4 and 4., ths mples Ths proves 4.9 snce Mn m M n m. e now prove 4.0. Usng 4.9, we have By defnton 4., Then wth 4.6 we get that n c x n = M n m C z z + ζ e, n ZN. 4.6 n Z N m M n = m n Z N M n = m + Re mx n + x n. m M n + O z z + ζ. 4.7 Mn m = [ Re mxn + x n ] = Re mx n + O z z + ζ. n By. and., we have whch mples that n ζ + Rez z T + r N 3ε /, 0 Im z Im z N ε, ζ + z z ζ + Rez z + Imz z N 3ε / + N ε Imz z. Then usng 4. and 4.5, we obtan that Mn m = n = n Summng 4.3 over, we get that recall that we tae g = 0 n n Re mx n + O N 3 ε + N ε Imz z Re mx n + O N 3 ε + N ε Imz z. 4.8 m x := m z z ζm 3 + = m z z ζm 3 + O, where we used that S 0 j = and Σe = + for,. Thus for the second term n the second lne of 4.8, we have z z m ζm Re mx n = m Re m + O n = m ζ Im z Im z N + O ε Im z + O, Re z where we have used the followng specal propertes of m z when z s a real number, n whch case m z has unt modulus: ma + Re m a + = 0, Im ma+ m a + =, Re m a + 4 a m a + =, a <, a+ := a Here the error O N ε Im z n 4.9 s due to m z mre z C Im z. Insertng 4.9 nto 4.8, we obtan that for some constant c > 0, n whch, together wth 4.7, proves 4.0. Mn m cim z Im z + ζ m + O N 3 ε + N ε Im z,

22 th Lemma 4.3, we now fnsh the proof of Lemma.7. Proof of Lemma.7. e frst assume that g = 0. th 4.3 and a perturbaton argument, we can show that M S + τ < c + τ L L for some constant c > 0. Then.6 can be proved as n 4.. Our man tas s to prove.7. Assume that M Su 0 = v for some vectors u 0, v 0 R N. Multplyng 4.3 wth u 0 M from the left and usng the defnton of S, we obtan that M u 0 + ζ + u 0 + S j u 0 u 0 j = u 0, M v e defne a symmetrc operator H : L T L T, where T := log N 4, log N 4 and wth and H 0 : u, H 0 v = 4,j H := H 0 + H, S j u u j v v j,,j T u, v L T, H : H j := δ j [ M + ζ + ]. For any vector u, we denote by u T the restrcton of u to L T. Then wth 4.9 and the fact that m, we can rewrte 4.3 as Frst we clam that u 0 T, Hu 0 T + S j u 0 4 u 0 j u 0, M v 0 + ON 0 u ,j H c Im zlog N for some constant c > 0. th Temple s nequalty, we have the followng estmate on the ground state energy of H: H E 0 H H φ H φ H φ E H H φ, 4.35 for any φ L T such that φ = and H φ < E H, where E 0 H and E H are the lowest two egenvalues of H. Applyng mn-max prncple to H H 0 H L L, we obtan that By 4.9, we have H L L E H E H 0 H L L = O z z + ζ + Im z. e then clam that for some constant c > 0. Recall that S S ζ = S 0 ζσ wth 4 E H 0 clog N Σ j u u j, u L T, u =.,j T Then agan by mn-max prncple, t suffces to prove the followng lemma. Lemma 4.4. For s j satsfyng.-.3, there exsts a constant c > 0 such that 4 s j u u j clog N 3, u L T, u =, u,,,.,j T

23 e postpone ts proof untl we fnsh the proof of Lemma.7. e now choose the tral state φ as a constant vector n 4.35,.e., φ 0 =,,,. T Then by defnton, H 0 φ 0 = 0 and H φ0 H L L E H by 4.36 and Then by 4.35 and 4.36, we have H H φ0 H L L E H H φ0 H φ0 H L L E H 0 H L L By the defnton of H, we have H φ0 = T = T [ Mn + ζ n, + ] n T m M n + m M n + T n T n T c Im z + ON 0 + m M n + T n Z N c Im zlog N 4 + O N 3 ε + N ε Im z, ζ + T ζ + T where we used 4.9 and m c Im z by 4. n the thrd step, and 4.0 n the last step. Together wth 4.38, H L L = ON 3 ε + N ε Im z and 4.37, ths proves th 4.34, 4.33 gves that for some c > 0, c Im zlog N 4 T u 0 + S j u 0 4 u 0 j u 0, M v 0 + ON 0 u 0.,j Now for some fxed 0 Z N, we choose v 0 = Se 0. Then the above nequalty becomes c Im zlog N 4 T u 0 + S j u 0 4 u 0 j S M u ON 0 u ,j In the followng, we suppose u 0, otherwse the proof s done. Snce for any Z N, u 0 M Su 0 = Se 0 = O, 4.40 we must have Now we decompose u 0 as follows: u 0 = u + ũ, u 0 Su 0. wth u = u 0, N Z N ũ = 0. Suppose u 0 ũ, then we have max u 0 mn u 0. Together wth 4.39, t mples that f u 0 ũ, then u 0 u C Im z. 4.4 On the other hand, f u 0 ũ, wth 4.3, 4.9 and the defnton of S n Defnton., we get that Then n ths case, wth 4.40 and 4.4 t s easy to see that ũ M Sũ = O + ζ + z z u. 4.4 u 0 Su 0 ũ Sũ S 0 ũ

24 By., we have whch mples S 0 ũ j = 0, Usng., for fxed j Z N we have S 0 ũ j S 0 ũ = S 0 x S 0 jy ũ x ũ y j S 0 ũ max S 0 ũ j S 0 ũ. 4.44,j x,y x,y S 0 x S 0 jy ũ x ũ y The lower bound n.3 shows that S 0 has a core,.e., there s a constant c s > 0 such that S 0 xy c s f x y. Then for any fxed j Z N, we choose x 0, x, x,, x n for some n = ON/ such that = x 0 x x x n x n = j, wth /3 x x + /,. Furthermore, set x 0 = x and x n = y. Clearly for any choces of x, n, we have ũ y ũ x = n ũ x ũ x = For our goal, we wll choose x s such that ũ y ũ x CN n ũ x ũ x. = x x /4, x + /4, n. Tang averagng over all x, n, n the above regons, we get that ũ y ũ x N n Average x,x, ũ,x n x ũ x. Note that by our choces, we always have x x and S x x c s for n, whch gves that Average x,x ũ x ũ x 4 ũ x ũ x 8c s = x,x x /4,x +/4 x,x x /4,x +/4 Together wth 4.45, we get that for some constant C > 0, S 0 ũ j S 0 ũ S 0 x S 0 jy CN n x,y + x,y S 0 x S 0 jy CN = x,x x /4,x +/4 For the frst term on the rght-hand sde, we have n = x,x x /4,x +/4 For the terms n the second lne, we notce that x : x x /4 S x x ũ x ũ x + ũ x ũ x C S x x S x x x x x x + x + x C s + ũ x ũ x. ũ x ũ x y : y x n /4,l Z N S l ũ ũ l. ũ y ũ y. 4

25 for all x such that x x /4, where C s s the constant appeared n.3. Then we can subdvde the nterval x, x or x, x nto subntervals wth lengths /, and proceed as above to get S 0 x x x x /4 ũ x ũ x C,l N S l ũ ũ l for some constant C > 0 that s ndependent of the choce of x. In sum, we have obtaned that S 0 ũ j S 0 ũ CN,l N Then from 4.43 and 4.44, we obtan that u 0 CN S l ũ ũ l = CN,l N Pluggng t nto 4.39, we get that f u 0 ũ, then N u0 C,l N S l u 0 u 0 l.,l N S l u 0 u 0 l. S l u 0 u 0 l C u 0 + ON 0 u 0 u 0 CN In sum, by our choce of v 0 = Se 0 and 4.3, we obtan from 4.4 and 4.46 that S M max S C Im z + N, whch completes the proof of.7 n the case wth g = 0. Gven any g R N such that g 3/4, we can wrte M g ζ = M 0 ζ + E, where E s a dagonal matrx wth max E = O g = O 3/4 by the Lpschtz contnuty estmate.0. Then.6 can be obtaned by combng.6 n the case g = 0 wth a standard perturbaton argument. For.7, we wrte S M g ζ S = S M 0 ζ S + S M 0 ζ S M g ζ Mζ 0 S M g ζ S Usng.7 n the case g = 0 and the bound S Mζ 0 L S N S M 0 ζ max S, L we get from 4.47 that S M g ζ S max S Mζ 0 max N S +O Im z + N 3/4. S M g ζ S. max Together wth.5, ths mples.7 for any g such that g 3/4. Proof of Lemma 4.4. Snce the matrx S 0 = s j has a core by.3, t suffces to prove that ŝ j u u j clog N 3, u L T, u =, u,,,, 4.48,j T where ŝ j := j. 5

26 Then we defne the followng two symmetrc operators F 0, : L T L T such that for any u, v L T, u, F 0 v = log N 5,j T, j T u u j v v j, where T denotes the perodc dstance on T, and u, F v = s j u u j v v j,,j T e frst show that for some constant c > 0, s j := ŝ j log N 5 j T. E F 0 clog N 3, 4.49 where E F 0 denotes the second lowest egenvalue of F 0. thout loss of generalty, we can regard F 0 as an operator on L T, C consstng of complex L vectors. Snce F 0 s a perodc operator on L T, C, ts egenvectors are the unt complex vectors wth Fourer components: Then for any p 0, we have w p, F 0 w p = = w p : w p := T e p, T, wth p = πn T, n T. log N 5 log N 5 l T n w p w p l = [ cospn] T log N 5 l T c 3 log N 5 T clog N 3. [ cosp l] Ths proves e now show that F defnes a postve operator. For smplcty of notatons, we let L = T and shft T to T :=, L. Then s j can be wrtten as s j = log N 5 ŝ j log N 5,L + j L + j,l +j L Fx any u L T. The followng proof s very smlar to the one below 4.45, so we shall omt some detals. For any fxed and L j L, we choose x 0, x,, x n for some n = Olog N 4 such that = x 0 x x x n x n = j, wth /3 x x + /,. Moreover, we set x 0 = and x n = j. Then we can get as before that n u u j Clog N 4 Average x,x, u,x n x u x, where we too average over all x x /4, x + /4, n. Note that by our choces, we always have x x and ŝ x = for n, whch gves that x + log N 5 log N 5 log N 5,L j L,L j L,L j L Clog N ŝ l u u l.,l T u u j Clog N4 Clog N 4 n = = x,x x /4,x +/4 x: x x /4 ŝ x u x u + ŝ x x y: y x n /4 u x u x ŝ jy u y u j 6

27 Then by 4.50, t s easy to see that F s a postve operator. Thus by mn-max prncple we have whch proves 4.48 together wth E F 0 + F E F 0, References [] P. Bourgade, L. Erdős, H.-T. Yau, and J. Yn, Unversalty for a class of random band matrces, Advances n Theoretcal and Mathematcal Physcs 07, no. 3, [] P. Bourgade, H.-T. Yau, and J. Yn, Random band matrces n the delocalzed phase, I: Quantum unque ergodcty and unversalty, n preparaton 08. [3] L. Erdős, A. Knowles, and H.-T. Yau, Averagng fluctuatons n resolvents of random band matrces, Ann. Henr Poncaré 4 03, [4] L. Erdős, A. Knowles, H.-T. Yau, and J. Yn, Delocalzaton and dffuson profle for random band matrces, Comm. Math. Phys , no., [5], The local semcrcle law for a general class of random matrces, Elect. J. Prob. 8 03, no. 59, 58. [6] L. Erdős, A. Knowles, H.-T. Yau, and J. Yn, Spectral statstcs of Erdős-Rény graphs II: Egenvalue spacng and the extreme egenvalues, Comm. Math. Phys. 34 0, [7] L. Erdős, B. Schlen, and H.-T. Yau, Local semcrcle law and complete delocalzaton for gner random matrces, Commun. Math. Phys , no., [8] L. Erdős, H.-T. Yau, and J. Yn, Bul unversalty for generalzed gner matrces, Probab. Theory Related Felds 54 0, no. -, [9] A. Knowles and J. Yn, The sotropc semcrcle law and deformaton of gner matrces, Comm. Pure Appl. Math , [0] F. Yang and J. Yn, Random band matrces n the delocalzed phase, III: Averagng fluctuatons, n preparaton 08. 7