aa r X i v : . [ m a t h - ph ] J un Bulk Universality for Unitary Matrix Models
M. PoplavskyiMathematical Division, B. Verkin Institute for Low Temperature Physics and EngineeringNational Academy of Sciences of Ukraine47 Lenin Ave., Kharkiv, 61103, UkraineE-mail:[email protected] April 25, 2008
Abstract
A proof of universality in the bulk of spectrum of unitary matrix models, assumingthat the potential is globally C and locally C function (see Theorem 1.2), is given. Theproof is based on the determinant formulas for correlation functions in terms of polynomialsorthogonal on the unit circle. The sin -kernel is obtained as a unique solution of a certainnonlinear integro-differential equation without using asymptotics of orthogonal polynomials. Key words : unitary matrix models, local eigenvalue statistics, universality.
Mathematics Subject Classification 2000 : 15A52, 15A57.
1. Introduction
In the paper we study a class of random matrix ensembles known as unitary matrix models.These models are defined by the probability law p n ( U ) dµ n ( U ) = Z − n, exp (cid:26) − n Tr V (cid:18) U + U ∗ (cid:19)(cid:27) dµ n ( U ) , (1.1)where U = { U jk } nj,k =1 is an n × n unitary matrix, µ n ( U ) is the Haar measure on the group U ( n ), Z n, is the normalization constant and V : [ − , → R + is a continuous function calledthe potential of the model. Denote e iλ j the eigenvalues of unitary matrix U . The joint probabilitydensity of λ j , corresponding to (1.1), is given by (see [1]) p n ( λ , . . . , λ n ) = 1 Z n Y ≤ j Assume that the potential V of the model (1.1) is a C ( − π, π ) function. Then: there exists a measure N ∈ M ([ − π, π ]) with a compact support σ such that NCM N n converges in probability to N ; • N has a bounded density ρ ; • denote ρ n := p ( n )1 the first marginal density, then for any φ ∈ H ( − π, π ) (cid:12)(cid:12)(cid:12)(cid:12)Z φ ( λ ) ρ n ( λ ) dλ − Z φ ( λ ) ρ ( λ ) dλ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k φ k / (cid:13)(cid:13) φ ′ (cid:13)(cid:13) / n − / ln / n, (1.4) where k·k denotes L norm on [ − π, π ]One of the main topics of local regime is a universality of local eigenvalue statistics. Let p ( n ) l ( λ , . . . , λ l ) = Z p n ( λ , . . . , λ l , λ l +1 , . . . , λ n ) dλ l +1 . . . dλ n (1.5)be the l -th marginal density of p n . Definition 1.1 We call by the bulk of the spectrum the set { λ ∈ σ : ρ ( λ ) > } , (1.6) where ρ is defined in Theorem 1.1. The main result of the paper is the proof of universality conjecture in the bulk of spectrumlim n →∞ [ nρ n ( λ )] − l p ( n ) l (cid:18) λ + x nρ n ( λ ) , . . . , λ + x l nρ n ( λ ) (cid:19) = det { S ( x j − x k ) } lj,k =1 , (1.7)where S ( x ) = sin πxπx . (1.8)By (1.7), the limiting local distributions of eigenvalues do not depend on potential V in (1.1),modulo some weak condition (see Theorem 1.2). The conjecture of universality of all correlationfunctions was suggested by F.J. Dyson (see [3]) in the early 60s who proved (1.7)–(1.8) for V ( x ) = 0. First rigorous proofs for Hermitian matrix models with nonquadratic V appearedonly in the 90s. The case of general V which is locally C function was studied in [4]. The caseof real analytic potential V was studied in [5], where the asymptotics of orthogonal polynomialswere obtained. For unitary matrix models the bulk universality was proved for V = 0 (see [3])and in the case of a linear V (see [6]).To prove the main result we need some properties of the polynomials orthogonal with respectto varying weight on the unit circle. Consider a system of functions (cid:8) e ikλ (cid:9) ∞ k =0 and use for themthe Gram–Schmidt procedure in L (cid:0) [ − π, π ] , e − nV ( λ ) (cid:1) . For any n we get the system of functions n P ( n ) k ( λ ) o ∞ k =0 which are orthogonal and normalized in L (cid:0) [ − π, π ] , e − nV ( λ ) (cid:1) . Since V is even,it is easy to see that all coefficients of these functions are real. Denote ψ ( n ) k ( λ ) = P ( n ) k ( λ ) e − nV ( λ ) / . (1.9)Then we obtain the orthogonal in L ( − π, π ) functions π Z − π ψ ( n ) k ( λ ) ψ ( n ) l ( λ ) dλ = δ kl . (1.10)2he reproducing kernel of the system (1.9) is given by K n ( λ, µ ) = n − X j =0 ψ ( n ) l ( λ ) ψ ( n ) l ( µ ) . (1.11)From (1.10) we obtain that the reproducing kernel satisfies the relation π Z − π K n ( λ, ν ) K n ( ν, µ ) dν = K n ( λ, µ ) , (1.12)and from the Cauchy inequality we have | K n ( λ, µ ) | ≤ K n ( λ, λ ) K n ( µ, µ ) . (1.13)We also use below the determinant form of the marginal densities (1.5) (see [1]) p ( n ) l ( λ , . . . , λ l ) = ( n − l )! n ! det k K n ( λ j , λ k ) k lj,k =1 . (1.14)In particular, ρ n ( λ ) = n − K n ( λ, λ ) , (1.15) p ( n )2 ( λ, µ ) = K n ( λ, λ ) K n ( µ, µ ) − | K n ( λ, µ ) | n ( n − . (1.16)The main result of the paper is Theorem 1.2 Assume that V ( λ ) is a C ( − π, π ) function, and there exists an interval ( a, b ) ⊂ σ such that sup λ ∈ ( a,b ) | V ′′′ ( λ ) | ≤ C , ρ ( λ ) ≥ C , λ ∈ ( a, b ) . (1.17) Then for any d > and λ ∈ [ a + d, b − d ] for K n defined in (1.11) we have lim n →∞ [ K n ( λ , λ )] − K n (cid:18) λ + xK n ( λ , λ ) , λ + yK n ( λ , λ ) (cid:19) = e i ( x − y ) / ρ ( λ ) sin π ( x − y ) π ( x − y ) (1.18) uniformly in ( x, y ) , varying on a compact set of R . R e m a r k 1.3 It is easy to see that the universality conjecture (1.7) follows from Theorem 1.2by (1.14).The method of the proof is a version of the one used in [4]. An important part of the proofis a uniform convergence of ρ n to ρ in a neighborhood of λ : Theorem 1.4 Under the assumptions of Theorem 1.2 for any d > there exists C ( d ) > suchthat for any λ ∈ [ a + d, b − d ] | ρ n ( λ ) − ρ ( λ ) | ≤ C ( d ) n − / . (1.19)3 . Proof of Basic Results P r o o f of Theorem 1.4. We will use some facts from the integral transformations theory(see [7]). Definition 2.1 Assume that g ( λ ) is a continuous function on the interval [ − π, π ] . Then itsGermglotz transformation is given by F [ g ] ( z ) = π Z − π e iλ + e iz e iλ − e iz g ( λ ) dλ, (2.1) where z ∈ C \ R . The inverse transformation is given by g ( µ ) = 12 π lim z → µ + i ℜ F [ g ] ( z ) . (2.2)For z = µ + iη , η = 0, we set f n ( z ) = π Z − π e iλ + e iz e iλ − e iz ρ n ( λ ) dλ. (2.3)Bellow we will derive a ”square” equation for f n . Denote I n ( z ) = π Z − π V ′ ( λ ) e iλ + e iz e iλ − e iz ρ n ( λ ) dλ. (2.4)Integrating by parts in (2.4), from (1.5) we obtain I n ( z ) = 1 Z n Z V ′ ( λ ) e iλ + e iz e iλ − e iz Y j Let K n ( λ, µ ) be defined by (1.11) . Then under the conditions of Theorem 1.2 forany δ > (cid:12)(cid:12)(cid:12)(cid:12)Z (cid:16) e iλ − e iµ (cid:17) | K n ( λ, µ ) | dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:20)(cid:12)(cid:12)(cid:12) ψ ( n ) n − ( λ ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( n ) n ( λ ) (cid:12)(cid:12)(cid:12) (cid:21) , (2.9) Z (cid:12)(cid:12)(cid:12) e iλ − e iµ (cid:12)(cid:12)(cid:12) | K n ( λ, µ ) | dµ ≤ (cid:20)(cid:12)(cid:12)(cid:12) ψ ( n ) n − ( λ ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( n ) n ( λ ) (cid:12)(cid:12)(cid:12) (cid:21) , (2.10) Z (cid:12)(cid:12)(cid:12) e iλ − e iµ (cid:12)(cid:12)(cid:12) | K n ( λ, µ ) | dλdµ ≤ , (2.11)5 | e iλ − e iµ | >δ | K n ( λ, µ ) | dµ ≤ δ − (cid:20)(cid:12)(cid:12)(cid:12) ψ ( n ) n − ( λ ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( n ) n ( λ ) (cid:12)(cid:12)(cid:12) (cid:21) , (2.12) Z | e iλ − e iµ | >δ | K n ( λ, µ ) | dλdµ ≤ δ − . (2.13)It is easy to see that (cid:12)(cid:12) e iλ − e iz (cid:12)(cid:12) > C | η | if | η | < C > 0. Hence, from (2.11)and (2.8) we derive f n ( z ) − iV ′ ( µ ) f n ( z ) − iQ n ( z ) − O (cid:0) n − η − (cid:1) . (2.14) Lemma 2.2 Under the conditions of Theorem 1.2 for any d > and λ ∈ [ a + d, b − d ] ρ n ( λ ) ≤ C, (2.15) (cid:12)(cid:12)(cid:12)(cid:12) dρ n ( λ ) dλ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18)(cid:12)(cid:12)(cid:12) ψ ( n ) n ( λ ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( n ) n − ( λ ) (cid:12)(cid:12)(cid:12) (cid:19) + C . (2.16)¿From the conditions of Theorem 1.2, we obtain that V ′′ ( λ ) is bounded on the interval [ a, b ].Hence, for µ ∈ [ a + d, b − d ] and sufficiently small η we have | Q n ( µ + iη ) − Q n ( µ ) | ≤ (cid:12)(cid:12) e − η − (cid:12)(cid:12) π Z − π | V ′ ( λ ) − V ′ ( µ ) | ρ n ( µ ) | e iλ − e iµ | | e iλ − e iz | dλ ≤ Cη Z | λ − µ | For d > , k = n − , n and µ ∈ [ a + d, b − d ] Z | λ − µ | Using (2.16) and (2.24), we get finally (cid:12)(cid:12)(cid:12)(cid:12) π f n ( z ) − ρ n ( µ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z | s | <η / (cid:12)(cid:12) ρ ′ n ( µ + s ) (cid:12)(cid:12) ds + Cη / ≤ Cη / . Theorem 1.4 is proved.Now we pass to the proof of Theorem 1.2. We will use the following representation of K n ,which can be derived from the well-known identities of random matrix theory (see [1])1 n K n ( λ, µ ) = 1 n n − X j =0 ψ ( n ) l ( λ ) ψ ( n ) l ( µ ) = Q − n, e − n ( V ( λ )+ V ( µ )) / × Z n Y j =2 (cid:16) e iλ − e iλ j (cid:17) (cid:16) e − iµ − e − iλ j (cid:17) e − nV ( λ j ) dλ j Y ≤ j For any d > we have uniformly in λ ∈ [ a + d, b − d ] and | x | , | y | ≤ nd/ (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x K n ( x, y ) + ∂∂y K n ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) n − / + | x − y | n − (cid:17) , (2.33) |K n ( x, y ) − K n (0 , y − x ) | ≤ C | x | (cid:16) n − / + | x − y | n − (cid:17) . (2.34) R e m a r k 2.8 Note that the last inequality with λ + x /n instead of λ , and x − x insteadof x and y , leads to the bound that is valid for any | x , | ≤ nd / |K n ( x , x ) − K n ( x , x ) | ≤ Cn − / | x − x | . (2.35)9 emma 2.9 For any | x | , | y | ≤ L (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x K n ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C, Z | x |≤L (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x K n ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ C. (2.36)Denote K ∗ n ( x ) = K n ( x, | x |≤L + K n ( L , L − x ) L 3, similarly to (2.32), we can restrict theintegration in (2.32) to | z | ≤ L / O ( L − ). This and Lemma 2.7 give us theequation ∂∂x K ∗ n ( x ) = Z | z |≤ L / K ∗ n ( z ) K ∗ n ( x − z ) z dz + r n ( x ) + O ( L − ) , (2.38)where r n ( x ) = Z | z |≤ L / K n ( z, K n ( x, z ) − K n (0 , x − z )) z dz, and by Lemma 2.7, for | x | ≤ L / r n ( x ) = O ( n − / log n ) . Now, using the estimates similar to (2.32), we can restrict the integration in (2.38) to the realaxis. From Lemma 2.9 and the relations (2.28), (2.29) we get Z |K ∗ n ( x ) | dx ≤ Z |K n ( x, | dx + C ′ ≤ C, Z (cid:12)(cid:12)(cid:12)(cid:12) ddx K ∗ n ( x ) (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ C. (2.39)Consider the Fourier transform b K ∗ n ( p ) = Z K ∗ n ( x ) e ipx dx, where the integral is defined in the L ( R ) sense, and write K ∗ n ( x ) as K ∗ n ( x ) = (2 π ) − Z b K ∗ n ( p ) e − ipx dp. (2.40)From (1.19) we have Z b K ∗ n ( p ) dp = 2 πρ ( λ ) + o (1) , (2.41)and from (2.39) and the Parseval equation we obtain Z p | b K ∗ n ( p ) | dp ≤ C. (2.42)From the definition of K n ( x, y ) we get that the kernel is positive definite L Z −L K n ( x, y ) f ( x ) f ( y ) dxdy ≥ , f ∈ L ( R ) , f ∈ L ( R ) Z b K ∗ n ( p ) | ˆ f ( p ) | dp ≥ − C || f || L ( R ) ( n − / log n + O ( L − )) . (2.43)From the Parseval equation and (2.34) there follows Z | b K ∗ n ( p ) − b K ∗ n ( − p ) | dp ≤ π Z |K ∗ n ( x ) − K ∗ n ( − x ) | dx ≤ Cn − / log n. (2.44)By the definition of singular integrals Z K ∗ n ( z ) K ∗ n ( x − z ) z dz = lim ε → +0 Z dz K ∗ n ( z ) K ∗ n ( y − z ) ℜ ( z + iε ) − . (2.45)In accordance with the relation Z e ipz ℜ ( z + iε ) − dz = πie − ε | p | sgn p and the Parseval equation, we can write the r.h.s. of (2.38) as i π lim ε → +0 Z dpdp ′ b K ∗ n ( p ) b K ∗ n ( p ′ ) e − ipx sign( p − p ′ ) e − ε | p − p ′ | = i π Z dpe − ipx b K ∗ n ( p ) p Z b K ∗ n ( p ′ ) dp ′ − i π Z dpe − ipx b K ∗ n ( p ) ∞ Z ( b K ∗ n ( p ′ ) − b K ∗ n ( − p ′ )) dp ′ . (2.46)Note that both integrals are absolutely convergent because b K ∗ n ∈ L ( R ) by (2.42). Now, usingthe Schwarz inequality and (2.42), we can estimate the second component (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Z ( b K ∗ n ( p ′ ) − b K ∗ n ( − p ′ )) dp ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L Z ( ˆ K ∗ n ( p ′ ) − b K ∗ n ( − p ′ )) dp ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + Z | p | > L | b K ∗ n ( p ′ ) | dp ′ ≤ L (cid:18)Z | b K ∗ n ( p ′ ) − b K ∗ n ( − p ′ ) | dp ′ (cid:19) / + C L − . Thus, from (2.44)–(2.46) we have uniformly in | x | < L / Z K ∗ n ( z ) K ∗ n ( x − z ) z dz = i π Z dp b K ∗ n ( p ) e − ipx p Z b K ∗ n ( p ′ ) dp ′ + O ( L − ) . This allows us to transform (2.38) into the following asymptotic relation that is valid for | x | ≤L / Z b K ∗ n ( p ) (cid:18) p Z b K ∗ n ( p ′ ) dp ′ − p (cid:19) e − ipx dp = O ( L − ) . (2.47)Consider the functions F n ( p ) = p Z b K ∗ n ( p ′ ) dp ′ . (2.48)Since p b K ∗ n ( p ) ∈ L ( R ), the sequence { F n ( p ) } consists of functions that are uniformly boundedand equicontinuous on R . Thus { F n ( p ) } is a compact family with respect to uniform convergence.Hence, the limit F of any subsequence { F n k } possesses the properties:11a) F is bounded and continuous;(b) F ( p ) = − F ( − p ) (see (2.44));(c) F ( p ) ≤ F ( p ′ ), if p ≤ p ′ (see (2.43));(d) F (+ ∞ ) − F ( −∞ ) = 2 πρ ( λ ) (see (2.41));(e) the following equation is valid for any smooth function g with the compact support (see(2.47)): Z ( F ( p ) − p ) g ( p ) dF ( p ) = 0 . (2.49)The last property implies that F ( p ) = p or F ( p ) = const, hence it follows from (a)–(c) that F ( p ) = p | p |≤ p + p sign( p ) | p |≥ p , where p = πρ ( λ ) from (d). We conclude that (2.49) is uniquely solvable, thus the sequence { F n } converges uniformly on any compact to the above F . This and (2.48) imply the weakconvergence of the sequence {K ∗ n } to the function ρ ( λ ) S ( ρ ( λ ) x ), where S ( x ) is defined in(1.8). But weak convergence combined with (2.29) and (2.36) implies the uniform convergenceof {K ∗ n } to K ∗ on any interval. Thus we have uniformly in ( x, y ), varying on a compact set of R , lim n →∞ K n ( x, y ) = ρ ( λ ) S ( ρ ( λ ) ( x − y )) . Recalling all definitions, we conclude that Theorem 1.2 is proved. Auxiliary Results for Theorem 1.2 P r o o f of Lemma 2.1. Denote r ( n ) k,j = π Z − π e iλ ψ ( n ) k ( λ ) ψ ( n ) j ( λ ) dλ. (2.50)Note that from the orthogonality (2.66) we have r ( n ) k,j = 0 for j > k + 1. Thus, e iλ ψ ( n ) k ( λ ) = k +1 X j =0 r ( n ) k,j ψ ( n ) j ( λ ) . (2.51)Multiplication on e iλ is isometric in L [ − π, π ], therefore k +1 X j =0 (cid:12)(cid:12)(cid:12) r ( n ) k,j (cid:12)(cid:12)(cid:12) = (cid:13)(cid:13)(cid:13) ψ ( n ) k ( λ ) (cid:13)(cid:13)(cid:13) = 1 . Finally we are ready to prove (2.9) π Z − π (cid:16) e iλ − e iµ (cid:17) | K n ( λ, µ ) | dµ = e iλ K n ( λ, λ ) − π Z − π e iµ n − X m =0 ψ ( n ) m ( µ ) ψ ( n ) m ( λ ) n − X l =0 ψ ( n ) l ( λ ) ψ ( n ) l ( µ ) dµ = e iλ K n ( λ, λ ) − n − X l,m =0 r ( n ) m,l ψ ( n ) l ( λ ) ψ ( n ) m ( λ )= r ( n ) n − ,n ψ ( n ) n − ( λ ) ψ ( n ) n ( λ ) . (2.52)12ow, using the Cauchy inequality and the bound (cid:12)(cid:12)(cid:12) r ( n ) n − ,n (cid:12)(cid:12)(cid:12) ≤ 1, we get (2.9). Similarly, it is easyto obtain the relation π Z − π (cid:12)(cid:12)(cid:12) e iλ − e iµ (cid:12)(cid:12)(cid:12) | K n ( λ, µ ) | dµ = 2 ℜ (cid:26) e iλ r ( n ) n − ,n ψ ( n ) n − ( λ ) ψ ( n ) n ( λ ) (cid:27) , which implies (2.10). The bounds (2.11),(2.12),(2.13) are evident consequences of (2.10). Thelemma is proved.P r o o f of Lemma 2.2. Observe that dρ n ( λ ) dλ = dρ n ( λ + t ) dt (cid:12)(cid:12)(cid:12)(cid:12) t =0 . Changing variables in (1.5) λ j = µ j + t , in view of periodicity of all functions in the consideration,we have the representation for ρ n ( λ + t ) ρ n ( λ + t ) = 1 Z n Z e − nV ( λ + t ) Y ≤ j For any function u : [ a , b ] → C with u ′ ∈ L ( a , b ) we have k u k ∞ ≤ (cid:13)(cid:13) u ′ (cid:13)(cid:13) + ( b − a ) − k u k , (2.55) where k · k , k · k ∞ are the L and uniform norms on the interval [ a , b ] . u ( λ ) = 1 b − a b Z a ( u ( λ ) − u ( µ )) dµ + 1 b − a b Z a u ( µ ) dµ. Using (2.55) for u = ρ n and the interval [ a + d, b − d ], we get (2.15).P r o o f of Lemma 2.3. From (1.4) and (2.21) we have for nonreal zf ( z ) − iV ′ ( µ ) f ( z ) − iQ ( z ) − , (2.56)where f ( z ) is the Germglotz transformation of the limiting density ρ ( λ ). By (2.19) and (2.2), Q ( µ + i 0) is an imaginary valued, bounded, continuous function. And from (2.2) we obtain ρ ( µ ) = 12 π ℜ f ( µ + i . Computing imaginary and real parts in (2.56), we get the relations ℑ f ( µ + i 0) = V ′ ( µ ) , (2.57) ℜ f ( µ + i 0) = q iQ ( µ ) + 1 − ( V ′ ( µ )) , (2.58)from which we obtain (2.22).P r o o f of Lemma 2.4. To prove (2.24) with k = n − p − n ( λ , . . . , λ n − ) = 1 Z − n Y ≤ j 1. Thus, Z | λ − µ | For any C function u : [ a , b ] → C k u k ∞ ≤ k u k (cid:13)(cid:13) u ′ (cid:13)(cid:13) + ( b − a ) − k u k , (2.64) where k · k , k · k ∞ are the L and uniform norms on the interval [ a , b ] . This inequality is a simple consequence of the relation u ( λ ) = 1 b − a b Z a (cid:0) u ( λ ) − u ( µ ) (cid:1) dµ + 1 b − a b Z a u ( µ ) dµ. (cid:2) λ − n − / , λ + n − / (cid:3) and the function ψ ( λ ) = ψ ( n ) n ( λ ). From theinequality we have | ψ ( λ ) | ≤ k ψ k , ∆ (cid:13)(cid:13) ψ ′ (cid:13)(cid:13) , ∆ + 12 n / k ψ k , ∆ , (2.65)where k·k , ∆ is L norm on the interval ∆. It is easy to see that k ψ k , ∆ ≤ k ψ k , [ − π,π ] = 1 . Denote P ( λ ) = P ( n ) n ( λ ) and ω ( λ ) = e − nV ( λ ) / , then ψ ( λ ) = P ( λ ) ω ( λ ). Now we estimate k ψ ′ k , [ − π,π ] : (cid:13)(cid:13) ψ ′ (cid:13)(cid:13) , [ − π,π ] = (cid:13)(cid:13) P ′ ω + P ω ′ (cid:13)(cid:13) , [ − π,π ] ≤ (cid:13)(cid:13) P ′ ω (cid:13)(cid:13) , [ − π,π ] + (cid:13)(cid:13) P ω ′ (cid:13)(cid:13) , [ − π,π ] , (cid:13)(cid:13) P ω ′ (cid:13)(cid:13) , [ − π,π ] = n (cid:13)(cid:13) P V ′ ω (cid:13)(cid:13) , [ − π,π ] ≤ Cn k P ω k , [ − π,π ] = Cn, (cid:13)(cid:13) P ′ ω (cid:13)(cid:13) , [ − π,π ] = Z P ′ ( λ ) P ′ ( λ ) ω ( λ ) dλ = − Z P ( λ ) P ′′ ( λ ) ω ( λ ) dλ + n Z P ( λ ) P ′ ( λ ) V ′ ( λ ) ω ( λ ) dλ. Using the orthogonality Z e − imλ ω ( λ ) ψ ( n ) k dλ = 0 , for m < k, (2.66)we obtain Z P ( λ ) P ′′ ( λ ) ω ( λ ) dλ = Z P ( λ ) γ ( n ) n ( − in ) e − inλ ω ( λ ) dλ = − in Z P ( λ ) P ′ ( λ ) ω ( λ ) dλ, where γ ( n ) n is defined in (2.26). Thus, (cid:13)(cid:13) P ′ ω (cid:13)(cid:13) , [ − π,π ] = n Z P ( λ ) P ′ ( λ ) (cid:0) V ′ ( λ ) + i (cid:1) ω ( λ ) dλ ≤ Cn (cid:13)(cid:13) P ′ ω (cid:13)(cid:13) , [ − π,π ] , and we obtain that k P ′ ω k , [ − π,π ] ≤ Cn . Combining all above bounds, we conclude that k ψ ′ k , [ − π,π ] ≤ Cn . Now, using (2.65) and (2.24), we obtain (2.25) for k = n . For k = n − η = n − / and µ ∈ [ a + d, b − d ] for f n ,defined in (2.3), we obtain (cid:12)(cid:12) ℑ f n ( µ + iη ) − V ′ ( µ ) (cid:12)(cid:12) ≤ Cn − / ln n. (2.67)Moreover, we estimate M = ℑ f n ( µ + iη ) + v.p. π Z − π cot s ρ n ( µ + s ) ds . Note that ℑ e iλ + e iz e iλ − e iz = − sin ( λ − µ )cosh η − cos ( λ − µ ) . M = v.p. Z (cid:18) cot s − sin s cosh η − cos s (cid:19) ρ n ( µ + s ) ds = Z | s |≤ d/ ln (cid:18) cosh η − cos s − cos s (cid:19) ρ ′ n ( µ + s ) ds + O ( η ) = I + I + I + O ( η ) , where I is the integral over | s | ≤ n − , I is the integral over n − ≤ | s | ≤ n − / and I is theintegral over n − / ≤ | s | ≤ d/ 2. We estimate every term: | I | (2.25) ≤ Cn / Z | s |≤ n − ln (cid:18) cosh η − cos s − cos s (cid:19) ds ≤ Cn − / ln n, | I | ≤ C ln n Z n − ≤| s |≤ n / (cid:12)(cid:12) ρ ′ n ( µ + s ) (cid:12)(cid:12) ds (2.24) ≤ Cn − / ln n, | I | (2.16) ≤ Cn − / Z | s |≤ d/ (cid:18)(cid:12)(cid:12)(cid:12) ψ ( n ) n ( µ + s ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( n ) n − ( µ + s ) (cid:12)(cid:12)(cid:12) (cid:19) ds ≤ Cn − / . Combining the above bounds with (2.67), we obtain that the lemma is proved.P r o o f of Lemma 2.7. To simplify notations we denote for t ∈ [0 , λ x = λ + x − txn , λ y = λ + y − txn . (2.68)Then, similarly to (2.30) and (2.54), we obtain ddt K n ( λ x , λ y ) = x π + λ Z − π + λ K n ( λ x , λ ) K n ( λ, λ y ) (cid:18) V ′ ( λ x ) + 12 V ′ ( λ y ) − V ′ ( λ ) (cid:19) dλ. (2.69)To get our estimates, we split this integral in two parts | λ − λ | ≤ d/ | λ − λ | ≥ d/ 2. Bythe assumption of the lemma, λ x , λ y are in [ a + d/ , b − d/ V ′ ( λ ) − V ′ ( λ x ) − V ′ ( λ y )= (cid:16) e iλ − e iλ x (cid:17) V ′′ ( λ x )2 ie iλ x + (cid:16) e iλ − e iλ y (cid:17) V ′′ ( λ y )2 ie iλ y + O (cid:18)(cid:12)(cid:12)(cid:12) e iλ − e iλ x (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) e iλ − e iλ y (cid:12)(cid:12)(cid:12) (cid:19) = (cid:16) e iλ − e iλ x (cid:17) V ′′ ( λ x )2 ie iλ x + (cid:16) e iλ − e iλ y (cid:17) V ′′ ( λ y )2 ie iλ y + O (cid:12)(cid:12)(cid:12) e iλ − e iλ x (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) e iλ − e iλ y (cid:12)(cid:12)(cid:12) + | x − y | n ! . Similarly to (2.52), we obtain π Z − π K n ( λ x , λ ) K n ( λ, λ y ) (cid:16) e iλ − e iλ x (cid:17) dλ = − r ( n ) n − ,n ψ ( n ) n ( λ x ) ψ ( n ) n − ( λ y ) . Z | λ − λ |≤ d/ K n ( λ x , λ ) K n ( λ, λ y ) (cid:16) e iλ − e iλ x (cid:17) dλ = − r n − ,n ψ ( n ) n ( λ x ) ψ ( n ) n − ( λ y ) − I d , where | I d | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z | λ − λ |≥ d/ K n ( λ x , λ ) K n ( λ, λ y ) (cid:16) e iλ − e iλ x (cid:17) dλ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z | λ − λ |≥ d/ | K n ( λ x , λ ) | dλ Z | λ − λ |≥ d/ | K n ( λ, λ y ) | dλ / ≤ C (cid:20)(cid:12)(cid:12)(cid:12) ψ ( n ) n − ( λ x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( n ) n ( λ x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( n ) n − ( λ y ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( n ) n ( λ y ) (cid:12)(cid:12)(cid:12) (cid:21) . The same bounds are valid for the term with the e iλ y instead of e iλ x . To estimate other terms,we use the Schwarz inequality Z | λ − λ |≤ d/ (cid:12)(cid:12)(cid:12) K n ( λ x , λ ) K n ( λ, λ y ) (cid:16) e iλ − e iλ x (cid:17) (cid:16) e iλ − e iλ y (cid:17)(cid:12)(cid:12)(cid:12) dλ ≤ π Z − π (cid:12)(cid:12)(cid:12) K n ( λ x , λ ) (cid:16) e iλ − e iλ x (cid:17)(cid:12)(cid:12)(cid:12) dλ π Z − π (cid:12)(cid:12)(cid:12) K n ( λ, λ y ) (cid:16) e iλ − e iλ y (cid:17)(cid:12)(cid:12)(cid:12) dλ / ≤ C (cid:20)(cid:12)(cid:12)(cid:12) ψ ( n ) n − ( λ x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( n ) n ( λ x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( n ) n − ( λ y ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( n ) n ( λ y ) (cid:12)(cid:12)(cid:12) (cid:21) , Z | λ − λ |≤ d/ | K n ( λ x , λ ) K n ( λ, λ y ) | dλ ≤ n ( ρ n ( λ x ) + ρ n ( λ y )) ≤ Cn. In the second integral we use the boundedness of V ′ ( λ ), the Cauchy inequality | K n ( λ x , λ ) K n ( λ, λ y ) | ≤| K n ( λ x , λ ) | + | K n ( λ, λ y ) | and (2.12). Thus, (cid:12)(cid:12)(cid:12)(cid:12) ddt K n ( λ x , λ y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | x | (cid:20)(cid:12)(cid:12)(cid:12) ψ ( n ) n − ( λ x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( n ) n ( λ x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( n ) n − ( λ y ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( n ) n ( λ y ) (cid:12)(cid:12)(cid:12) + | x − y | n (cid:21) . (2.70)Now, using (2.25), we obtain (cid:12)(cid:12)(cid:12)(cid:12) ddt K n ( λ x , λ y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | x | (cid:16) n / + | x − y | n − (cid:17) . (2.71)Finally, observing that ∂∂x K n ( x, y ) + ∂∂y K n ( x, y ) = − ( xn ) − e − i ( n − x − y ) / n ddt K n ( λ x , λ y ) | t =0 , K n ( x, y ) − K n (0 , y − x ) = e − i ( n − x − y ) / n · n (cid:0) K n ( λ x , λ y ) | t =0 − K n ( λ x , λ y ) | t =1 (cid:1) , | x | ≤ nd / Z − K n ( x, x ) K n ( x + t, x + t ) − |K n ( x, x + t ) | t dt ≤ C. (2.72)Denote Ω = [ − π + λ , π + λ ] , Ω +0 = Ω / Ω − , (2.73)Ω − = (cid:26) λ ∈ Ω : (cid:12)(cid:12)(cid:12)(cid:12) sin λ − λ (cid:12)(cid:12)(cid:12)(cid:12) ≤ sin 12 n (cid:27) = [ λ − /n, λ + 1 /n ] , and consider the quantity W = * n Y j =2 (cid:12)(cid:12)(cid:12)(cid:12) − sin / n sin ( λ j − λ ) / (cid:12)(cid:12)(cid:12)(cid:12)+ , (2.74)where the symbol < . . . > denotes the average with respect to p n ( λ , λ , . . . , λ n ). We willestimate W from above. To do this we use the relation1 − sin n sin µ − λ (cid:0) e i ( λ +1 /n ) − e iµ (cid:1) (cid:0) e i ( λ − /n ) − e iµ (cid:1) ( e iλ − e iµ ) , (1.2) and the Schwarz inequality. We get that W is not larger than the product of two integrals I + and I − , where I ± = Z − n Z Ω n − e − nV ( λ ) Y ≤ j 0, it follows from (2.76) that W can be estimated bellow as W ≥ ( n − Z Ω φ ( λ ) * δ ( λ − λ ) exp n X j =3 ln φ ( λ j ) + dλ. Note that h δ ( λ − λ ) i = p ( n )2 ( λ , λ ). Therefore the Jensen inequality implies W ≥ ( n − Z Ω − φ ( λ ) p ( n )2 ( λ , λ ) × exp * δ ( λ − λ ) n X j =3 ln φ ( λ j ) + h p ( n )2 ( λ , λ ) i − dλ = ( n − Z Ω − φ ( λ ) p ( n )2 ( λ , λ ) × exp ( n − Z Ω ln φ (cid:0) λ ′ (cid:1) p ( n )3 (cid:0) λ , λ, λ ′ (cid:1) dλ ′ h p ( n )2 ( λ , λ ) i − dλ, where p ( n ) k is defined in (1.5). Using (1.14) for l = 2 , 3, we have p ( n )3 (cid:0) λ , λ, λ ′ (cid:1) = nn − ρ n (cid:0) λ ′ (cid:1) p ( n )2 ( λ , λ )+ " ℜ ( K n ( λ , λ ) K n ( λ, λ ′ ) K n ( λ ′ , λ )) n ( n − 1) ( n − − K n ( λ , λ ) | K n ( λ, λ ′ ) | + K n ( λ, λ ) | K n ( λ , λ ′ ) | n ( n − 1) ( n − . (2.79)By the Cauchy inequality,2 (cid:12)(cid:12) K n ( λ , λ ) K n (cid:0) λ, λ ′ (cid:1) K n (cid:0) λ ′ , λ (cid:1)(cid:12)(cid:12) ≤ K / n ( λ , λ ) K / n ( λ, λ ) (cid:12)(cid:12) K n (cid:0) λ, λ ′ (cid:1) K n (cid:0) λ ′ , λ (cid:1)(cid:12)(cid:12) ≤ K n ( λ , λ ) (cid:12)(cid:12) K n (cid:0) λ, λ ′ (cid:1)(cid:12)(cid:12) + K n ( λ, λ ) (cid:12)(cid:12) K n (cid:0) λ , λ ′ (cid:1)(cid:12)(cid:12) , we obtain that the second term in (2.79) is nonpositive, hence p ( n )3 (cid:0) λ , λ, λ ′ (cid:1) ≤ nn − ρ n (cid:0) λ ′ (cid:1) p ( n )2 ( λ , λ ) . Taking into account that ln φ ( λ ′ ) ≤ 0, finally we get W ≥ ( n − Z Ω − φ ( λ ) p ( n )2 ( λ , λ ) dλ · exp n Z Ω ρ n (cid:0) λ ′ (cid:1) ln φ (cid:0) λ ′ (cid:1) dλ ′ . (2.80)20ow we will show that the second multiplier in (2.80) is bounded from below n Z Ω ρ n (cid:0) λ ′ (cid:1) ln φ (cid:0) λ ′ (cid:1) dλ ′ = Z | s |≤ + Z ≤| s |≤ nd / + Z nd / ≤| s |≤ nπ ρ n ( λ + s/n ) ln φ ( λ + s/n ) ds ≥ C Z | s |≤ ln (cid:18) − sin s/ (2 n )sin / (2 n ) (cid:19) ds + Z ≤| s |≤ nd / ln (cid:18) − sin / (2 n )sin s/ (2 n ) (cid:19) ds + ln (cid:18) − sin / (2 n )sin d / (cid:19) Z | s |≤ nπ ρ n ( λ + s/n ) ds ≥ C ( I + I ) + O (cid:0) n − (cid:1) .I = Z ln (cid:18) cos ( s/n ) − cos (1 /n )1 − cos (1 /n ) (cid:19) ds = − n /n Z sin t sin ( t + 1 /n ) t − /n t − /n dt ≥ − CI = n d / Z /n ln (cid:18) cos (1 /n ) − cos t − cos t (cid:19) dt = ( nd / − 1) ln (cid:18) − sin / n sin d / (cid:19) − n (1 − cos 1 /n ) d / Z /n cot t/ t − /n t − /n t + 1 /n dt ≥ − C − Cn − d / Z /n dtt ( t + 1 /n ) ≥ − C. Thus, from (2.75) and (2.80) we obtain n Z Ω − φ ( λ ) p ( n )2 ( λ , λ ) dλ ≥ − C. (2.81)Then, using (1.14), (2.27), (2.15), (2.77), and the inequality 1 t ≤ C sin / n sin t/ n , we obtain (2.72)for x = 0 from (2.81). Substituting λ by λ + x/n , we get (2.72) for any | x | ≤ nd / C n = sup (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x K n ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) . In view of (2.32) C n ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v.p. Z | z − x |≤ + Z | z − x |≥ K n ( x, z ) K n ( z, y ) z − x dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + o (1) ≤ | I ( x, y ) | + | I ( x, y ) | + o (1) . Using the Schwarz inequality and (2.28) with (2.29), we can estimate I as follows: | I ( x, y ) | ≤ K / n ( x, x ) K / n ( y, y ) ≤ C. 21o estimate I denoteˆ t ∗ n = sup { t > | x − y | ≤ t ⇒ K n ( x, y ) ≥ ρ n ( λ ) / } ,t ∗ n = min (cid:8) ˆ t ∗ n , (cid:9) . (2.82)We will prove that the sequence t ∗ n is bounded from below by some nonzero constant. Represent I in the form I ( x, y ) = v.p. Z | t |≤ t ∗ n K n ( x, x + t ) K n ( x + t, y ) − K n ( x, x ) K n ( x, y ) t dt + Z t ∗ n ≤| t |≤ K n ( x, x + t ) K n ( x + t, y ) t dt = I ′ + I ′′ . Using (2.29), we have | I ′′ | ≤ C | ln t ∗ n | . On the other hand, from (1.11) and the Cauchy inequalitywe obtain for any x, y, z |K n ( x, z ) − K n ( y, z ) | ≤ ( K n ( x, x ) + K n ( y, y ) − K n ( x, y )) K n ( z, z )= (cid:18)(cid:16) K / n ( x, x ) − K / n ( y, y ) (cid:17) + 2 (cid:16) K / n ( x, x ) K / n ( y, y ) − K n ( x, y ) (cid:17)(cid:19) K n ( z, z ) . (2.83)From (2.35) we get that the first term of (2.83) is bounded by Cn − / | x − y | . The second termwe rewrite as K / n ( x, x ) K / n ( y, y ) − K n ( x, y ) = K n ( x, x ) K n ( y, y ) − K n ( x, y ) K / n ( x, x ) K / n ( y, y ) + K n ( x, y ) . Thus, for | x − y | ≤ t ∗ n we get |K n ( x, z ) − K n ( y, z ) | ≤ C (cid:16) n − / | x − y | / + K n ( x, x ) K n ( y, y ) − |K n ( x, y ) | (cid:17) . (2.84)Hence, using (2.84), (2.72) and the Schwarz inequality, we obtain (cid:12)(cid:12) I ′ (cid:12)(cid:12) ≤ C Z | t |≤ t ∗ n |K n ( x, x + t ) − K n ( x, x ) | + |K n ( x + t, y ) − K n ( x, y ) || t | dt ≤ C ( t ∗ n ) / . Finally, from the above estimates we have C n ≤ C (cid:16) | ln t ∗ n | + ( t ∗ n ) / (cid:17) . (2.85)Note that if the sequence t ∗ n is not bounded from below, then we have C ≤ ρ n ( λ ) / ≤ |K n ( x + t ∗ n , x ) − K n ( x, x ) | ≤ C n t ∗ n ≤ Ct ∗ n ln t ∗ n + Ct ∗ n , and we get a contradiction. Thus t ∗ n ≥ d ∗ for some n -independent d ∗ > 0. Therefore, from (2.85)we obtain the first inequality of (2.36).To prove the second inequality of (2.36), we observe that by (2.33) we have Z | x |≤L (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x K n ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) dx = Z | x |≤L (cid:12)(cid:12)(cid:12)(cid:12) ∂∂y K n ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) dx + o (1) . ∂∂y K n ( x, y ) as ∂∂y K n ( x, y ) = v.p. Z | z − y |≤ d ∗ + Z | z |≤ L | z − y |≥ d ∗ K n ( x, z ) K n ( z, y ) y − z dz + O (cid:0) L − (cid:1) = I ( x, y ) + I ( x, y ) + O (cid:0) L − (cid:1) . To complete the proof, it is enough to estimate I , . Since in I the domain of integration issymmetric with respect to y , we can write I ( x, y ) = Z | z − y |≤ d ∗ ( K n ( x, z ) − K n ( x, y )) K n ( z, y ) y − z dz + Z | z − y |≤ d ∗ ( K n ( z, y ) − K n ( y, y )) K n ( x, y ) y − z dz. Now, using the Schwarz inequality and (2.28), we obtain (cid:12)(cid:12) I ( x, y ) (cid:12)(cid:12) ≤ d ∗ C Z | z − y |≤ d ∗ |K n ( x, z ) − K n ( x, y ) | ( z − y ) dz + 2 d ∗ K n ( x, y ) Z | z − y |≤ d ∗ |K n ( z, y ) − K n ( y, y ) | ( z − y ) dz. Integrating the above inequality with respect to x and using (2.28) with (2.29), we get Z (cid:12)(cid:12) I ( x, y ) (cid:12)(cid:12) dx ≤ C Z | z − y |≤ d ∗ |K n ( z, y ) − K n ( y, y ) | ( z − y ) dz + C Z | z − y |≤ d ∗ K n ( z, z ) + K n ( y, y ) − K n ( z, y )( z − y ) dz. Using the bounds (2.83) in the second integral and (2.84) in the first one, in view of (2.72) weobtain the bound for I . To estimate I , we write Z (cid:12)(cid:12) I ( x, y ) (cid:12)(cid:12) dx ≤ Z | z | , | z ′ |≤ L | z − y | >d ∗ | z ′ − y | >d ∗ (cid:12)(cid:12)(cid:12)(cid:12) K n ( y, z ) K n ( z, z ′ ) K n ( z ′ , y )( z − y ) ( z ′ − y ) (cid:12)(cid:12)(cid:12)(cid:12) dzdz ′ ≤ C Z | z | , | z ′ |≤ L | z − y | >d ∗ | z ′ − y | >d ∗ (cid:12)(cid:12)(cid:12)(cid:12) K n ( y, z ) z − y (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) K n ( y, z ′ ) z ′ − y (cid:12)(cid:12)(cid:12)(cid:12) ! dzdz ′ ≤ C. Above bounds for I and I prove the second inequality of (2.36). Thus, Lemma 2.9 is proved. Acknowledgement. The author is grateful to Dr. M.V. Shcherbina for the problemstatement and fruitful discussions. References [1] M.L. Mehta , Random Matrices. Acad. Press, New York, 1991.232] A. 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