Bulk Universality for Unitary Matrix Models

Bulk Universality for Unitary Matrix Models

A proof of universality in the bulk of spectrum of unitary matrix models, assuming that the potential is globally C² and locally C³ function (see Theorem 1.2), is given. The proof is based on the determinant formulas for correlation functions in terms of polynomials orthogonal on the unit circle. Th...

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Date:2009
Main Author: Poplavskyi, M.
Format: Article
Language:English
Published: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2009
Series:Журнал математической физики, анализа, геометрии
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/106543
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Cite this:Bulk Universality for Unitary Matrix Models / M. Poplavskyi // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 3. — С. 245-274. — Бібліогр.: 10 назв. — англ.

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spelling nasplib_isofts_kiev_ua-123456789-1065432025-02-09T18:18:39Z Bulk Universality for Unitary Matrix Models Poplavskyi, M. A proof of universality in the bulk of spectrum of unitary matrix models, assuming that the potential is globally C² and locally C³ function (see Theorem 1.2), is given. The proof is based on the determinant formulas for correlation functions in terms of polynomials orthogonal on the unit circle. The sin-kernel is obtained as a unique solution of a certain nonlinear integrodifferential equation without using asymptotics of orthogonal polynomials. The author is grateful to Dr. M.V. Shcherbina for the problem statement and fruitful discussions. 2009 Article Bulk Universality for Unitary Matrix Models / M. Poplavskyi // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 3. — С. 245-274. — Бібліогр.: 10 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106543 en Журнал математической физики, анализа, геометрии application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
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description A proof of universality in the bulk of spectrum of unitary matrix models, assuming that the potential is globally C² and locally C³ function (see Theorem 1.2), is given. The proof is based on the determinant formulas for correlation functions in terms of polynomials orthogonal on the unit circle. The sin-kernel is obtained as a unique solution of a certain nonlinear integrodifferential equation without using asymptotics of orthogonal polynomials.
format Article
author Poplavskyi, M.
spellingShingle Poplavskyi, M.
Bulk Universality for Unitary Matrix Models
Журнал математической физики, анализа, геометрии
author_facet Poplavskyi, M.
author_sort Poplavskyi, M.
title Bulk Universality for Unitary Matrix Models
title_short Bulk Universality for Unitary Matrix Models
title_full Bulk Universality for Unitary Matrix Models
title_fullStr Bulk Universality for Unitary Matrix Models
title_full_unstemmed Bulk Universality for Unitary Matrix Models
title_sort bulk universality for unitary matrix models
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2009
url https://nasplib.isofts.kiev.ua/handle/123456789/106543
citation_txt Bulk Universality for Unitary Matrix Models / M. Poplavskyi // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 3. — С. 245-274. — Бібліогр.: 10 назв. — англ.
series Журнал математической физики, анализа, геометрии
work_keys_str_mv AT poplavskyim bulkuniversalityforunitarymatrixmodels
first_indexed 2025-11-29T13:07:08Z
last_indexed 2025-11-29T13:07:08Z
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fulltext Journal of Mathematical Physics, Analysis, Geometry 2009, vol. 5, No. 3, pp. 245–274 Bulk Universality for Unitary Matrix Models M. Poplavskyi Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkiv, 61103, Ukraine E-mail:poplavskiymihail@rambler.ru Received April 25, 2008 A proof of universality in the bulk of spectrum of unitary matrix mod- els, assuming that the potential is globally C2 and locally C3 function (see Theorem 1.2), is given. The proof is based on the determinant formulas for correlation functions in terms of polynomials orthogonal on the unit circle. The sin-kernel is obtained as a unique solution of a certain nonlinear integro- differential equation without using asymptotics of orthogonal polynomials. Key words: unitary matrix models, local eigenvalue statistics, univer- sality. 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 pn (U) dµn (U) = Z−1 n,2 exp { −nTrV ( U + U∗ 2 )} dµn (U) , (1.1) where U = {Ujk}n j,k=1 is an n × n unitary matrix, µn (U) is the Haar measure on the group U(n), Zn,2 is the normalization constant and V : [−1, 1] → R + is a continuous function called the potential of the model. Denote eiλj the eigen- values of unitary matrix U . The joint probability density of λj, corresponding to (1.1), is given by (see [1]) pn (λ1, . . . , λn) = 1 Zn ∏ 1≤j<k≤n ∣∣∣eiλj − eiλk ∣∣∣2 exp −n n∑ j=1 V (cos λj)  . (1.2) c© M. Poplavskyi, 2009 M. Poplavskyi To simplify notations, below we will write V (x) instead of V (cos x). Normalized Counting Measure of eigenvalues (NCM) is given by Nn (∆) = n−1� { λ (n) l ∈ ∆, l = 1, . . . , n } , ∆ ⊂ [−π, π]. (1.3) The random matrix theory deals with several asymptotic regimes of the eigen- value distribution. The global regime is centered around weak convergence of NCM (1.3). Global regime for unitary matrix models was studied in [2]. We will use the main result of [2]: Theorem 1.1. Assume that the potential V of the model (1.1) is a C2 (−π, π) function. Then: • there exists a measure N ∈ M1 ([−π, π]) with a compact support σ such that NCM Nn converges in probability to N ; • N has a bounded density ρ; • denote ρn := p (n) 1 the first marginal density, then for any φ ∈ H1 (−π, π)∣∣∣∣∫ φ (λ) ρn (λ) dλ− ∫ φ (λ) ρ (λ) dλ ∣∣∣∣ ≤ C ‖φ‖1/2 2 ∥∥φ′∥∥1/2 2 n−1/2 ln1/2 n, (1.4) where ‖·‖2 denotes L2 norm on [−π, π] One of the main topics of local regime is a universality of local eigenvalue statistics. Let p (n) l (λ1, . . . , λl) = ∫ pn (λ1, . . . , λl, λl+1, . . . , λn) dλl+1 . . . dλn (1.5) be the l-th marginal density of pn. Definition 1.1. We call by the bulk of the spectrum the set {λ ∈ σ : ρ (λ) > 0} , (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 spectrum lim n→∞ [nρn (λ)] −l p (n) l ( λ+ x1 nρn (λ) , . . . , λ+ xl nρn (λ) ) = det {S (xj − xk)}l j,k=1 , (1.7) 246 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 Bulk Universality for Unitary Matrix Models where S (x) = sinπx πx . (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 correlation functions 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 appeared only in the 90s. The case of general V which is locally C3 function was studied in [4]. The case of real analytic potential V was studied in [5], where the asymptotics of orthogonal polynomials were 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 or- thogonal with respect to varying weight on the unit circle. Consider a sys- tem of functions { eikλ }∞ k=0 and use for them the Gram–Schmidt procedure in L2 ( [−π, π] , e−nV (λ) ) . For any n we get the system of functions { P (n) k (λ) }∞ k=0 which are orthogonal and normalized in L2 ( [−π, π] , e−nV (λ) ) . 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)/2. (1.9) Then we obtain the orthogonal in L2(−π, π) functions π∫ −π ψ (n) k (λ)ψ(n) l (λ) dλ = δkl. (1.10) The reproducing kernel of the system (1.9) is given by Kn (λ, µ) = n−1∑ 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∫ −π Kn (λ, ν)Kn (ν, µ) dν = Kn (λ, µ) , (1.12) and from the Cauchy inequality we have |Kn (λ, µ)|2 ≤ Kn (λ, λ)Kn (µ, µ) . (1.13) Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 247 M. Poplavskyi We also use below the determinant form of the marginal densities (1.5) (see [1]) p (n) l (λ1, . . . , λl) = (n− l)! n! det ‖Kn (λj , λk)‖l j,k=1 . (1.14) In particular, ρn (λ) = n−1Kn (λ, λ) , (1.15) p (n) 2 (λ, µ) = Kn (λ, λ)Kn (µ, µ)− |Kn (λ, µ)|2 n(n− 1) . (1.16) The main result of the paper is Theorem 1.2. Assume that V (λ) is a C2 (−π, π) function, and there exists an interval (a, b) ⊂ σ such that sup λ∈(a,b) |V ′′′ (λ) | ≤ C1, ρ (λ) ≥ C2, λ ∈ (a, b) . (1.17) Then for any d > 0 and λ0 ∈ [a+ d, b− d] for Kn defined in (1.11) we have lim n→∞ [Kn (λ0, λ0)] −1 Kn ( λ0 + x Kn (λ0, λ0) , λ0 + y Kn (λ0, λ0) ) = ei(x−y)/2ρ(λ0) sinπ (x− y) π (x− y) (1.18) uniformly in (x, y), varying on a compact set of R 2. R e m a r k 1.3. It is easy to see that the universality conjecture (1.7) follows from Theorem 1.2 by (1.14). The method of the proof is a version of the one used in [4]. An important part of the proof is a uniform convergence of ρn to ρ in a neighborhood of λ0: Theorem 1.4. Under the assumptions of Theorem 1.2 for any d > 0 there exists C (d) > 0 such that for any λ ∈ [a+ d, b− d] |ρn (λ)− ρ (λ)| ≤ C (d)n−2/9. (1.19) 248 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 Bulk Universality for Unitary Matrix Models 2. Proof of Basic Results P r o o f of Theorem 1.4. We will use some facts from the integral transfor- mations theory (see [7]). Definition 2.1. Assume that g (λ) is a continuous function on the interval [−π, π]. Then its Germglotz transformation is given by F [g] (z) = π∫ −π eiλ + eiz eiλ − eiz g (λ) dλ, (2.1) where z ∈ C\R. The inverse transformation is given by g (µ) = 1 2π lim z→µ+i0 �F [g] (z) . (2.2) For z = µ+ iη, η �= 0, we set fn (z) = π∫ −π eiλ + eiz eiλ − eiz ρn (λ) dλ. (2.3) Bellow we will derive a ”square” equation for fn. Denote In (z) = π∫ −π V ′(λ) eiλ + eiz eiλ − eiz ρn (λ) dλ. (2.4) Integrating by parts in (2.4), from (1.5) we obtain In (z) = 1 Zn ∫ V ′ (λ1) eiλ1 + eiz eiλ1 − eiz ∏ j<k ∣∣∣eiλj − eiλk ∣∣∣2exp −n n∑ j=1 V (λj)  n∏ j=1 dλj = 1 nZn ∫ e−nV (λ1) d dλ1 eiλ1 + eiz eiλ1 − eiz ∏ j<k ∣∣∣eiλj − eiλk ∣∣∣2exp −n n∑ j=2 V (λj)   n∏ j=1 dλj . The integrated term equals 0, because all functions here are 2π -periodic. After differentiation we have the sum of n terms under integral sign. Denote I0 (z) = 1 nZn ∫ d dλ1 ( eiλ1 + eiz eiλ1 − eiz )∏ j<k ∣∣∣eiλj − eiλk ∣∣∣2 exp −n n∑ j=1 V (λj)  n∏ j=1 dλj , Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 249 M. Poplavskyi Im (z) = 1 nZn ∫ eiλ1 + eiz eiλ1 − eiz ∏ 2≤j<k≤n ∣∣∣eiλj − eiλk ∣∣∣2 d dλ1 ∣∣∣eiλ1 − eiλm ∣∣∣2 × ∏ k �=m ∣∣∣eiλ1 − eiλk ∣∣∣2 exp −n n∑ j=1 V (λj)  n∏ j=1 dλj, m = 2, n. From symmetry with respect to λj we obtain that all Im (z), except I0(z), are equal, hence In (z) = I0 (z) + (n− 1) I2 (z) . I0 (z) = 1 n π∫ −π d dλ1 ( eiλ1 + eiz eiλ1 − eiz ) ρn (λ1) dλ1 = −2i n π∫ −π eiλ1eiz (eiλ1 − eiz)2 ρn (λ1) dλ1 = − i 2n π∫ −π ( eiλ1 + eiz eiλ1 − eiz )2 ρn (λ1) dλ1 + i 2n . To transform I2, we use the symmetry of p(n) 2 ( p(n) 2 (λ1, λ2) = p (n) 2 (λ2, λ1) ). I2 (z) = 1 n ∫ eiλ1 + eiz eiλ1 − eiz d dλ1 ∣∣∣eiλ1 − eiλ2 ∣∣∣2 |eiλ1 − eiλ2 |2 p (n) 2 (λ1, λ2) dλ1dλ2 = i n ∫ eiλ1 + eiz eiλ1 − eiz eiλ1 + eiλ2 eiλ1 − eiλ2 p (n) 2 (λ1, λ2) dλ1dλ2 = i 2n ∫ ( eiλ1 + eiz eiλ1 − eiz − eiλ2 + eiz eiλ2 − eiz ) eiλ1 + eiλ2 eiλ1 − eiλ2 p (n) 2 (λ1, λ2) dλ1dλ2 = − i 2n ∫ 2 ( eiλ1 + eiλ2 ) eiz (eiλ1 − eiz) (eiλ2 − eiz) p (n) 2 (λ1, λ2) dλ1dλ2 = i 2n − i 2n ∫ eiλ1 + eiz eiλ1 − eiz eiλ2 + eiz eiλ2 − eiz p (n) 2 (λ1, λ2) dλ1dλ2. Therefore, from (1.5) and (1.14) we obtain In (z) = i 2 − i 2 f2 n (z)− i n2 ∫ |Kn (λ1, λ2)|2 ( eiλ1 − eiλ2 )2 e2iz (eiλ1 − eiz)2 (eiλ2 − eiz)2 dλ1dλ2. (2.5) On the other hand, denoting Qn (z) = π∫ −π eiλ + eiz eiλ − eiz ( V ′ (λ)− V ′ (µ) ) ρn (λ) dλ, (2.6) 250 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 Bulk Universality for Unitary Matrix Models for z = µ+ iη, from (2.3) we get In (z) = Qn (z) + V ′ (µ) fn (z) . (2.7) Finally, from (2.5) and (2.7) we obtain the ”square” equation f2 n (z)− 2iV ′ (µ) fn (z)− 2iQn (z)− 1 = − 2 n2 Gn (z) , (2.8) with Gn (z) = ∫ |Kn (λ1, λ2)|2 ( eiλ1 − eiλ2 )2 e2iz (eiλ1 − eiz)2 (eiλ2 − eiz)2 dλ1dλ2. To proceed further we have to prove the following properties of the reproducing kernel Kn. Lemma 2.1. Let Kn (λ, µ) be defined by (1.11). Then under the conditions of Theorem 1.2 for any δ > 0∣∣∣∣∫ (eiλ − eiµ ) |Kn (λ, µ)|2 dµ ∣∣∣∣ ≤ 1 2 [∣∣∣ψ(n) n−1 (λ) ∣∣∣2 + ∣∣∣ψ(n) n (λ) ∣∣∣2] , (2.9)∫ ∣∣∣eiλ − eiµ ∣∣∣2 |Kn (λ, µ)|2 dµ ≤ [∣∣∣ψ(n) n−1 (λ) ∣∣∣2 + ∣∣∣ψ(n) n (λ) ∣∣∣2] , (2.10)∫ ∣∣∣eiλ − eiµ ∣∣∣2 |Kn (λ, µ)|2 dλdµ ≤ 2, (2.11) ∫ |eiλ−eiµ|>δ |Kn (λ, µ)|2 dµ ≤ δ−2 [∣∣∣ψ(n) n−1 (λ) ∣∣∣2 + ∣∣∣ψ(n) n (λ) ∣∣∣2] , (2.12) ∫ |eiλ−eiµ|>δ |Kn (λ, µ)|2 dλdµ ≤ 2δ−2. (2.13) It is easy to see that ∣∣eiλ − eiz ∣∣ > C |η| if |η| < 1 for some C > 0. Hence, from (2.11) and (2.8) we derive f2 n (z)− 2iV ′ (µ) fn (z)− 2iQn (z)− 1 = O ( n−2η−4 ) . (2.14) Lemma 2.2. Under the conditions of Theorem 1.2 for any d > 0 and λ ∈ [a+ d, b− d] ρn (λ) ≤ C, (2.15)∣∣∣∣dρn (λ) dλ ∣∣∣∣ ≤ C1 (∣∣∣ψ(n) n (λ) ∣∣∣2 + ∣∣∣ψ(n) n−1 (λ) ∣∣∣2)+ C2. (2.16) Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 251 M. Poplavskyi 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 |Qn (µ+ iη)−Qn (µ)| ≤ ∣∣e−η − 1 ∣∣ π∫ −π |V ′ (λ)− V ′ (µ)| ρn (µ) |eiλ − eiµ| |eiλ − eiz | dλ ≤ Cη  ∫ |λ−µ|<d/2 dλ∣∣∣(λ− µ)2 + η2 ∣∣∣1/2 + ∫ |λ−µ|>d/2 ρn (λ) dλ∣∣∣(λ− µ)2 + η2 ∣∣∣1/2 |λ− µ|  ≤ Cη ln−1 η + Cηd−2 ≤ Cη ln−1 η. (2.17) Besides, applying (1.4), for φ (λ) = eiλ + eiµ eiλ − eiµ ( V ′(λ)− V ′(µ) ) we get Qn (µ) = Q (µ) +O ( n−1/2 ln1/2 n ) , (2.18) where Q (µ) = π∫ −π eiλ + eiµ eiλ − eiµ ( V ′ (λ)− V ′ (µ) ) ρ (λ) dλ. (2.19) Combining (2.17) and (2.18), we find Qn (µ+ iη) = Q (µ) +O ( η ln−1 η ) +O ( n−1/2 ln1/2 n ) . (2.20) From (2.20) and (2.14) for z = µ+ in−4/9 we have f2 n (z)− 2iV ′ (µ) fn (z)− 2iQ (µ)− 1 = O(n−2/9). (2.21) Lemma 2.3. ρ (µ) = 1 2π √ 2iQ (µ) + 1− (V ′ (µ))2. (2.22) Lemma 2.3 and the equation (2.21) imply that for z = µ+ in−4/9 1 2π �fn (z) = ρ (µ) +O ( n−2/9 ) ρ−1 (µ) . (2.23) Lemma 2.4. For d > 0, k = n− 1, n and µ ∈ [a+ d, b− d]∫ |λ−µ|<n−1/4 ∣∣∣ψ(n) k (λ) ∣∣∣2 dλ ≤ Cn−1/4, (2.24) ∣∣∣ψ(n) k (λ) ∣∣∣2 ≤ Cn7/8, |µ− λ| ≤ n−1/4. (2.25) 252 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 Bulk Universality for Unitary Matrix Models Taking into account (2.23), to prove Theorem 1.4 it is enough to show that 1 2π �fn (z) = ρn (µ) +O ( n−2/9 ) . We use an evident relation �eiλ + eiz eiλ − eiz = sinh η cosh η − cos (λ− µ) = d dλ 2 arctan ( tan ( λ− µ 2 ) coth (η 2 )) . Combining the relation 1 2π ∫ �eiλ + eiz eiλ − eiz dλ = 1 with (2.15), we obtain ∣∣∣∣ 12πfn (z)− ρn (µ) ∣∣∣∣ = (2π)−1 ∣∣∣∣∣ ( ∫ |µ−λ|≤η1/2 + ∫ η1/2≤|µ−λ|≤d/2 + ∫ |µ−λ|≥d/2 ) sinh η cosh η − cos (λ− µ) × (ρn (λ)− ρn (µ)) dλ ∣∣∣∣∣ ≤ C ∣∣∣∣∣ ∫ |s|≤η1/2 sinh η cosh η − cos s (ρn (s+ µ)− ρn (µ)) ds ∣∣∣∣∣+ Cη1/2 + Cη. Using (2.16) and (2.24), we get finally∣∣∣∣ 12πfn (z)− ρn (µ) ∣∣∣∣ ≤ C ∫ |s|<η1/2 ∣∣ρ′n (µ+ s) ∣∣ ds+ Cη1/2 ≤ Cη1/2. Theorem 1.4 is proved. Now we pass to the proof of Theorem 1.2. We will use the following repre- sentation of Kn, which can be derived from the well-known identities of random matrix theory (see [1]) 1 n Kn (λ, µ) = 1 n n−1∑ j=0 ψ (n) l (λ)ψ(n) l (µ) = Q−1 n,2e −n(V (λ)+V (µ))/2 × ∫ n∏ j=2 ( eiλ − eiλj )( e−iµ − e−iλj ) e−nV (λj)dλj ∏ 2≤j<k≤n ∣∣∣eiλj − eiλk ∣∣∣2 , (2.26) where Qn,2 = n! n−1∏ j=0 ∣∣∣γ(n) l ∣∣∣−2 , and γ (n) l is the coefficient in front of eilλ in the function P (n) l . Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 253 M. Poplavskyi R e m a r k 2.5. Consider the determinant (see (1.2)) det { eikλj }n−1 k,j=0 = ei(n−1) ∑ λj/2 det { ei(k−(n−1)/2)λj }n−1 k,j=0 . Taking the complex conjugate, we obtain det { eikλj }n−1 k,j=0 = e−i(n−1) ∑ λj/2 det { e−i(k−(n−1)/2)λj }n−1 k,j=0 = (−1)[n/2]e−i(n−1) ∑ λj/2 det { ei(k−(n−1)/2)λj }n−1 k,j=0 . Thus, from (2.26) we get that the function e−i(n−1)(λ−µ)/2Kn (λ, µ) is real valued. Now denote K̃n (x, y) = 1 n Kn ( λ0 + x n , λ0 + y n ) , Kn (x, y) = e−i(n−1)(x−y)/2nK̃n (x, y) . (2.27) From the above we have that Kn(x, y) is a real-valued and symmetric function. We get from (1.11)–(1.13) nπ∫ −nπ Kn (x, z)Kn (z, y) dz = Kn (x, y) , |Kn (x, y)|2 ≤ Kn (x, x)Kn (y, y) , (2.28) Kn (x, x) = ρn (λ0 + x/n) ≤ C, |Kn (x, y)| ≤ C, for |x| , |y| ≤ nd0/2 (2.29) Differentiating in (2.26) K̃n (x, y) with respect to x for λ = λ0+x/n, µ = µ0+y/n, we get ∂ ∂x K̃n (x, y) = −1 2 V ′ (λ) K̃n (x, y) + n− 1 Qn,2 e−n(V (λ)+V (µ))/2 × ∫ ieiλ eiλ − eiλ2 n∏ j=2 ( eiλ − eiλj )( e−iµ − e−iλj ) dλj ∏ 2≤j<k≤n ∣∣∣eiλj − eiλk ∣∣∣2 = −1 2 V ′ (λ) K̃n (x, y) + i n2 π∫ −π eiλ eiλ − eiλ2 (Kn (λ2, λ2)Kn (λ, µ)−Kn (λ, λ2)Kn (λ2, µ)) dλ2 = −1 2 V ′ (λ) K̃n (x, y) 254 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 Bulk Universality for Unitary Matrix Models i 2n2 π∫ −π eiλ + eiλ2 eiλ − eiλ2 (Kn (λ2, λ2)Kn (λ, µ)−Kn (λ, λ2)Kn (λ2, µ)) dλ2 + i(n − 1) 2n2 Kn (λ, µ) = −1 2 V ′ (λ) K̃n (x, y) + 1 2n nπ∫ −nπ cot ( x− z 2n )( K̃n (z, z) K̃n (x, y)− K̃n (x, z) K̃n (z, y) ) dz + i(n− 1) 2n K̃n (x, y) . (2.30) Lemma 2.6. Denote D (λ) = V ′ (λ) + v.p. π∫ −π cot s 2 ρn (λ+ s) ds. Then for any d > 0 we have uniformly in [a+ d, b− d] |D (λ)| ≤ Cn−1/4 lnn. The definition of Kn (2.27), the above Lemma, and the bound (2.29) yield ∂ ∂x Kn (x, y) = 1 2n v.p. nπ∫ −nπ cot ( z − x 2n ) Kn (x, z)Kn (z, y) dz +O(n−1/4 lnn). (2.31) Below we take |x| , |y| ≤ L = lnn. Then from the inequality |z| ≤ nπ we get∣∣∣∣x− z 2n ∣∣∣∣ ≤ 3π/4. The function x cot x is bounded on [0, 3π/4], thus ∣∣∣∣ 12n cot ( x− z 2n )∣∣∣∣ ≤ C ∣∣∣∣ 1 x− z ∣∣∣∣ . For |x| , |y| ≤ L we can restrict integration in (2.31) by the domain |z| ≤ 2L, substituting O(n−1/4 lnn) by O (L−1 ) . This follows from the bound∣∣∣∣∣∣∣ 1 2n ∫ 2L≤|z|≤nπ cot ( x− z 2n ) Kn (x, z)Kn (z, y) dz ∣∣∣∣∣∣∣ ≤ CL−1 ∫ |Kn (x, z)| |Kn (z, y)| dz ≤ CL−1. Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 255 M. Poplavskyi Note that 1 2n cot x 2n − 1 x = O ( n−2 lnn ) , forx = O (lnn) . Hence, from the above estimates and (2.31) we get ∂ ∂x Kn (x, y) = v.p. ∫ |z|≤2L Kn (x, z)Kn (z, y) z − x dz +O (L−1 ) . (2.32) The following lemma shows that Kn behaves almost like a difference kernel. Lemma 2.7. For any d > 0 we have uniformly in λ0 ∈ [a+ d, b− d] and |x| , |y| ≤ nd/4∣∣∣∣ ∂∂xKn (x, y) + ∂ ∂y Kn (x, y) ∣∣∣∣ ≤ C ( n−1/8 + |x− y|n−2 ) , (2.33) |Kn (x, y)−Kn (0, y − x)| ≤ C |x| ( n−1/8 + |x− y|n−2 ) . (2.34) R e m a r k 2.8. Note that the last inequality with λ0+x1/n instead of λ0, and x2 − x1 instead of x and y, leads to the bound that is valid for any |x1,2| ≤ nd0/8 |Kn (x2, x2)−Kn (x1, x1)| ≤ Cn−1/8 |x2 − x1| . (2.35) Lemma 2.9. For any |x| , |y| ≤ L∣∣∣∣ ∂∂xKn (x, y) ∣∣∣∣ ≤ C, ∫ |x|≤L ∣∣∣∣ ∂∂xKn (x, y) ∣∣∣∣2 dx ≤ C. (2.36) Denote K∗ n(x) = Kn(x, 0)1|x|≤L +Kn(L, 0)(1 + L− x)1L<x≤L+1 (2.37) + Kn(−L, 0)(1 + L+ x)1−L−1≤x<−L, and observe that for y = 0 and for any |x| ≤ L/3, similarly to (2.32), we can restrict the integration in (2.32) to |z| ≤ 2L/3 with a mistake O(L−1). This and Lemma 2.7 give us the equation ∂ ∂x K∗ n(x) = ∫ |z|≤2L/3 K∗ n(z)K∗ n(x− z) z dz + rn(x) +O(L−1), (2.38) 256 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 Bulk Universality for Unitary Matrix Models where rn(x) = ∫ |z|≤2L/3 Kn(z, 0)(Kn(x, z) −Kn(0, x− z)) z dz, and by Lemma 2.7, for |x| ≤ L/3 we have rn(x) = O(n−1/8 log n). Now, using the estimates similar to (2.32), we can restrict the integration in (2.38) to the real axis. From Lemma 2.9 and the relations (2.28), (2.29) we get∫ |K∗ n(x)|2dx ≤ ∫ |Kn(x, 0)|2dx+ C ′ ≤ C, ∫ ∣∣∣∣ ddxK∗ n(x) ∣∣∣∣2dx ≤ C. (2.39) Consider the Fourier transform K̂∗ n(p) = ∫ K∗ n(x)e ipxdx, where the integral is defined in the L2(R) sense, and write K∗ n(x) as K∗ n(x) = (2π)−1 ∫ K̂∗ n(p)e −ipxdp. (2.40) From (1.19) we have ∫ K̂∗ n(p)dp = 2πρ(λ0) + o(1), (2.41) and from (2.39) and the Parseval equation we obtain∫ p2|K̂∗ n(p)|2dp ≤ C. (2.42) From the definition of Kn(x, y) we get that the kernel is positive definite L∫ −L Kn(x, y)f(x)f(y)dxdy ≥ 0, f ∈ L2(R), therefore from (2.34) we have for any function f ∈ L2(R)∫ K̂∗ n(p)|f̂(p)|2dp ≥ −C||f ||2L2(R)(n −1/8 log4 n+O(L−1)). (2.43) From the Parseval equation and (2.34) there follows∫ |K̂∗ n(p)− K̂∗ n(−p)|2dp ≤ 2π ∫ |K∗ n(x)−K∗ n(−x)|2dx ≤ Cn−1/8 log3 n. (2.44) Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 257 M. Poplavskyi By the definition of singular integrals∫ K∗ n(z)K∗ n(x− z) z dz = lim ε→+0 ∫ dzK∗ n(z)K∗ n(y − z)�(z + iε)−1. (2.45) In accordance with the relation∫ eipz�(z + iε)−1dz = πie−ε|p|sgn p and the Parseval equation, we can write the r.h.s. of (2.38) as i 4π lim ε→+0 ∫ dpdp′K̂∗ n(p)K̂∗ n(p ′)e−ipxsign(p − p′)e−ε|p−p′| = i 2π ∫ dpe−ipxK̂∗ n(p) p∫ 0 K̂∗ n(p ′)dp′ − i 4π ∫ dpe−ipxK̂∗ n(p) ∞∫ 0 (K̂∗ n(p ′)− K̂∗ n(−p′))dp′. (2.46) Note that both integrals are absolutely convergent because K̂∗ n ∈ L1(R) by (2.42). Now, using the Schwarz inequality and (2.42), we can estimate the second com- ponent∣∣∣∣∣∣ ∞∫ 0 (K̂∗ n(p ′)− K̂∗ n(−p′))dp′ ∣∣∣∣∣∣ ≤ ∣∣∣∣∣∣∣ L2∫ 0 (K̂∗ n(p ′)− K̂∗ n(−p′))dp′ ∣∣∣∣∣∣∣ + ∫ |p|>L2 |K̂∗ n(p ′)|dp′ ≤ L (∫ |K̂∗ n(p ′)− K̂∗ n(−p′)|2dp′ )1/2 + CL−1. Thus, from (2.44)–(2.46) we have uniformly in |x| < L/3 ∫ K∗ n(z)K∗ n(x− z) z dz = i 2π ∫ dpK̂∗ n(p)e −ipx p∫ 0 K̂∗ n(p ′)dp′ +O(L−1). This allows us to transform (2.38) into the following asymptotic relation that is valid for |x| ≤ L/3: ∫ K̂∗ n(p) ( p∫ 0 K̂∗ n(p ′)dp′ − p ) e−ipxdp = O(L−1). (2.47) 258 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 Bulk Universality for Unitary Matrix Models Consider the functions Fn(p) = p∫ 0 K̂∗ n(p ′)dp′. (2.48) Since pK̂∗ n(p) ∈ L2(R), the sequence {Fn(p)} consists of functions that are uni- formly bounded and equicontinuous on R. Thus {Fn(p)} is a compact family with respect to uniform convergence. Hence, the limit F of any subsequence {Fnk } possesses the properties: (a) 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 (−∞) = 2πρ(λ0) (see (2.41)); (e) the following equation is valid for any smooth function g with the compact support (see (2.47)): ∫ (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) = p1|p|≤p0 + p0 sign(p)1|p|≥p0 , where p0 = πρ(λ0) from (d). We conclude that (2.49) is uniquely solvable, thus the sequence {Fn} converges uniformly on any compact to the above F . This and (2.48) imply the weak convergence of the sequence {K∗ n} to the function ρ (λ0)S (ρ (λ0)x), where S(x) is defined in (1.8). But weak convergence combined with (2.29) and (2.36) implies the uniform convergence of {K∗ n} to K∗ on any interval. Thus we have uniformly in (x, y), varying on a compact set of R 2, lim n→∞Kn(x, y) = ρ (λ0)S (ρ (λ0) (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∫ −π eiλψ (n) k (λ)ψ(n) j (λ) dλ. (2.50) Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 259 M. Poplavskyi Note that from the orthogonality (2.66) we have r(n) k,j = 0 for j > k + 1. Thus, eiλ ψ (n) k (λ) = k+1∑ j=0 r (n) k,jψ (n) j (λ) . (2.51) Multiplication on eiλ is isometric in L2 [−π, π], therefore k+1∑ j=0 ∣∣∣r(n) k,j ∣∣∣2 = ∥∥∥ψ(n) k (λ) ∥∥∥ 2 = 1. Finally we are ready to prove (2.9) π∫ −π ( eiλ − eiµ ) |Kn (λ, µ)|2 dµ = eiλKn (λ, λ)− π∫ −π eiµ n−1∑ m=0 ψ(n) m (µ)ψ(n) m (λ) n−1∑ l=0 ψ (n) l (λ)ψ(n) l (µ) dµ = eiλKn (λ, λ)− n−1∑ l,m=0 r (n) m,lψ (n) l (λ)ψ(n) m (λ) = r (n) n−1,nψ (n) n−1 (λ)ψ (n) n (λ). (2.52) Now, using the Cauchy inequality and the bound ∣∣∣r(n) n−1,n ∣∣∣ ≤ 1, we get (2.9). Similarly, it is easy to obtain the relation π∫ −π ∣∣∣eiλ − eiµ ∣∣∣2 |Kn (λ, µ)|2 dµ = 2� { eiλr (n) n−1,nψ (n) n−1 (λ)ψ (n) n (λ) } , which implies (2.10). The bounds (2.11),(2.12),(2.13) are evident consequences of (2.10). The lemma is proved. P r o o f of Lemma 2.2. Observe that dρn (λ) dλ = dρn (λ+ t) dt ∣∣∣∣ t=0 . Changing variables in (1.5) λj = µj + t, in view of periodicity of all functions in the consideration, we have the representation for ρn (λ+ t) ρn (λ+ t) = 1 Zn ∫ e−nV (λ+t) ∏ 2≤j<k≤n ∣∣eiµj − eiµk ∣∣2 n∏ j=2 e−nV (µj+t) ∣∣∣eiλ − eiµj ∣∣∣2 dµj . 260 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 Bulk Universality for Unitary Matrix Models After differentiating with respect to t, for t = 0 we get dρn (λ) dλ = −nV ′ (λ) p(n) 1 (λ)− n (n− 1) π∫ −π V ′ (µ) p(n) 2 (λ, µ) dµ = −V ′ (λ)Kn (λ, λ)− π∫ −π V ′ (µ) [ Kn (λ, λ)Kn (µ, µ)− |Kn (λ, µ)|2 ] dµ. (2.53) Since V ′ (λ) is an odd function, and Kn (λ, λ) is an even function, we obtain π∫ −π V ′ (λ)Kn (λ, λ) dλ = 0. Thus, from (2.53) we get ρ′n (λ) = π∫ −π ( V ′ (µ)− V ′ (λ) ) |Kn (λ, µ)|2 dµ. (2.54) We split this integral in two parts corresponding to the domains |µ− λ| ≤ d/2 and |µ− λ| ≥ d/2. In the second integral we use (2.12). It follows from (1.17) that in the first integral we can rewrite V ′ (λ) as V ′ (µ)− V ′ (λ) = (µ− λ)V ′′ (λ) +O ( |µ− λ|2 ) = ( eiµ − eiλ ) V ′′ (λ) ieiλ +O (( eiµ − eiλ )2 ) , and using (2.9) and (2.10), we get (2.16). To prove (2.15) we use the following well-known inequality. Proposition 2.10. For any function u : [a1, b1] → C with u′ ∈ L1(a1, b1) we have ‖u‖∞ ≤ ∥∥u′∥∥ 1 + (b1 − a1)−1 ‖u‖1 , (2.55) where ‖ · ‖1, ‖ · ‖∞ are the L1 and uniform norms on the interval [a1, b1]. This inequality can be obtained easily from the relation u (λ) = 1 b1 − a1 b1∫ a1 (u (λ)− u (µ)) dµ+ 1 b1 − a1 b1∫ a1 u (µ) dµ. Using (2.55) for u = ρn and the interval [a+ d, b− d], we get (2.15). Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 261 M. Poplavskyi P r o o f of Lemma 2.3. From (1.4) and (2.21) we have for nonreal z f2 (z)− 2iV ′ (µ) f (z)− 2iQ (z)− 1 = 0, (2.56) where f (z) is the Germglotz transformation of the limiting density ρ (λ). By (2.19) and (2.2), Q (µ+ i0) is an imaginary valued, bounded, continuous function. And from (2.2) we obtain ρ (µ) = 1 2π �f (µ+ i0) . Computing imaginary and real parts in (2.56), we get the relations �f (µ+ i0) = V ′ (µ) , (2.57) �f (µ+ i0) = √ 2iQ (µ) + 1− (V ′ (µ))2, (2.58) from which we obtain (2.22). P r o o f of Lemma 2.4. To prove (2.24) with k = n − 1 we introduce the probability density p−n (λ1, . . . , λn−1) = 1 Z− n ∏ 1≤j<k≤n−1 ∣∣∣eiλj − eiλk ∣∣∣2 exp −n n−1∑ j=1 V (λj)  . (2.59) Denote ρ−n (λ) = n− 1 n ∫ p−n (λ, λ2 . . . , λn−1) dλ2 . . . dλn−1 = 1 n n−2∑ j=0 ∣∣∣ψ(n) j (λ) ∣∣∣2 . (2.60) Thus we get ∣∣∣ψ(n) n−1 (λ) ∣∣∣2 = n ( ρn (λ)− ρ−n (λ) ) . (2.61) Analogously to the equation (2.8), we can obtain the ”square” equation i 2 [ f− n (z) ]2 + π∫ −π eiλ + eiz eiλ − eiz V ′ (λ) ρ−n (λ) dλ = i 2 +O ( n−2η−4 ) , (2.62) for the Germglotz transformation f− n (z) of the function ρ−n (λ). Denote ∆n (z) = n ( fn (z)− f− n (z) ) = π∫ −π eiλ + eiz eiλ − eiz ∣∣∣ψ(n) n−1 (λ) ∣∣∣2 dλ. (2.63) 262 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 Bulk Universality for Unitary Matrix Models Subtracting (2.62) from (2.8), we obtain for z = µ+ in−1/4 i 2 ∆n (z) ( fn (z) + f− n (z) ) = − π∫ −π eiλ + eiz eiλ − eiz V ′ (λ) ∣∣∣ψ(n) n−1 (λ) ∣∣∣2 dλ+O (1) , i 2 ∆n (z) ( fn (z) + f− n (z)− 2iV ′ (µ) ) = π∫ −π eiλ + eiz eiλ − eiz ( V ′ (µ)− V ′ (λ) ) ∣∣∣ψ(n) n−1 (λ) ∣∣∣2 dλ+O (1) = O (1) . Note that �f− n (z) > 0 for �z > 0 therefore �∆n ( µ+ in−1/4 ) ≤ C �fn ( µ+ in−1/4 ) Analogously to (2.23), we can obtain for z = µ+ in−1/4 1 2π �fn (z) = ρ (µ) +O ( n−1/8 ) ρ−1 (µ) , hence �fn (z) ≥ C2 for sufficiently large n, where C2 is defined in (1.17). Thus, �∆n ( µ+ in−1/4 ) ≤ C. Note that �eiλ + eiz eiλ − eiz = sinh η cosh η − cos (µ− λ) ≥ C η η2 + (µ− λ)2 , for η2 + (µ− λ)2 < 1. Thus, ∫ |λ−µ|<n−1/4 ∣∣∣ψ(n) n−1 (λ) ∣∣∣2 dλ ≤ 2n−1/2 ∫ |λ−µ|<n−1/4 ∣∣∣ψ(n) n−1 (λ) ∣∣∣2 n−1/2 + (µ− λ)2 dλ ≤ Cn−1/4�∆n ( µ+ in−1/4 ) ≤ Cn−1/4. A similar bound can be obtained for ψ(n) n (λ) by using the densities: p+ n (λ1, . . . , λn+1) = 1 Q+ n,2 ∏ 1≤j≤n+1 e−nV (λj) ∏ 1≤j<k≤n+1 ∣∣∣eiλj − eiλk ∣∣∣2 , ρ+ n (λ) = n+ 1 n ∫ p+ n (λ, λ2, . . . , λn+1) dλ2 . . . dλn+1 = 1 n n∑ j=0 ∣∣∣ψ(n) j (λ) ∣∣∣2 . Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 263 M. Poplavskyi Analogously, we will have ∣∣∣ψ(n) n (λ) ∣∣∣2 = n (ρ+ n (λ)− ρn (λ)). Thus, the esti- mate (2.24) is proved. Now we proceed to prove (2.25) for k = n. We use the inequality Proposition 2.11. For any C1 function u : [a1, b1] → C ‖u‖2 ∞ ≤ 2 ‖u‖2 ∥∥u′∥∥ 2 + (b1 − a1)−1 ‖u‖2 2 , (2.64) where ‖ · ‖2, ‖ · ‖∞ are the L2 and uniform norms on the interval [a1, b1]. This inequality is a simple consequence of the relation u2 (λ) = 1 b1 − a1 b1∫ a1 ( u2 (λ)− u2 (µ) ) dµ+ 1 b1 − a1 b1∫ a1 u2 (µ) dµ. Consider the interval ∆ = [ λ− n−1/4, λ+ n−1/4 ] and the function ψ (λ) = ψ (n) n (λ). From the inequality we have |ψ (λ)|2 ≤ 2 ‖ψ‖2,∆ ∥∥ψ′∥∥ 2,∆ + 1 2 n1/4 ‖ψ‖2,∆ , (2.65) where ‖·‖2,∆ is L2 norm on the interval ∆. It is easy to see that ‖ψ‖2,∆ ≤ ‖ψ‖2,[−π,π] = 1. Denote P (λ) = P (n) n (λ) and ω (λ) = e−nV (λ)/2, then ψ (λ) = P (λ)ω (λ). Now we estimate ‖ψ′‖2,[−π,π]:∥∥ψ′∥∥ 2,[−π,π] = ∥∥P ′ω + Pω′∥∥ 2,[−π,π] ≤ ∥∥P ′ω ∥∥ 2,[−π,π] + ∥∥Pω′∥∥ 2,[−π,π] ,∥∥Pω′∥∥ 2,[−π,π] = n 2 ∥∥PV ′ω ∥∥ 2,[−π,π] ≤ Cn ‖Pω‖2,[−π,π] = Cn, ∥∥P ′ω ∥∥2 2,[−π,π] = ∫ P ′ (λ)P ′ (λ)ω2 (λ) dλ = − ∫ P (λ)P ′′ (λ)ω2 (λ) dλ + n ∫ P (λ)P ′ (λ)V ′ (λ)ω2 (λ) dλ. Using the orthogonality∫ e−imλω (λ)ψ(n) k dλ = 0, for m < k, (2.66) 264 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 Bulk Universality for Unitary Matrix Models we obtain∫ P (λ)P ′′ (λ)ω2 (λ) dλ = ∫ P (λ) γ(n) n (−in)2 e−inλω2 (λ) dλ = −in ∫ P (λ)P ′ (λ)ω2 (λ) dλ, where γ(n) n is defined in (2.26). Thus,∥∥P ′ω ∥∥2 2,[−π,π] = n ∫ P (λ)P ′ (λ) ( V ′ (λ) + i ) ω2 (λ) dλ ≤ Cn ∥∥P ′ω ∥∥ 2,[−π,π] , and we obtain that ‖P ′ω‖2,[−π,π] ≤ Cn. Combining all above bounds, we conclude that ‖ψ′‖2,[−π,π] ≤ Cn. Now, using (2.65) and (2.24), we obtain (2.25) for k = n. For k = n− 1 the proof is the same. P r o o f of Lemma 2.6. Similarly to (2.21) for η = n−3/8 and µ ∈ [a+ d, b− d] for fn, defined in (2.3), we obtain∣∣�fn (µ+ iη)− V ′ (µ) ∣∣ ≤ Cn−3/8 lnn. (2.67) Moreover, we estimate M = �fn (µ+ iη) + v.p. π∫ −π cot s 2 ρn(µ+ s) ds. Note that �eiλ + eiz eiλ − eiz = − sin (λ− µ) cosh η − cos (λ− µ) . Hence, M = v.p. ∫ ( cot s 2 − sin s cosh η − cos s ) ρn (µ+ s) ds = ∫ |s|≤d/2 ln ( cosh η − cos s 1− cos s ) ρ′n (µ+ s) ds+O (η) = I1 + I2 + I3 +O (η) , where I1 is the integral over |s| ≤ n−2, I2 is the integral over n−2 ≤ |s| ≤ n−1/4 and I3 is the integral over n−1/4 ≤ |s| ≤ d/2. We estimate every term: |I1| (2.25) ≤ Cn7/8 ∫ |s|≤n−2 ln ( cosh η − cos s 1− cos s ) ds ≤ Cn−9/8 lnn, |I2| ≤ C lnn ∫ n−2≤|s|≤n1/4 ∣∣ρ′n (µ+ s) ∣∣ ds (2.24) ≤ Cn−1/4 lnn, Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 265 M. Poplavskyi |I3| (2.16) ≤ Cn−1/4 ∫ |s|≤d/2 (∣∣∣ψ(n) n (µ+ s) ∣∣∣2 + ∣∣∣ψ(n) n−1 (µ+ s) ∣∣∣2) ds ≤ Cn−1/4. 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, 1] λx = λ0 + x− tx n , λy = λ0 + y − tx n . (2.68) Then, similarly to (2.30) and (2.54), we obtain d dt Kn (λx, λy) = x π+λ0∫ −π+λ0 Kn (λx, λ)Kn (λ, λy) ( 1 2 V ′ (λx) + 1 2 V ′ (λy)− V ′ (λ) ) dλ. (2.69) To get our estimates, we split this integral in two parts |λ− λ0| ≤ d/2 and |λ− λ0| ≥ d/2. By the assumption of the lemma, λx, λy are in [a+ d/2, b − d/2], thus in the first integral we can write V ′ (λ)− 1 2 V ′ (λx)− 1 2 V ′ (λy) = ( eiλ − eiλx ) V ′′ (λx) 2ieiλx + ( eiλ − eiλy ) V ′′ (λy) 2ieiλy +O (∣∣∣eiλ − eiλx ∣∣∣2 + ∣∣∣eiλ − eiλy ∣∣∣2) = ( eiλ − eiλx ) V ′′ (λx) 2ieiλx + ( eiλ − eiλy ) V ′′ (λy) 2ieiλy +O (∣∣∣eiλ − eiλx ∣∣∣ ∣∣∣eiλ − eiλy ∣∣∣+ |x− y|2 n2 ) . Similarly to (2.52), we obtain π∫ −π Kn (λx, λ)Kn (λ, λy) ( eiλ − eiλx ) dλ = −r(n) n−1,nψ (n) n (λx)ψ (n) n−1 (λy). Hence,∫ |λ−λ0|≤d/2 Kn (λx, λ)Kn (λ, λy) ( eiλ − eiλx ) dλ = −rn−1,nψ (n) n (λx)ψ (n) n−1 (λy)− Id, 266 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 Bulk Universality for Unitary Matrix Models where |Id| = ∣∣∣∣∣∣∣ ∫ |λ−λ0|≥d/2 Kn (λx, λ)Kn (λ, λy) ( eiλ − eiλx ) dλ ∣∣∣∣∣∣∣ ≤ C  ∫ |λ−λ0|≥d/2 |Kn (λx, λ)|2 dλ ∫ |λ−λ0|≥d/2 |Kn (λ, λy)|2 dλ  1/2 (2.12) ≤ C [∣∣∣ψ(n) n−1 (λx) ∣∣∣2 + ∣∣∣ψ(n) n (λx) ∣∣∣2 + ∣∣∣ψ(n) n−1 (λy) ∣∣∣2 + ∣∣∣ψ(n) n (λy) ∣∣∣2] . The same bounds are valid for the term with the eiλy instead of eiλx . To estimate other terms, we use the Schwarz inequality∫ |λ−λ0|≤d/2 ∣∣∣Kn (λx, λ)Kn (λ, λy) ( eiλ − eiλx )( eiλ − eiλy )∣∣∣ dλ ≤  π∫ −π ∣∣∣Kn (λx, λ) ( eiλ − eiλx )∣∣∣2 dλ π∫ −π ∣∣∣Kn (λ, λy) ( eiλ − eiλy )∣∣∣2 dλ 1/2 (2.11) ≤ C [∣∣∣ψ(n) n−1 (λx) ∣∣∣2 + ∣∣∣ψ(n) n (λx) ∣∣∣2 + ∣∣∣ψ(n) n−1 (λy) ∣∣∣2 + ∣∣∣ψ(n) n (λy) ∣∣∣2] , ∫ |λ−λ0|≤d/2 |Kn (λx, λ)Kn (λ, λy)| dλ ≤ n (ρn (λx) + ρn (λy)) ≤ Cn. In the second integral we use the boundedness of V ′ (λ), the Cauchy inequality |Kn (λx, λ)Kn (λ, λy)| ≤ |Kn (λx, λ)|2 + |Kn (λ, λy)|2 and (2.12). Thus,∣∣∣∣ ddtKn (λx, λy) ∣∣∣∣ ≤ C |x| [∣∣∣ψ(n) n−1 (λx) ∣∣∣2 + ∣∣∣ψ(n) n (λx) ∣∣∣2 + ∣∣∣ψ(n) n−1 (λy) ∣∣∣2 + ∣∣∣ψ(n) n (λy) ∣∣∣2 + |x− y| n ] . (2.70) Now, using (2.25), we obtain∣∣∣∣ ddtKn (λx, λy) ∣∣∣∣ ≤ C |x| ( n7/8 + |x− y|n−1 ) . (2.71) Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 267 M. Poplavskyi Finally, observing that ∂ ∂x Kn (x, y) + ∂ ∂y Kn (x, y) = − (xn)−1 e−i(n−1)(x−y)/2n d dt Kn (λx, λy)|t=0 , Kn (x, y)−Kn (0, y − x) = e−i(n−1)(x−y)/2n· 1 n ( Kn (λx, λy)|t=0 − Kn (λx, λy)|t=1 ) , and using (2.71), we conclude that the lemma is proved. P r o o f of Lemma 2.9. First, show that for any |x| ≤ nd0/2 we have the bound 1∫ −1 Kn (x, x)Kn (x+ t, x+ t)− |Kn (x, x+ t)|2 t2 dt ≤ C. (2.72) Denote Ω0 = [−π + λ0, π + λ0] , Ω+ 0 = Ω0/Ω− 0 , (2.73) Ω− 0 = { λ ∈ Ω0 : ∣∣∣∣sin λ− λ0 2 ∣∣∣∣ ≤ sin 1 2n } = [λ0 − 1/n, λ0 + 1/n] , and consider the quantity W = 〈 n∏ j=2 ∣∣∣∣1− sin2 1/2n sin2 (λj − λ0) /2 ∣∣∣∣ 〉 , (2.74) where the symbol < . . . > denotes the average with respect to pn (λ0, λ2, . . . , λn). We will estimate W from above. To do this we use the relation 1− sin2 1 2n sin2 µ− λ 2 = ( ei(λ+1/n) − eiµ ) ( ei(λ−1/n) − eiµ ) (eiλ − eiµ)2 , (1.2) and the Schwarz inequality. We get that W 2 is not larger than the product of two integrals I+ and I−, where I± = Z−1 n ∫ Ωn−1 0 e−nV (λ0) ∏ 2≤j<k≤n ∣∣∣eiλj − eiλk ∣∣∣2 × exp −n n∑ j=2 V (λj)  n∏ j=2 ∣∣∣ei(λ0±1/n) − eiλj ∣∣∣2 dλj . 268 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 Bulk Universality for Unitary Matrix Models Moreover, the expression n (V (λ0)− V (λ0 ± 1/n)) is bounded in view of (1.17). Hence, from (1.15) we obtain W ≤ Cρ1/2 n (λ0 + 1/n) ρ1/2 n (λ0 − 1/n) ≤ C. (2.75) On the other hand, W can be represented as follows: W = 〈 n∏ j=2 (φ1 (λj) + φ2 (λj)) 〉 , (2.76) where φ1 (λ) = ( sin2 1 2n − sin2 λ− λ0 2 )2 sin2 1 2n sin2 λ− λ0 2 1Ω− 0 , (2.77) φ2 (λ) = 1− sin2 λ− λ0 2 sin2 1 2n 1Ω− 0 + 1− sin2 1 2n sin2 λ− λ0 2 1Ω+ 0 . (2.78) Since 0 ≤ φ2 (λ) ≤ 1 and φ1 (λ) ≥ 0, it follows from (2.76) that W can be estimated bellow as W ≥ (n− 1) ∫ Ω0 φ1 (λ) 〈 δ (λ2 − λ) exp  n∑ j=3 lnφ2 (λj)  〉 dλ. Note that 〈δ (λ2 − λ)〉 = p (n) 2 (λ0, λ). Therefore the Jensen inequality implies W ≥ (n− 1) ∫ Ω− 0 φ1 (λ) p (n) 2 (λ0, λ) × exp  〈 δ (λ2 − λ) n∑ j=3 lnφ2 (λj) 〉[ p (n) 2 (λ0, λ) ]−1  dλ = (n− 1) ∫ Ω− 0 φ1 (λ) p (n) 2 (λ0, λ) × exp (n− 2) ∫ Ω0 lnφ2 ( λ′ ) p (n) 3 ( λ0, λ, λ ′) dλ′ [p(n) 2 (λ0, λ) ]−1  dλ, Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 269 M. Poplavskyi where p(n) k is defined in (1.5). Using (1.14) for l = 2, 3, we have p (n) 3 ( λ0, λ, λ ′) = n n− 2 ρn ( λ′ ) p (n) 2 (λ0, λ) + [ 2� (Kn (λ0, λ)Kn (λ, λ′)Kn (λ′, λ0)) n (n− 1) (n− 2) − Kn (λ0, λ0) |Kn (λ, λ′)|2 +Kn (λ, λ) |Kn (λ0, λ ′)|2 n (n− 1) (n− 2) ] . (2.79) By the Cauchy inequality, 2 ∣∣Kn (λ0, λ)Kn ( λ, λ′ ) Kn ( λ′, λ0 )∣∣ ≤ 2K1/2 n (λ0, λ0)K1/2 n (λ, λ) ∣∣Kn ( λ, λ′ ) Kn ( λ′, λ0 )∣∣ ≤ Kn (λ0, λ0) ∣∣Kn ( λ, λ′ )∣∣2 +Kn (λ, λ) ∣∣Kn ( λ0, λ ′)∣∣2 , we obtain that the second term in (2.79) is nonpositive, hence p (n) 3 ( λ0, λ, λ ′) ≤ n n− 2 ρn ( λ′ ) p (n) 2 (λ0, λ) . Taking into account that lnφ2 (λ′) ≤ 0, finally we get W ≥ (n− 1) ∫ Ω− 0 φ1 (λ) p (n) 2 (λ0, λ) dλ · exp n ∫ Ω0 ρn ( λ′ ) lnφ2 ( λ′ ) dλ′  . (2.80) Now we will show that the second multiplier in (2.80) is bounded from below n ∫ Ω0 ρn ( λ′ ) lnφ2 ( λ′ ) dλ′ =  ∫ |s|≤1 + ∫ 1≤|s|≤nd0/2 + ∫ nd0/2≤|s|≤nπ  ρn (λ0 + s/n) lnφ2 (λ0 + s/n) ds ≥ C  ∫ |s|≤1 ln ( 1− sin2 s/ (2n) sin2 1/ (2n) ) ds+ ∫ 1≤|s|≤nd0/2 ln ( 1− sin2 1/ (2n) sin2 s/ (2n) ) ds  + ln ( 1− sin2 1/ (2n) sin2 d0/4 ) ∫ |s|≤nπ ρn (λ0 + s/n) ds ≥ C (I1 + I2) +O ( n−1 ) . 270 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 Bulk Universality for Unitary Matrix Models I1 = 1∫ 0 ln ( cos (s/n)− cos (1/n) 1− cos (1/n) ) ds = −n 1/n∫ 0 sin t sin (t+ 1/n) t− 1/n 2 sin t− 1/n 2 dt ≥ −C I2 = n d0/2∫ 1/n ln ( cos (1/n)− cos t 1− cos t ) dt = (nd0/2− 1) ln ( 1− sin2 1/2n sin2 d0/2 ) − n (1− cos 1/n) d0/2∫ 1/n cot t/2 t− 1/n 2 sin t− 1/n 2 1 sin t+ 1/n 2 dt ≥ −C − Cn−1 d0/2∫ 1/n dt t (t+ 1/n) ≥ −C. Thus, from (2.75) and (2.80) we obtain n ∫ Ω− 0 φ1 (λ) p (n) 2 (λ0, λ) dλ ≥ −C. (2.81) Then, using (1.14), (2.27), (2.15), (2.77), and the inequality 1 t2 ≤ C sin2 1/2n sin2 t/2n , we obtain (2.72) for x = 0 from (2.81). Substituting λ0 by λ0 + x/n, we get (2.72) for any |x| ≤ nd0/2. Now we are ready to prove (2.36). Denote Cn = sup ∣∣∣∣ ∂∂xKn (x, y) ∣∣∣∣. In view of (2.32) Cn ≤ ∣∣∣∣∣∣∣ v.p. ∫ |z−x|≤1 + ∫ |z−x|≥1  Kn (x, z)Kn (z, y) z − x dz ∣∣∣∣∣∣∣ + o (1) ≤ |I1 (x, y)|+ |I2 (x, y)|+ o (1) . Using the Schwarz inequality and (2.28) with (2.29), we can estimate I2 as follows: |I2 (x, y)| ≤ K1/2 n (x, x)K1/2 n (y, y) ≤ C. To estimate I1 denote t̂∗n = sup {t > 0 : |x− y| ≤ t ⇒ Kn (x, y) ≥ ρn(λ0)/2} , t∗n = min { t̂∗n, 1 } . (2.82) Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 271 M. Poplavskyi We will prove that the sequence t∗n is bounded from below by some nonzero constant. Represent I1 in the form I1 (x, y) = v.p. ∫ |t|≤t∗n Kn (x, x+ t)Kn (x+ t, y)−Kn (x, x)Kn (x, y) t dt + ∫ t∗n≤|t|≤1 Kn (x, x+ t)Kn (x+ t, y) t dt = I ′1 + I ′′1 . Using (2.29), we have |I ′′1 | ≤ C |ln t∗n|. On the other hand, from (1.11) and the Cauchy inequality we obtain for any x, y, z |Kn (x, z) −Kn (y, z)|2 ≤ (Kn (x, x) +Kn (y, y)− 2Kn (x, y))Kn (z, z) = (( K1/2 n (x, x)−K1/2 n (y, y) )2 + 2 ( K1/2 n (x, x)K1/2 n (y, y)−Kn (x, y) )) Kn (z, z) . (2.83) From (2.35) we get that the first term of (2.83) is bounded by Cn−1/4 |x− y|2. The second term we rewrite as K1/2 n (x, x)K1/2 n (y, y)−Kn (x, y) = Kn (x, x)Kn (y, y)−K2 n (x, y) K1/2 n (x, x)K1/2 n (y, y) +Kn (x, y) . Thus, for |x− y| ≤ t∗n we get |Kn (x, z)−Kn (y, z)|2 ≤ C ( n−1/4 |x− y|3/2 +Kn (x, x)Kn (y, y)− |Kn (x, y)|2 ) . (2.84) Hence, using (2.84), (2.72) and the Schwarz inequality, we obtain ∣∣I ′1∣∣ ≤ C ∫ |t|≤t∗n |Kn (x, x+ t)−Kn (x, x)|+ |Kn (x+ t, y)−Kn (x, y)| |t| dt ≤ C (t∗n) 1/2 . Finally, from the above estimates we have Cn ≤ C ( |ln t∗n|+ (t∗n) 1/2 ) . (2.85) Note that if the sequence t∗n is not bounded from below, then we have C ≤ ρn (λ0) /2 ≤ |Kn (x+ t∗n, x)−Kn (x, x)| ≤ Cnt ∗ n ≤ Ct∗n ln t ∗ n + Ct∗n, 272 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 Bulk Universality for Unitary Matrix Models 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∫ |x|≤L ∣∣∣∣ ∂∂xKn (x, y) ∣∣∣∣2 dx = ∫ |x|≤L ∣∣∣∣ ∂∂yKn (x, y) ∣∣∣∣2 dx+ o(1). Then we rewrite the analog of (2.32) for ∂ ∂y Kn (x, y) as ∂ ∂y Kn (x, y) = v.p. ∫ |z−y|≤d∗ + ∫ |z|≤2L 1|z−y|≥d∗  Kn (x, z)Kn (z, y) y − z dz +O (L−1 ) = I1 (x, y) + I2 (x, y) +O (L−1 ) . To complete the proof, it is enough to estimate I2 1,2. Since in I1 the domain of integration is symmetric with respect to y, we can write I1 (x, y) = ∫ |z−y|≤d∗ (Kn (x, z)−Kn (x, y))Kn (z, y) y − z dz + ∫ |z−y|≤d∗ (Kn (z, y)−Kn (y, y))Kn (x, y) y − z dz. Now, using the Schwarz inequality and (2.28), we obtain ∣∣I2 1 (x, y) ∣∣ ≤ 2d∗C ∫ |z−y|≤d∗ |Kn (x, z) −Kn (x, y)|2 (z − y)2 dz + 2d∗K2 n (x, y) ∫ |z−y|≤d∗ |Kn (z, y)−Kn (y, y)|2 (z − y)2 dz. Integrating the above inequality with respect to x and using (2.28) with (2.29), we get ∫ ∣∣I2 1 (x, y) ∣∣ dx ≤ C ∫ |z−y|≤d∗ |Kn (z, y)−Kn (y, y)|2 (z − y)2 dz + C ∫ |z−y|≤d∗ Kn (z, z) +Kn (y, y)− 2Kn (z, y) (z − y)2 dz. Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 273 M. Poplavskyi Using the bounds (2.83) in the second integral and (2.84) in the first one, in view of (2.72) we obtain the bound for I2 1 . To estimate I2, we write∫ ∣∣I2 2 (x, y) ∣∣ dx ≤ ∫ |z|,|z′|≤2L 1|z−y|>d∗1|z′−y|>d∗ ∣∣∣∣Kn (y, z)Kn (z, z′)Kn (z′, y) (z − y) (z′ − y) ∣∣∣∣ dzdz′ ≤ C ∫ |z|,|z′|≤2L 1|z−y|>d∗1|z′−y|>d∗ (∣∣∣∣Kn (y, z) z − y ∣∣∣∣2 + ∣∣∣∣Kn (y, z′) z′ − y ∣∣∣∣2 ) dzdz′ ≤ C. Above bounds for I1 and I2 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 problem statement and fruitful discussions. References [1] M.L. Mehta, Random Matrices. Acad. Press, New York, 1991. [2] A. Kolyandr, On Eigenvalue Distribution of Invariant Ensembles of Random Matrices. — Dop. Ukr. Ac. Sci. Math. (1997), No. 7, 14–20. (Ukrainian) [3] F.J. Dyson, Statistical Theory of Energy Levels of Complex Systems. I–III. — J. Math. Phys. 3 (1962), 140–175. [4] L. Pastur and M. Shcherbina, Universality of the Local Eigenvalue Statistics for a Class of Unitary Invariant Matrix Ensembles. — J. Stat. Phys. 86 (1997), 109–147. [5] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach — CIMS. New York Univ., New York, 1999. [6] K. Johansson, The Longest Increasing Subsequence in a Random Permutation and a Unitary Random Matrix Model. — Math. Res. Lett. 5 (1998), 63–82. [7] N.I. Muskhelishvili, Singular Integral Equations. P. Noordhoff, Groningen, 1953. [8] L. Pastur and M. Shcherbina, Bulk Universality and Related Properties of Hermitian Matrix Model. — J. Stat. Phys. 130 (2007), 205–250. [9] F.J. Dyson, A Class of Matrix Ensembles. — J. Math. Phys. 13 (1972), 90–107. [10] P. Deift, T. Kriecherbauer, K.T.-K. McLaughlin, S. Venakides, and X. Zhou, Uni- form Asymptotics for Polynomials Orthogonal with Respect to Varying Exponential Weights and Applications to Universality Questions in Random Matrix Theory. — Comm. Pure Appl. Math. 52 (1999), 1335–1425. 274 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3

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