nlin.SI 2011-09-06 0

A centerless representation of the Virasoro algebra associated with the unitary circular ensemble

We consider the 2-dimensional Toda lattice tau functions $\tau_n(t,s;\eta,\theta)$ deforming the probabilities $\tau_n(\eta,\theta)$ that a randomly chosen matrix from the unitary group U(n), for the Haar measure, has no eigenvalues within an arc $(\…

Authors: Luc Haine, Didier V, erstichelen

A CENTERLESS REPRESENT A TION OF THE VIRASORO A LGEBRA ASSOCIA TED WITH THE UNIT AR Y CIR CULAR ENSEMBLE LUC HAINE AND DIDIER V ANDERSTICHELEN Abstract. W e consider t he 2-dimensional T o da lattice tau func- tions τ n ( t, s ; η , θ ) deforming the probabilities τ n ( η , θ ) that a ran- domly c hosen matrix from the unitary group U ( n ), fo r the Haar measure, has no eigenv alues within an arc ( η , θ ) of the unit circle. W e sho w that thes e tau functions satisfy a centerless Viraso ro al- gebra of constraints, with a b oundar y part in the sens e of Adler, Shiota and v an Mo erb eke. As an application, w e obtain a new deriv ation of a differential equation due to T racy and Widom, sat- isfied b y these probabilities, linking it to t he Painlev ´ e VI equation. 1. Introduction Consider the group U ( n ) of n × n unitary matrices, with the nor- malized Haar measure as a probabilit y measure. The W eyl in tegral form ula giv es the induc ed de nsit y distribution on t he eigenv alues of the matrices on the unit circle in the complex plane, a nd is giv en by 1 n ! | ∆ n ( z ) | 2 n Y k =1 d z k 2 π iz k ; z k = e iϕ k and ∆ n ( z ) = Y 1 ≤ k < l ≤ n ( z k − z l ) . Th us, for η , θ ∈ ] − π , π [, with η ≤ θ , the probability that a randomly c hosen matrix from U ( n ) has no eigen v alues within an arc of circle ( η , θ ) = { z ∈ S 1 | η < a rg( z ) < θ } is given b y τ n ( η , θ ) = 1 (2 π ) n n ! Z 2 π + η θ . . . Z 2 π + η θ Y 1 ≤ k < l ≤ n | e iϕ k − e iϕ l | 2 d ϕ 1 . . . d ϕ n . Ob viously , this probability dep ends only on the length θ − η . All o f this is w ell kno wn and w e refer the reader to Meh ta [1] for details. W e Date : Ja n uary 18 2 010. 2000 Mathematics Subje ct Classific ation. 15A52, 17B68. Key wor ds and ph r ases. Random matrices, Virasoro algebra. The author s ackno wledge the par tia l suppor t of the Belgia n Interuniv ers it y At- traction P ole P06/0 2. The second author is a Research F ellow a t FNRS, Belgium. 1 2 L. HAINE A ND D. V AND ERSTICHELEN shall denote by R ( θ ) = − 1 2 d d θ log τ n ( − θ , θ ) , (1.1) the logarithmic deriv ativ e of the probabilit y tha t an arc of circle of length 2 θ con tains no eigenv alues of a randomly c hosen unitary matrix. The starting motiv ation for our work was to understand a differen tial equation satisfied b y the function R ( θ ) R ( θ ) 2 + 2 sin θ cos θ R ( θ ) R ′ ( θ ) + sin 2 θ R ′ ( θ ) 2 = 1 2  1 4 sin 2 θ R ′′ ( θ ) 2 R ′ ( θ ) + sin θ cos θ R ′′ ( θ ) +  cos 2 θ + n 2 sin 2 θ  R ′ ( θ )  , (1.2) obtained by T racy and Widom in [2], from the p oin t of view of the Adler-Shiota-v an Mo erb eke [3] approac h, in terms of Virasoro con- strain ts. Introducing the 2 -T o da time-dep endent tau f unctions τ n ( t, s ; η , θ ) = 1 n ! Z [ θ , 2 π + η ] n | ∆ n ( z ) | 2 n Y k =1  e P ∞ j =1 ( t j z j k + s j z − j k ) d z k 2 π iz k  , (1.3) with z k = e iϕ k , deforming the probabilities τ n ( η , θ ) = τ n (0 , 0; η , θ ), w e disco v er that the y satisfy a set of Virasoro cons train ts indexed b y al l in tegers, decoupling into a b oundary-part and a t ime-part 1 i  e ik θ ∂ ∂ θ + e ik η ∂ ∂ η  τ n ( t, s ; η , θ ) = L ( n ) k τ n ( t, s ; η , θ ) , k ∈ Z , i = √ − 1 , with the time-dep endent op erators L ( n ) k pro viding a cen terless repre- sen tation of the ful l Virasoro a lgebra, see Section 2 (Theorem 2.2 ) for a precise statemen t and the pro of of the result. In their study of P ainlev ´ e e quations s atisfied (as functions of x ) by in tegrals o f Gessel t yp e E U ( n ) e x tr ( M + M ) , where the exp ectation E U ( n ) refers to in tegration with res p ect t o the Haar measure o v er the whole of U ( n ), Adler and v a n Mo erb ek e [4] found the sl 2 subalgebra corre- sp onding to k = − 1 , 0 , 1, without b oundary terms. The app earance of b oundary terms and of a ful l cente rless Virasoro algebra is to the b est of our kno wledge new . F rom this result, it is easy to obtain equation (1.2), using the algorithmic metho d of [3]. F ina lly , similarly to a result b y the first author and Semengue [5] on t he Jacobi p olynomial ensem- ble, we sho w that R ( θ ) is the restriction to the unit circle of a function r ( z ) defined in the complex plane, so that σ ( z ) = − i ( z − 1) r ( z ) − n 2 z / 4 satisfies a special case of the Ok amoto-Jim b o-Miw a form of the P ainlev ´ e VI equation. This will b e explained in Section 3 of the pap er. A CENTERLESS REPRESENT A TION OF THE VIRA SORO ALGEBRA 3 2. A cente r le ss re p resent a tion o f the Virasor o algebra The pro of of the Vira soro constrain ts satisfied b y the in tegral (1.3) is a non- trivial adapta tion of the self-similarit y argumen t exploited in the case of the Gaussian ensem bles, based on the inv ariance of the inte grals with resp ect to translatio ns, see [6] and references therein. Here, w e replace translations by appropriate rotations. More precisely , setting d I n ( t, s, z ) = | ∆ n ( z ) | 2 n Y α =1  e P ∞ j =1 ( t j z j α + s j z − j α ) d z α 2 π iz α  , (2.1) with z α = e iϕ α and | ∆ n ( z ) | 2 = Q 1 ≤ α<β ≤ n | z α − z β | 2 , w e ha v e the fun- damen tal next prop osition. Prop osition 2.1. The fol lowing varia tional formulas hold d d ε d I n  z α 7→ z α e ε ( z k α − z − k α )    ε =0 =  L ( n ) k − L ( n ) − k  d I n , (2.2) d d ε d I n  z α 7→ z α e iε ( z k α + z − k α )    ε =0 = i  L ( n ) k + L ( n ) − k  d I n , (2.3) for al l k ≥ 0 , with L ( n ) k = k − 1 X j =1 ∂ 2 ∂ t j ∂ t k − j + n ∂ ∂ t k + ∞ X j =1 j t j ∂ ∂ t j + k − ∞ X j = k + 1 j s j ∂ ∂ s j − k − k − 1 X j =1 j s j ∂ ∂ t k − j − nk s k , k ≥ 1 , (2.4) L ( n ) 0 = ∞ X j =1 j t j ∂ ∂ t j − ∞ X j =1 j s j ∂ ∂ s j , (2.5) L ( n ) − k = − k − 1 X j =1 ∂ 2 ∂ s j ∂ s k − j − n ∂ ∂ s k − ∞ X j =1 j s j ∂ ∂ s j + k + ∞ X j = k + 1 j t j ∂ ∂ t j − k + k − 1 X j =1 j t j ∂ ∂ s k − j + nk t k , k ≥ 1 . (2.6) Pr o of. W e shall only giv e the pro of of (2.2), the pro of of (2.3) is similar. Up on setting E = n Y α =1 e P ∞ j =1 ( t j z j α + s j z − j α ) , 4 L. HAINE A ND D. V AND ERSTICHELEN the following four relations hold, for k ≥ 0,  ∂ ∂ t k + nδ k , 0  E =  n X α =1 z k α  E  ∂ ∂ s k + nδ k , 0  E =  n X α =1 z − k α  E , (2.7) 1 2 X i + j = k i,j > 0 ∂ 2 ∂ t i ∂ t j − n 2 δ k , 0 ! E = X 1 ≤ α<β ≤ n i + j = k i,j > 0 z i α z j β + k − 1 2 n X α =1 z k α ! E 1 2 X i + j = k i,j > 0 ∂ 2 ∂ s i ∂ s j − n 2 δ k , 0 ! E = X 1 ≤ α<β ≤ n i + j = k i,j > 0 z − i α z − j β + k − 1 2 n X α =1 z − k α ! E . (2.8) W e split the computation in to f o ur con tributions, corresponding to v arious factors in (2.1). Contribution 1 : F or k > 0, w e ha v e ∂ ∂ ε   ∆ n  z e ε ( z k − z − k )    2    ε =0 = | ∆ n ( z ) | 2 X 1 ≤ α<β ≤ n ( z α + z β )( z k α − z k β − ( z − k α − z − k β )) z α − z β = | ∆ n ( z ) | 2 X 1 ≤ α<β ≤ n ( z α + z β )  k − 1 X i =0 z i α z k − 1 − i β + k − 1 X i =0 z − i − 1 α z i − k β  = | ∆ n ( z ) | 2 E − 1 " 2 X 1 ≤ α<β ≤ n i + j = k i,j > 0 ( z i α z j β + z − i α z − j β ) + ( n − 1 ) n X α =1 ( z k α + z − k α ) # E . Using the fo ur relations (2.7) and (2.8), we obtain ∂ ∂ ε   ∆ n  z e ε ( z k − z − k )    2    ε =0 = 2 | ∆ n ( z ) | 2 E − 1 " 1 2 X i + j = k i,j > 0 ∂ 2 ∂ t i ∂ t j + 1 2 X i + j = k i,j > 0 ∂ 2 ∂ s i ∂ s j + n − k 2 ∂ ∂ t k + n − k 2 ∂ ∂ s k # E , (2.9) A CENTERLESS REPRESENT A TION OF THE VIRA SORO ALGEBRA 5 whic h is also trivially satisfied fo r k = 0. Contribution 2 : F or k ≥ 0, using the relations (2.7), we ha ve ∂ ∂ ε n Y α =1 d  z α e ε ( z k α − z − k α )     ε =0 = E − 1 n X α =1  ( k + 1 ) z k α + ( k − 1) z − k α  E n Y α =1 d z α = E − 1 h ( k + 1 ) ∂ ∂ t k + ( k − 1) ∂ ∂ s k i E n Y α =1 d z α . (2.10) Contribution 3 : F or k ≥ 0, using the relations (2.7), we ha ve ∂ ∂ ε n Y α =1 e P ∞ j =1  t j  z α e ε ( z k α − z − k α )  j + s j  z α e ε ( z k α − z − k α )  − j     ε =0 = n X α =1 h ∞ X j =1 j t j z j α ( z k α − z − k α ) − ∞ X j =1 j s j z − j α ( z k α − z − k α ) i E = " ∞ X j =1 j t j n X α =1 z j + k α − k − 1 X j =1 j t j n X α =1 z j − k α − ∞ X j = k j t j n X α =1 z j − k α − k − 1 X j =1 j s j n X α =1 z k − j α − ∞ X j = k j s j n X α =1 z k − j α + ∞ X j =1 j s j n X α =1 z − k − j α # E = " ∞ X j =1 j t j ∂ ∂ t k + j − k − 1 X j =1 j t j ∂ ∂ s k − j − ∞ X j = k + 1 j t j ∂ ∂ t j − k − nk t k − k − 1 X j =1 j s j ∂ ∂ t k − j − ∞ X j = k + 1 j s j ∂ ∂ s j − k − nk s k + ∞ X j =1 j s j ∂ ∂ s k + j # E . (2.11) Contribution 4 : F or k ≥ 0, using the relations (2.7), we ha ve ∂ ∂ ε n Y α =1 1 2 π iz α e ε ( z k α − z − k α )    ε =0 = E − 1 h − n X α =1 z k α + n X α =1 z − k α i E n Y α =1 1 2 π iz α = E − 1 h − ∂ ∂ t k + ∂ ∂ s k i E n Y α =1 1 2 π iz α . (2.12) Adding up (2.9), (2.10), (2.11) a nd (2.12) giv es (2.2). This concludes the pro o f of Prop osition 2.1.  6 L. HAINE A ND D. V AND ERSTICHELEN W e are now able to state our main result. Theorem 2.2. (i) The tau functions 1 τ n ( t, s ; η , θ ) , n ≥ 1 , define d in (1.3) , satisfy B k ( η , θ ) τ n ( t, s ; η , θ ) = L ( n ) k τ n ( t, s ; η , θ ) , k ∈ Z , (2.13) with L ( n ) k , k ∈ Z , de fi ne d as in ( 2 .4) , (2.5) , (2.6) , and B k ( η , θ ) = 1 i  e ik θ ∂ ∂ θ + e ik η ∂ ∂ η  ; i = √ − 1 . (2.14) (ii) Th e op er ators L ( n ) k , k ∈ Z , satisfy the c ommutation r elations of the c enterless Vir asor o algebr a, that is  L ( n ) k , L ( n ) l  = ( k − l ) L ( n ) k + l , k , l ∈ Z . (2.15) Pr o of. (i) Denoting z α = e iϕ α , the c hange of v ariable z α 7→ z α e ε ( z k α − z − k α ) in t he in tegral ( 1.3) giv es the fo llo wing t r a nsformation on the angle ϕ α 7→ ϕ α + 2 ε sin( k ϕ α ), inducing a change in the limits of integration giv en b y the inv erse map ϕ α 7→ ϕ α − 2 ε s in( k ϕ α ) + O ( ε 2 ) , (2.16) for ε small enough. Making the c hange of v ariable in the inte gral (1.3 ), with the corresp onding c hange in the limits o f in t egr a tion, lea v es it in v ariant. Th us, by differen tia t ing the result with respect to ε and ev alua ting it at ε = 0, us ing the chain rule together with (2.2) and (2.16), w e obtain 0 =  − 2 sin( k θ ) ∂ ∂ θ − 2 sin( k η ) ∂ ∂ η + L ( n ) k − L ( n ) − k  τ n ( t, s ; η , θ ) . (2.17) Similarly , the change of v a r iable z α 7→ z α e iε ( z k α + z − k α ) corresp onds to the transformation ϕ α 7→ ϕ α + 2 ε cos( k ϕ α ), with in v erse ϕ α 7→ ϕ α − 2 ε c os( k ϕ α ) + O ( ε 2 ) , whic h, using ( 2 .3), leads to 0 =  − 2 i cos( k θ ) ∂ ∂ θ − 2 i cos( k η ) ∂ ∂ η + L ( n ) k + L ( n ) − k  τ n ( t, s ; η , θ ) . (2.18) Adding and subtracting (2.17) and (2.18) giv es the constrain ts (2 .13), with B k ( η , θ ) defined as in (2.14). 1 See the beginning of Section 3, for a justification of the terminolo gy . A CENTERLESS REPRESENT A TION OF THE VIRA SORO ALGEBRA 7 (ii) Consider the complex Lie algebra A giv en b y the direct sum of t wo comm uting copies of the Heisen b erg algebra with bases { ~ a , a j | j ∈ Z } and { ~ b , b j | j ∈ Z } a nd defining commutation relations [ ~ a , a j ] = 0 , [ a j , a k ] = j δ j, − k ~ a , [ ~ b , b j ] = 0 , [ b j , b k ] = j δ j, − k ~ b , (2.19) [ ~ a , ~ b ] = 0 , [ a j , b k ] = 0 , [ ~ a , b j ] = 0 , [ ~ b , a j ] = 0 , with j, k ∈ Z . Let B b e the space of formal p ow er series in the v ar iables t 1 , t 2 , . . . and s 1 , s 2 , . . . , and consider the follow ing repres en tatio n of A in B : a j = ∂ ∂ t j , a − j = j t j , b j = ∂ ∂ s j , b − j = j s j , a 0 = b 0 = µ , ~ a = ~ b = 1 , (2.20) for j > 0 , and µ ∈ C . Define the op erators A ( n ) k = 1 2 X j ∈ Z : a − j a j + k : , B ( n ) k = 1 2 X j ∈ Z : b − j b j + k : , where k ∈ Z , a j , b j are as in (2.20) with µ = n , and where the colons indicate normal ordering, defined b y : a j a k :=  a j a k if j ≤ k , a k a j if j > k , and a similar definition for : b j b k :, obtained b y changing the a ’s in b ’s in the former. Using these notations, w e can rewrite (2.4), (2.5) and (2.6) as follows L ( n ) k = A ( n ) k − B ( n ) − k + 1 2 k − 1 X j =1 ( a j − b − j )( a k − j − b j − k ) , k ≥ 1 L ( n ) 0 = A ( n ) 0 − B ( n ) 0 , L ( n ) − k = A ( n ) − k − B ( n ) k − 1 2 k − 1 X j =1 ( a − j − b j )( a j − k − b k − j ) , k ≥ 1 . As sho wn in [7] (see Lecture 2 ) the op erators A ( n ) k , k ∈ Z , pro vide a represen tation of the Virasoro algebra in B with cen tral charge c = 1, that is [ A ( n ) k , A ( n ) l ] = ( k − l ) A ( n ) k + l + δ k , − l k 3 − k 12 , (2.21) 8 L. HAINE A ND D. V AND ERSTICHELEN for k , l ∈ Z . Similarly , the op erators B ( n ) k satisfy the comm utation relations [ B ( n ) k , B ( n ) l ] = ( k − l ) B ( n ) k + l + δ k , − l k 3 − k 12 , (2.22) for k , l ∈ Z . F urthermore w e ha ve for k , l ∈ Z [ a k , A ( n ) l ] = k a k + l , [ b k , B ( n ) l ] = k b k + l , [ a k , B ( n ) l ] = 0 , [ b k , A ( n ) l ] = 0 . (2.23) Let us no w establish the comm utatio n relations (2.15). W e give the pro of for k , l ≥ 0, the o ther cases b eing similar. As [ A ( n ) i , B ( n ) j ] = 0 , i, j ∈ Z , w e hav e using (2.1 9 ), (2.21), (2 .2 2) and ( 2 .23) [ L ( n ) k , L ( n ) l ] = ( k − l )  A ( n ) k + l − B ( n ) − k − l  − 1 2 l − 1 X j =1 j ( a j + k − b − j − k )( a l − j − b j − l ) − 1 2 l − 1 X j =1 ( l − j )( a j − b − j )( a k + l − j − b j − k − l )+ 1 2 k − 1 X j =1 j ( a j + l − b − j − l )( a k − j − b j − k ) + 1 2 k − 1 X j =1 ( k − j ) ( a j − b − j )( a k + l − j − b j − k − l ) . Relab eling the indices in t he sums, we ha v e [ L ( n ) k , L ( n ) l ] =( k − l )  A ( n ) k + l − B ( n ) − k − l  − 1 2 k + l − 1 X j = k + 1 ( j − k )( a j − b − j )( a k + l − j − b j − k − l ) − 1 2 l − 1 X j =1 ( l − j )( a j − b − j )( a k + l − j − b j − k − l ) + 1 2 k + l − 1 X j = l +1 ( j − l )( a j − b − j )( a k + l − j − b j − k − l ) + 1 2 k − 1 X j =1 ( k − j )( a j − b − j )( a k + l − j − b j − k − l ) =( k − l ) L ( n ) k + l . This concludes the pro of o f Theorem 2.2.  A CENTERLESS REPRESENT A TION OF THE VIRA SORO ALGEBRA 9 3. The unit ar y circular ensemble and the P ainlev ´ e VI equa tion It is well kno wn, see fo r instance [1], that the integral τ n ( t, s ; η , θ ) defined in (1.3) can b e represen ted as a T o eplitz determinan t τ n ( t, s ; η , θ ) = det  µ k − l ( t, s ; η , θ )  0 ≤ k , l ≤ n − 1 , (3.1) with µ k ( t, s ; η , θ ) = Z 2 π + η θ z k  e P ∞ j =1 ( t j z j + s j z − j ) d z 2 π iz  ; z = e iϕ , k ∈ Z . A nic e consequence of this represen tatio n is that τ n ( t, s ; η , θ ) is a tau function of a reduc tion of the 2-T o da lattice hierarc h y , that w as called the T o eplitz hierarc h y in [4]. Therefore, as with an y 2-T o da tau func- tion (see [8]), it satisfies the KP equation in t he t = ( t 1 , t 2 , . . . ) (or s = ( s 1 , s 2 , . . . )) v ariables separately  ∂ 4 ∂ t 4 1 + 3 ∂ 2 ∂ t 2 2 − 4 ∂ 2 ∂ t 1 ∂ t 3  log τ n + 6  ∂ 2 ∂ t 2 1 log τ n  2 = 0 . (3.2) As announced in the in tro duction, in this s ection, using the metho d of [3], w e establish the following result. Theorem 3.1. The Vir asor o c onstr aints (2.13) , c ombine d with the K P e quation (3.2) in the t variables (or the KP e quation in the s variab les), imply that the function R ( θ ) define d in (1.1) satisfies ( 1 .2) . Pr o of. Remem b ering the definition of L ( n ) 0 in (2.5), the Virasoro con- strain t in (2.13) fo r k = 0, ev aluated along the lo cus t = s = 0, giv es ∂ log τ n ( t, s ; η , θ ) ∂ θ      t = s =0 = − ∂ log τ n ( t, s ; η , θ ) ∂ η      t = s =0 , (3.3) whic h is a reformulation of the fact that the gap probabilit y τ n (0 , 0; η , θ ) only dep ends on the length θ − η . Define the op erator D = ∂ ∂ θ − ∂ ∂ η and put for a fixed n f ( t, s ; η , θ ) = log τ n ( t, s ; η , θ ) , g ( η , θ ) = − 1 2 D log τ n ( t, s ; η , θ )   t = s =0 . (3.4) Notice that for k ≥ 0 D k log τ n ( t, s ; η , θ )   t = s =0 η = − θ = d k d θ k log τ n ( t, s ; − θ , θ )   t = s =0 . 10 L. HAINE A ND D. V AND ERSTICHELEN Clearly , from the definition of R ( θ ) in (1 .1), w e ha ve R ( θ ) = g ( − θ , θ ) = − 1 2 d d θ log τ n ( t, s ; − θ , θ )   t = s =0 . Remem b ering the definition of L ( n ) k in (2.4), the constraints in (2.13) for k = 1 , 2, ev aluated at s = ( s 1 , s 2 , s 3 , . . . ) = (0 , 0 , 0 , . . . ), can b e written B 1 ( η , θ ) f    s =0 = X j ≥ 1 j t j ∂ f ∂ t j +1      s =0 + n ∂ f ∂ t 1      s =0 , (3.5) B 2 ( η , θ ) f    s =0 = X j ≥ 1 j t j ∂ f ∂ t j +2      s =0 + ∂ 2 f ∂ t 2 1      s =0 +  ∂ f ∂ t 1  2      s =0 + n ∂ f ∂ t 2      s =0 . (3.6) Using (3 .3 ) and the defi nition of g ( η , θ ) (3.4), the constrain t (3.5) ev al- uated along t he lo cus t = s = 0 gives ∂ f ∂ t 1      t = s =0 = 1 in ( e iη − e iθ ) g ( η , θ ) . (3.7) Consequen tly , alo ng the lo cus η = − θ , w e hav e ∂ f ∂ t 1      t = s =0 η = − θ = − 2 n sin( θ ) R ( θ ) . W e then pro ceed b y induction. W e call ∂ n f ∂ t j 1 ∂ t j 2 . . . ∂ t j n , a t deriv ativ e o f we igh ted degree | j | = j 1 + j 2 + · · · + j n . Then, for k ≥ 1, w e compute the system formed by ( all t -deriv ativ es of weigh ted degree k of (3 .5) , all t -deriv ativ es of weigh ted degree k − 1 of (3.6) , (3.8) ev alua ted at t = s = 0. F or instance, for k = 1, (3.8) reduces to B 1 ( η , θ )  ∂ f ∂ t 1    t = s =0  = ∂ f ∂ t 2      t = s =0 + n ∂ 2 f ∂ t 2 1      t = s =0 , B 2 ( η , θ ) f    t = s =0 = ∂ 2 f ∂ t 2 1      t = s =0 + n ∂ f ∂ t 2      t = s =0 + ∂ f ∂ t 1      t = s =0 ! 2 . A CENTERLESS REPRESENT A TION OF THE VIRA SORO ALGEBRA 11 After substitution of (3.7), this system of e quations can be solved for ∂ 2 f ∂ t 2 1    t = s =0 and ∂ f ∂ t 2    t = s =0 in terms of η , θ , g ( η , θ ) and D g ( η , θ ), whenev er n 6 = 1. Conseque n tly , on the lo cus η = − θ , the pa rtials ∂ 2 f ∂ t 2 1    t = s =0 η = − θ and ∂ f ∂ t 2    t = s =0 η = − θ can b e expressed in terms of θ , R ( θ ) a nd R ′ ( θ ). F or general k ≥ 1, supp ose all the t -deriv ative s of f of we igh t ed degree k , ev aluated at t = s = 0, ha ve b een express ed in terms of η , θ and g ( η , θ ), . . . , D k − 1 g ( η , θ ), whenev er n 6 = 1 , . . . , k − 1. Then (3.8) is a system of linear equations where the unkno wns are all t he t -deriv ativ es of f of w eigh ted degree k + 1, ev a lua ted at t = s = 0, and the co efficien ts can b e expres sed in t erms of η , θ and g ( η , θ ) , . . . , D k − 1 g ( η , θ ). This is a system of p ( k ) + p ( k − 1) linear equations in p ( k + 1) unknowns , where p ( k ) is the n umber o f part it ions of the natural n umber k . As p ( k + 1) ≤ p ( k ) + p ( k − 1), this system can b e solv ed and all the t - deriv a t ives of f of we igh t ed degree k + 1, ev aluated at t = s = 0 can b e expresse d in terms of η , θ , and g ( η , θ ), . . . , D k g ( η , θ ), whenev er n 6 = k . Consequen tly , on the lo cus η = − θ , the t -deriv ativ es o f f of we igh ted degree k + 1, ev alua t ed at t = s = 0 and on the lo cus η = − θ , can b e expresse d in terms of θ , R ( θ ), R ′ ( θ ), . . . , R ( k ) ( θ ). Since the KP equation (3.2) con tains t -deriv ativ es of f of w eigh ted degree les s or equal to 4, by p erforming the ab o ve sc heme up to k = 3, w e can express a ll these deriv atives , ev a lua ted at t = s = 0 and η = − θ , in terms of θ , R ( θ ) and its first three deriv at ives , whe nev er n ≥ 4. This giv es us a third order differen tial equation for R ( θ ): 0 = 4 R ( θ ) 2 − 2  n 2 + (1 − n 2 ) cos 2 θ  R ′ ( θ ) + 8 sin 2 θ R ( θ ) R ′ ( θ ) − 2 sin 2 θ R ′′ ( θ ) + sin 2 θ  12 R ′ ( θ ) 2 − R ′′′ ( θ )  . Multiplying the left-hand and the right-hand side of this equation with 1 4 sin θ  2 cos θ R ′ ( θ ) + sin θ R ′′ ( θ )  , w e obtain 0 = d d θ  sin 2 θ R ′ ( θ ) W ( θ )  , (3.9) with W ( θ ) = R ( θ ) 2 + 2 sin θ cos θ R ( θ ) R ′ ( θ ) + sin 2 θ R ′ ( θ ) 2 − 1 2  1 4 sin 2 θ R ′′ ( θ ) 2 R ′ ( θ ) + sin θ cos θ R ′′ ( θ ) +  cos 2 θ + n 2 sin 2 θ  R ′ ( θ )  . 12 L. HAINE A ND D. V AND ERSTICHELEN Equation (3.9) imp lies tha t W ( θ ) = 0, whic h is the equation ( 1 .2), obtained b y T racy and Widom in [2]. This concludes the pro of of Theorem 3.1.  Remark 3.2. In the ab ov e pro o f, we ha d to assume that n ≥ 4, where n is the size of the random unitary matrices. F or n = 1 , 2 , 3, the func- tion R ( θ ) also satisfies (1.2), as can be sho wn b y direct computation, using the repre sen tation (3.1) of the probabilit y τ n ( η , θ ) as a T o eplitz determinan t. It w ould b e in teresting to relate the pro of with the or ig i- nal deriv ation in [2]. F or the Gaussian ensem bles, the relation b etw een the tw o metho ds has b een studied in [9] . Finally , similarly to the case of the Jacobi p olynomial ensem ble ( see [5]), we observ e that R ( θ ) in (1.1) is link ed to the P ainlev ´ e VI equation. Precisely , w e show that it is the restriction to the unit circle of a solution of (a sp ecial case of ) the P ainlev ´ e VI equation, defined for z ∈ C . Corollary 3.3. Put R ( θ ) = r ( e − 2 iθ ) . The n , the function σ ( z ) = − i ( z − 1) r ( z ) − n 2 4 z satisfies the Okamoto-Jim b o- Miw a form of the Painlev´ e VI e quation [ z ( z − 1 ) σ ′′ ] 2 + 4 z ( z − 1)( σ ′ ) 3 + 4 σ ′ σ 2 + 4(1 − 2 z ) σ ( σ ′ ) 2 − c 1 ( σ ′ ) 2 + [2(1 − 2 z ) c 4 − c 2 ] σ ′ + 4 c 4 σ − c 3 = 0 , (3.10) with c 1 = n 2 , c 2 = 3 n 4 8 , c 3 = n 6 16 , c 4 = − n 4 16 . (3.11) Pr o of. F r o m (1.2), by a straightforw ard computation, putting R ( θ ) = r ( e − 2 iθ ), w e obtain that r ( z ) satisfies [ z ( z − 1 ) r ′′ ] 2 + 4 z 2 ( z − 1) r ′ r ′′ − 4 iz ( z − 1) 2 ( r ′ ) 3 − 4 i ( z 2 − 1) r ( r ′ ) 2 + [4 z 2 − n 2 ( z − 1) 2 ]( r ′ ) 2 − 4 ir 2 r ′ = 0 . (3.12) Substituting in (3.12) r ( z ) = i σ ( z ) + xz z − 1 for some constant x , and annihilating the co efficien t of σ 2 , one finds that x = n 2 / 4. With this choice of x , the new function σ ( z ) satisfies the P ainlev ´ e VI equation (3.10) if we pic k c 1 , c 2 , c 3 and c 4 as in (3.11), whic h establishes Corolla r y 3.3.  A CENTERLESS REPRESENT A TION OF THE VIRA SORO ALGEBRA 13 4. Dis cussion o f the resul ts and some fur the r directions Our starting mo t iv a tion w as to understand a differential equation (1.2) due to T racy and Widom [2], satisfied by the logarithmic deriv a- tiv e of the g a p probabilit y that an ar c of circ le of length 2 θ con tains no eigen v alues of a randomly chosen unita ry n × n matrix, from the po int of view of the a lg ebraic approach initiated by Adler, Shiota and v an Mo erb ek e [3]. The main surprise is that the 2 -dimensional T o da tau functions (1 .3) defor ming these ga p probabilities, satisfy a cen terless ful l Virasoro algebra of cons train ts. The result stands in con trast with the corresp onding in tegrals for the Gaussian or the orthogonal p olyno- mial ensem bles, which roughly satisfy only ”half of” a Virasoro t yp e algebra of constraints, see [3], [5], [6] a nd [9]. As men tio ned at the b eginning of Section 3, the in tegrals (1.3) can b e express ed as T o eplitz determinan ts, see (3.1). As suc h, they are v ery sp ecial instances of tau functions for the so-called T o eplitz lattices [4], that is τ n ( t, s ) = det  µ k − l ( t, s )  0 ≤ k , l ≤ n − 1 , (4.1) where µ k ( t, s ) = Z S 1 z k e P ∞ j =1 ( t j z j + s j z − j ) w ( z ) d z 2 π iz , k ∈ Z , (4 .2) and w ( z ) is some (complex -v alued) w eight function defi ned o n t he unit circle S 1 , suc h that t he trigonometric moments µ k = µ k (0 , 0) = Z S 1 z k w ( z ) d z 2 π iz , k ∈ Z , satisfy det  µ k − l  0 ≤ k , l ≤ n − 1 6 = 0 , ∀ n ≥ 1. In the sp ecial case (3.1) that w e consider in this pap er, w ( z ) = χ ( η,θ ) c ( z ) is the characteristic function of the complemen t of the arc of circle ( η , θ ) = { z ∈ S 1 | η < arg( z ) < θ } . As it immediately follo ws from ( 4 .2), a t the lev el of the trigonometric momen ts, the T o eplitz hierarch y is giv en by the simple equations T j µ k ≡ ∂ µ k ∂ t j = µ k + j , T − j µ k ≡ ∂ µ k ∂ s j = µ k − j , ∀ j ≥ 1 . Ob viously [ T i , T j ] = 0 , ∀ i, j ∈ Z , if w e define T 0 µ k = µ k . F ollowing an idea in tro duced in [5] in the contex t of the 1-dimensional T o da lattices, w e define the following v ector fields on the trigonometric momen ts V j µ k = ( k + j ) µ k + j , ∀ j ∈ Z . (4.3) 14 L. HAINE A ND D. V AND ERSTICHELEN These v ector fields trivially satisfy the comm utatio n relations [ V i , V j ] = ( j − i ) V i + j (4.4) [ V i , T j ] = j T i + j , ∀ i, j ∈ Z , (4.5) from whic h it follo ws that [[ V i , T j ] , T j ] = j [ T i + j , T j ] = 0 , ∀ i, j ∈ Z . (4.6) Equations (4.4 ), (4.5) and (4.6) mean that the v ector fields V j , j ∈ Z , fo r m a Virasoro algebra of master symme tries, in the sense of F uc hssteiner [10], for the T o eplitz hierarc hy . The tau functions (4.1) a dmit the f o llo wing expansion τ n ( t, s ) = X 0 ≤ i 0 < ···

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