Poincaré Lelong Approach to Universality and Scaling of Correlations Between Zeros

Transcription

1 Commun. Math. Phys. 208, (2000 Communcatons n Mathematcal Physcs Sprnger-Verlag 2000 Poncaré Lelong Approach to Unversalty an Scalng of Correlatons Between Zeros Pavel Bleher 1,, Bernar Shffman 2,, Steve Zeltch 2, 1 Department of Mathematcal Scences, IUPUI, Inanapols, I 46202, USA. E-mal: bleher@math.upu.eu 2 Department of Mathematcs, Johns Hopkns Unversty, Baltmore, MD 21218, USA. E-mal: shffman@math.jhu.eu; zel@math.jhu.eu Receve: 17 March 1999 / Accepte: 5 August 1999 Abstract: Ths note s concerne wth the scalng lmt as of n-pont correlatons between zeros of ranom holomorphc polynomals of egree n m varables. More generally we stuy correlatons between zeros of holomorphc sectons of powers L of any postve holomorphc lne bunle L over a compact Kähler manfol. Dstances are rescale so that the average ensty of zeros s nepenent of. Our man result s that the scalng lmts of the correlaton functons an, more generally, of the correlaton forms are unversal,.e. nepenent of the bunle L, manfol M or pont on M. Introucton Ths note s a companon to our artcle [BSZ], n whch we stuy the correlatons between the zeros of a ranom holomorphc secton s H 0 (M, L of a power L of a postve lne bunle L M over a compact m-mensonal complex manfol M. Snce the hypersurface volume of the zeros of a secton of L n a ball U aroun a gven pont z 0 s Vol (U, we rescale U U to get a ensty of zeros nepenent of. After expanng U ths way, all manfols an lne bunles appear asymptotcally alke, an t s natural to ask f the local statstcs of zeros are unversal,.e. nepenent of L, M, ω an z 0. To efne our statstcs, we frst prove H 0 (M, L wth a natural Gaussan measure (see Sects The local statstcs are measure by the scale n-pont zero correlaton forms K n ( z1,..., zn, z j U (see Sect They are smooth off the agonal, an ther norms efne scale zero correlaton measures K n ( z1,..., zn. (The scale correlaton forms exten to all of ( U n as currents Research partally supporte by SF grant #DMS Research partally supporte by SF grant #DMS Research partally supporte by SF grant #DMS

2 772 P. Bleher, B. Shffman, S. Zeltch of orer 0. The correlaton measures K n gve the expecte value of the prouct of the volumes of the zero set n n omans n M; see Eq. (12. In [BSZ], we use geometrc probablty methos an a (unversal scale Szegö kernel to prove that there exst unversal lmts as of these correlaton measures an more generally of the correlatons between smultaneous zeros of k m sectons. Here we use a complex analytc approach base on the Poncaré Lelong formula for the currents of ntegraton over the zero set of a secton, together wth the scale Szegö kernel from [BSZ], to prove unversalty for the correlaton forms. Ths approach, although lmte to the hypersurface case, allows for a result on the level of forms an a somewhat smpler proof. Our unversalty theorem s as follows: Man Theorem. There s a unversal current K n D (m 1n,(m 1n (C mn such that the followng hols: suppose that (L, h s a postve Hermtan lne bunle on an m- mensonal compact complex manfol M, an let K n be the n-pont zero correlaton current on M n. Suppose z 0 M an choose local holomorphc coornates n M about z 0 such that h z 0 = z 2. Then K n ( z 1,..., z n = K n (z1,...,z n + O ( 1. Furthermore, K n s a smooth form on the off-agonal oman Gm n consstng of n-tuples of stnct ponts z 1,...,z n n C m. The error term has k th orer ervatves C A,k on each compact subset A Gn m, k 0. Ths result s new n all mensons, an the proof oes not smplfy n any essental way n the case m = 1 of Remann surfaces. Our metho leas to ntegral formulae for the unversal lmt forms, although the etals raply become complcate as the number n of ponts ncrease. For the case n = 2 of the par correlaton functon, we carry out the calculaton n complete etal n menson m = 1 an also use the metho to obtan an explct formula for the scalng lmt par correlaton measures n all mensons (Theorems 4.1 an 4.2. In partcular, our formula gves the scalng lmt par correlatons for SU(m + 1-polynomals (whch are sectons of powers of the hyperplane bunle over complex projectve space CP m. The unversal formula n menson m = 1 agrees, as t must, wth that of Hannay [Ha] n the case of ranom SU(2-polynomals. (The unversalty of ths formula was not prove prevously. Smlar formulas for correlatons of zeros of real polynomals were gven n [BD]. Before we get starte on the proof, the followng heurstc remark on correlaton measures an forms may be helpful. Although the efnton of the correlaton measures s formulate n terms of expecte volumes of the zero set (or n the one-mensonal case, expecte numbers of zeros n omans of M, t can also be gven a probablstc nterpretaton: the probablty that the zero vsor of a ranom secton s smultaneously ntersects balls of raus ε aroun z 1,...,z n, respectvely, s cε 2n Kn (z1,...,z n (where Kn s the correlaton functon gven by K n V M = K n. The correlaton form K n gves a more refne probablty that the zero vsor of s has tangent hyperplanes close to n fxe complex hyperplanes n T M. A fnal remark on the term unversalty : n ths paper t means nepenence from the etals of the geometrc settng,.e. from the complex manfol, lne bunle, metrc, connecton, etc. In ranom matrx theory (cf. [D] t has a somewhat fferent meanng.

3 Poncaré Lelong Approach to Correlatons Between Zeros 773 There, the settng conssts of a fxe class of matrces; what vares s the probablty measure ν on the space of matrces. Unversalty then means that the correlaton functons n the scalng lmt are the same for a broa class of measures ν. Ths noton of unversalty makes sense as well for ranom polynomals an sectons. We coul stuy the statstcs of zeros relatve to more general measures on H 0 (M, L than Gaussan ones, e.g. measures of the form ν (s = e M ( s(z 2 +a s(z 4 +b s(z 2 V (z s. Such measures are base aganst sectons wth strong oscllatons, an one coul ask how that affects the statstcs of zeros. However, we o not conser unversalty n the measure aspect n ths paper. 1. otaton We summarze here the notaton from complex analyss that we wll nee n the proof. Ths notaton s the same as n [SZ] an [BSZ], except that fferent normalzatons for the metrc an volume form are use n [SZ] Complex geometry. We enote by (L, h M a holomorphc lne bunle wth smooth Hermtan metrc h whose curvature form h = log e L 2 h, (1 s a postve (1,1-form. Here, e L s a local non-vanshng holomorphc secton of L over an open set U M, an e L h = h(e L,e L 1/2 s the h-norm of e L. As n [BSZ], we gve M the Hermtan metrc corresponng to the Kähler form ω = 1 2 h an the nuce Remannan volume form V M = 1 m! ωm. (2 We enote by H 0 (M, L the space of holomorphc sectons of L = L L. The metrc h nuces Hermtan metrcs h on L gven by s h = s h.wegve H 0 (M, L the Hermtan nner prouct s 1,s 2 = h (s 1,s 2 V M (s 1,s 2 H 0 (M, L, (3 M an we wrte s = s, s 1/2. For a holomorphc secton s H 0 (M, L,weletZ s enote the current 1 of ntegraton over the zero vsor of s: (Z s,ϕ= ϕ, ϕ D m 1,m 1 (M. Z s The Poncaré Lelong formula (see e.g., [GH] expresses the ntegraton current of a holomorphc secton s = gel n the form: Z s = π log g = π log s h + ω. (4 1 Here, D p,q ( enotes the space of compactly supporte (p, q forms on a complex manfol. A current s an element of the ual space D p,q ( = D m p,m q (.

4 774 P. Bleher, B. Shffman, S. Zeltch We also enote by Z s the Remannan (2m 2-volume along the regular ponts of Z s, regare as a measure on M: 1 ( Z s,ϕ= ϕvol 2m 2 = ϕω m 1 ; (5 Zs reg (m 1! Zs reg.e., Z s s the total varaton measure of the current of ntegraton over Z s : Z s =Z s 1 (m 1! ωm 1. ( Ranom sectons an Gaussan measures. We now gve H 0 (M, L the complex Gaussan probablty measure µ(s = 1 π e c 2 c, s = j=1 c j S j, (7 where {S j : 1 j } s an orthonormal bass for H 0 (M, L an c s 2 - mensonal Lebesgue measure. Ths Gaussan s characterze by the property that the 2 real varables Rc j, Ic j (j = 1,..., are nepenent ranom varables wth mean 0 an varance 1 2 ;.e., E c j = 0, E c j c k = 0, E c j c k = δ jk. Here an throughout ths paper, E enotes expectaton: E ϕ = ϕµ. We then regar the currents Z s (resp. measures Z s, as current-value (resp. measure-value ranom varables on the probablty space (H 0 (M, L, µ;.e., for each test form (resp. functon ϕ, (Z s,ϕ(resp. ( Z s,ϕ s a complex-value ranom varable. Snce the zero current Z s s unchange when s s multple by an element of C, our results are the same f we nstea regar Z s as a ranom varable on the unt sphere SH 0 (M, L wth Haar probablty measure. We prefer to use Gaussan measures n orer to facltate our computatons Correlaton currents an measures. The n-pont correlaton current of the zeros s the current on M n = M M M (n tmes gven by K n (z1,...,z n := E (Z s (z 1 Z s (z 2 Z s (z n (8 n the sense that for any test form ϕ 1 (z 1 ϕ n (z n D m 1,m 1 (M D m 1,m 1 (M, ( K n (z 1,...,z n, ϕ 1 (z 1 ϕ n (z n = E [( Z s,ϕ 1 ( Zs,ϕ 2 ( Zs,ϕ n ]. (9 In a smlar way we efne the n-pont correlaton measures K n measures of the n-pont correlaton currents: K n (z1,...,z n = K n (z1,...,z n as the total varaton 1 1 (m 1! ωm 1 z 1 (m 1! ωm 1 z n. (10

5 Poncaré Lelong Approach to Correlatons Between Zeros 775 By (6 an (10, we have ( K n (z1,...,z n, ϕ 1 (z 1...ϕ n (z n = E [ ( Z s,ϕ 1 ( Z s,ϕ 2 ( Z s,ϕ n ], (11 where ϕ j C 0 (M. Equvalently, n K n (U 1 U n = E Vol(Z s U j, U 1,...,U n M. (12 j=1 Remark. In the case of par correlaton on a Remann surface (n = 2, m M = 1, the correlaton measures take the form K 2 (z, w =[ ] (K 1 (z 1 + κ (z, wω z ω w ( 0, where [ ] enotes the current of ntegraton along the agonal ={(z, z} M M, an κ C (M M Szegö kernels. As n [Ze,SZ,BSZ] an elsewhere, we analyze the lmt by lftng t to a prncpal S 1 bunle π : X M. Let us recall how ths goes. We enote by L the ual lne bunle to L, an efne X as the crcle bunle X = {λ L : λ h = 1}, where h s the norm on L ual to h. We can vew X as the bounary of the sc bunle D ={λ L : ρ(λ > 0}, where ρ(a = 1 λ 2 h. The sc bunle D s strctly pseuoconvex n L, snce h s postve, an hence X nherts the structure of a strctly pseuoconvex CR manfol. Assocate to X s the contact form α = ρ X = ρ X. We also gve X the volume form V X = 1 m! α (αm = α π V M. (13 The settng for our analyss of the Szegö kernel s the Hary space H 2 (X L 2 (X of square ntegrable CR functons on X, where we use the nner prouct F 1,F 2 = 1 F 1 F 2 V X, F 1,F 2 L 2 (X. (14 2π X We let r θ x = e θ x (x X enote the S 1 acton on X. The acton r θ commutes wth the Cauchy-Remann operator b ; hence H 2 (X = =0 H 2 (X, where H 2 (X ={F H 2 (X : F(r θ x = e θ F(x}. A secton s of L etermnes an equvarant functon ŝ on X: ( ŝ (z, λ = λ,s (z, (z,λ X; (15 then ŝ (r θ x = e θ s (x. The map s ŝ s a untary equvalence between H 0 (M, L an H 2 (X. We let : L 2 (X H 2 (X enote the orthogonal projecton. The Szegö kernel (x, y s efne by F(x = (x, yf (yv X (y, F L 2 (X. (16 X

6 776 P. Bleher, B. Shffman, S. Zeltch It can be gven as (x, y = j=1 Ŝ j (xŝ j (y, (17 where S 1,...,S form an orthonormal bass of H 0 (M, L. 2. Scalng In orer that we may stuy the local nature of the ranom varable Z s, we fx a pont z 0 M an choose a holomorphc coornate chart :,0 U,z 0 ( C m,u M such that ω z 0 = m z j z j 2. (18 0 j=1 For example, f L s the hyperplane secton bunle O(1 over CP m wth the Fubn-Stuy metrc h FS, an z 0 = (1 : 0 : :0, then the coornate chart : C m U ={w CP m : w 0 = 0}, (z = (1 : z 1 : :z m (.e., z j = w j /w 0 satsfes (18. To smplfy notaton, we entfy U wth. For a current T D p,q (, we wrte ( z ( T = τ T D p,q (, (τ λ (z = λz. (In partcular, f T = T jk (zz j z k, then T( z = 1 Tjk ( z z j z k. We efne the rescale zero current of s H 0 (M, L by ( z Ẑs (z := Z s. (19 The scale n-pont correlaton currents are then efne by: E (Ẑ s (z 1 Ẑs (z2 Ẑs (zn = K n ( z 1,..., z n D n,n (M n. Followng the approach of [SZ], we fx an orthonormal bass {Sj } of H 0 (M, L an wrte Sj = fj e L over U. Any secton n H 0 (M, L may then be wrtten as s = j=1 c j fj e L. To smplfy the notaton we let f = (f1,...,f : U C an we put j=1 c j f j = c f. (20

7 Poncaré Lelong Approach to Correlatons Between Zeros 777 Hence Z s = an therefore 1 1 π log c f, Ẑs = π z z log c f ( z, (21 Ẑ s (z 1 Ẑ s (z n = ( n π z 1 z 1 ] z n z n [log c f ( z1 log c f ( zn. (22 We then can wrte the rescale correlaton currents n the form K n ( z 1 z n,..., = E ( Ẑ s (z 1 Ẑ s (z n ( n = π z 1 z 1 z n z n ( ( C log c f z 1 c log f z n e c 2 c. π ( Scalng lmt of the Szegö kernel. The asymptotcs of the Szegö kernel along the agonal were gven by [T] an [Ze]: π m m (x, x = 1 + O( 1. (24 For our proof of the Man Theorem, we nee the followng lemma from [BSZ], whch gves the near-agonal asymptotcs of the Szegö kernel. Lemma 2.1. Let z 0 M an choose local coornates {z j } n a neghborhoo of z 0 so that z 0 = 0 an h (z 0 = z j z j. Then π m m ( z, θ ; w, ϕ = e(θ ϕ+i(z w 2 1 z w 2 + O( 1/2. Here, (z, θ enotes the pont e θ e L (z h el (z X, an smlarly for (w, ϕ. In (24 an Lemma 2.1, the expresson O( α means a term wth k th orer ervatves C k α, for all k 0. Lemma 2.1 says that the Szegö kernel has a unversal scalng lmt. In fact, ts scalng lmt s the frst Szegö kernel of the reuce Hesenberg group; see [BSZ]. 3. Unversalty All the eas of the proof of the Man Theorem occur n the smplest case n = 2. So frst we prove unversalty n that case an then exten the proof to general n.

8 778 P. Bleher, B. Shffman, S. Zeltch Thus, our frst object s to prove unversalty of the large lmt of the rescale par correlaton current (from (23 wth n = 2 ( z w, = E (Ẑ s (z Ẑs (w K 2 = 1 π 2 z z w w C log c f ( z log c f ( w e c 2 π c. As n [SZ], we wrte f = f u an expan the ntegran n (25: log c f ( z log c f ( w =log f ( z log f ( w + log f ( z log c u ( w + log f ( w log c u ( z + log c u ( z log c u ( w. (26 Let us enote the terms resultng from ths expanson by E 1,E 2,E 3,E 4, respectvely. In partcular, E 1 = 1 [ π 2 z z w w log f ( z log f ( w ]. (27 By (15, Ŝj (z, θ = eθ e L (z h f j (z, where (z, θ are the coornates n X gven n Sect By (17, (25 (z, w = e L (z h e L(w h f (z, f (w, (28 where we wrte (z, w = (z, 0; w, 0. Snce (z, z 1/2 = e L (z h f (z, each factor n (27 has the form 2 1 log ( z, z log e L ( z h. By (24, z z w w log ( z z, 0 as. On the other han z z log e L ( z h = ω( z. Hence the frst term converges to the normalze Euclean (ouble Kähler form: E 1 = 2π z 2 2π w 2 + O( 1. (29 The mle two terms vansh snce the ntegrals n E 2 an E 3 are nepenent of w an z respectvely (see [SZ, 3.2]. The nterestng term s therefore E 4 = 1 π 2 z z w w log c u ( z log c u ( w e c 2 C π c. (30

9 Poncaré Lelong Approach to Correlatons Between Zeros 779 To evaluate E 4, we conser the ntegral G 2 (x1,x 2 := log c x 1 log c x 2 e c 2 C π c (x1,x 2 C (31 wth x 1 = u ( z, x 2 = u ( w. To smplfy t, we construct a Hermtan orthonormal bass {e 1,...,e } for C such that x 1 = e 1 an x 2 = ξ 1 e 1 + ξ 2 e 2, ξ 1 = x 2,x 1, ξ 2 = 1 ξ 1 2. (32 Ths s possble because we can always multply e 2 by a phase e θ so that ξ 2 s postve real. We then make a untary change of varables to express the ntegral n the {e j } coornates. Snce the Gaussan s U( -nvarant, (31 smplfes to G 2 (x1,x 2 = G 2 (ξ 1,ξ 2 = 1 π 2 e ( c1 2 + c 2 2 log ξ 1 log c 1 ξ 1 + c 2 ξ 2 c 1 c 2 C 2 (33 (where we use the fact that the Gaussan ntegral n each c j,j 3, equals 1 by constructon. By performng a rotaton of the c 1 varable, we may replace ξ 1 wth ξ 1 an replace G 2 (ξ 1,ξ 2 wth G(cos θ := G 2 (cos θ,sn θ, (34 where cos θ = ξ 1 = x 1,x 2,0 θ π/2. Hence (30 becomes E 4 = 1 π 2 z z w w G(cos θ, cos θ = u ( z, u ( w. (35 By the unversal scalng formula for the Szegö kernel (Lemma 2.1 an (28, we have (z, w cos θ = (z, z 1/2 (w, w 1/2 = e 2 1 z w 2 + O( 2 1. (36 Thus we get the unversal formula: K 2 (z, w = 2π z 2 2π w π 2 z z w w G(e 2 1 z w 2. (37 Ths completes the proof for the par correlaton case n = 2. (otce that the formula has the same form n all mensons. The proof for general n s smlar. We agan wrte f = f u an expan the ntegran n (23: log c f ( z1 log c f ( z2 log c f ( zn = log f ( z1 log f ( z2 log f ( zn + log f ( z1 log f ( z2 log f ( zn 1 log c u ( zn + + log c u ( z1 log c u ( z2 log c u ( zn.

10 780 P. Bleher, B. Shffman, S. Zeltch We enote the terms resultng from ths expanson by E 1,...,E 2 n, respectvely. As before, the frst term converges to the normalze Euclean n-fol Kähler form: E 1 = 2π z 1 2 The E 2 n term s obtane from the functon G n (x1,x 2,...,x n := C x 1,x 2,...,x n C. Precsely, we substtute 2π z n 2 + O( 1. log c x 1 log c x 2 log c x n e c 2 π c, (38 x j = u ( zj (39 n (38 an apply the operator ( n π z 1 z 1 z n z n. As above, we efne a specal Hermtan orthonormal bass {e 1,...,e n } for the n-mensonal complex subspace spanne by {x 1,...,x n }. We put: x 1 = e 1 x 2 = ξ 21 e 1 + ξ 22 e 2 ξ 22 = 1 ξ x n = ξ n1 e 1 + +ξ nn e n ξ nn = 1 j n 1 ξ nj 2. Such a bass exsts because we can always multply e j by a phase e θ so that the last component ξ jj s postve real. We complete {e j } to a bass of C, an we now let c j enote coornates relatve to ths bass. As above, we rewrte the Gaussan ntegral n these coornates. After ntegratng out the varables {c n+1,...,c }, (38 smplfes to the n-mensonal complex Gaussan ntegral G n (x1,...,x n = G n (ξ 21,ξ 22,...,ξ nn = 1 π n C n e c 2 log c 1 log c 1 ξ 21 + c 2 ξ 22 log c 1 ξ n1 +...c n ξ nn c. ote that the varables ξ jk epen on ; we wrte ξ jk = ξjk when we nee to ncate ths epenence. To prove unversalty, we observe that the ξ jk are unversal algebrac functons of the nner proucts x a,x b. Inee, ξ j1 ξ k1 + +ξ jk ξ kk = x j,x k, 1 k j n, (41 where we set ξ 11 = 1. These algebrac functons are obtane by nucton (lexcographcally usng (41. (The trangular matrx (ξ jk s just the nverse of the matrx escrbng the Gram-Schmt process. By (39, t follows that the ξjk are unversal algebrac functons of the varables u ( zj, u ( zk = ( zj, zk ( zj, zj 1/2 ( zk, zk 1/2 = e I(zj z k 1 2 zj z k 2 + O( 1. (40

11 Poncaré Lelong Approach to Correlatons Between Zeros 781 We note here that x 1 x n 2 = et( x j,x k et (e I(zj z k 2 1 zj z k 2 = e z j 2 et (e z j z k. (42 When the z j are stnct (.e., (z 1,...,z n Gn m, the lmt etermnant n (42 s nonzero (see [BSZ] an thus ξjk = ξ jk + O( 1, where the ξ jk are unversal realanalytc functons of z Gn m. We conclue that the E 2n term converges to a unversal current: ( n E 2 n = π z 1 z 1 z n z ng n (ξ21,...,ξ kk + O( 1. Conser now a general term E a. Suppose wthout loss of generalty that E a comes from log c u ( z1 log c u ( zk log f ( zk+1 log f ( zn. As above we obtan E a = ( π k z 1 z 1 z k z kg k (ξ 21,...,ξ kk 2π z k+1 2 2π z n 2 + O( 1. Hence ths term also approaches a unversal current. (As n the par correlaton case, terms wth only one u vansh. 4. Explct Formulae We now calculate explctly the lmt par correlaton measures K 2 (z, w Prelmnares. The frst step s to compute G(e 1 2 r2, where s the Euclean Laplacan on C m an r = ζ (ζ C m. To begn ths computaton, we wrte a j = r j e ϕ j an then rewrte (33 (34 as G(cos θ = 2 π 2π r 1 r 2 e (r2 1 +r2 2 log r 1 log r 1 cos θ + r 2 e ϕ sn θ ϕr 1 r 2. (43 We now evaluate the nner ntegral by Jensen s formula, whch gves 2π 2π log(r 1 cos θ for r 2 sn θ r 1 cos θ log r 1 cos θ + r 2 sn θe ϕ ϕ =. 0 2π log(r 2 sn θ for r 2 sn θ r 1 cos θ (44

12 782 P. Bleher, B. Shffman, S. Zeltch Hence G(cos θ = r 1 r 2 e (r2 1 +r2 2 log r 1 log max(r 1 cos θ,r 2 sn θr 1 r 2. (45 ow change varables agan wth r 1 = ρ cos ϕ,r 2 = ρ sn ϕ to get G(cos θ = 4 Snce π/2 0 0 ρ 3 e ρ2 log(ρ cos ϕlog max(ρ cos ϕ cos θ,ρ sn ϕ sn θ cos ϕ sn ϕϕρ. (46 log max(ρ cos ϕ cos θ,ρ sn ϕ sn θ = log(ρ cos ϕ cos θ+ log + (tan ϕ tan θ, we can wrte G = G 1 + G 2, where G 1 (cos θ = 4 G 2 (cos θ = 4 π/2 0 0 π/2 0 π/2 θ ρ 3 e ρ2 log(ρ cos ϕlog(ρ cos ϕ cos θcos ϕ sn ϕϕρ, (47 ρ 3 e ρ2 log(ρ cos ϕlog(tan ϕ tan θcos ϕ sn ϕϕρ. (48 From (47, G 1 (cos θ = C 1 + C 2 log cos θ an thus G 1 (e 2 1 r2 = C C 2r 2, so that ( G 1 (e r2 = r 2 + 2m 1 (C 1 1 r r 2 C 2r 2 = 2mC 2. (49 We now evaluate G 2 (e 2 1 r2. Snce the ntegran n (48 vanshes when ϕ = π/2 θ,wehave ( π/2 r G 2(cos θ = 4 log tan θ ρ 3 e ρ2 log(ρ cos ϕcos ϕ sn ϕϕρ. r 0 π/2 θ Substtutng tan 2 θ = e r2 1, we have r log tan θ = r 1 e. r2 Thus where I 1 = I 2 = π/2 0 π/2 θ π/2 0 π/2 θ r G 2(e 2 1 4r r2 = 1 e (I r2 1 + I 2, ρ 3 e ρ2 (log ρcos ϕ sn ϕϕρ = C sn 2 θ = C(1 e r2, ρ 3 e ρ2 (log cos ϕcos ϕ sn ϕϕρ.

13 Poncaré Lelong Approach to Correlatons Between Zeros 783 Thus We compute π/2 sn θ I 2 = 1 (log cos ϕcos ϕ sn ϕϕ = 1 t log tt 2 π/2 θ 2 0 = 1 8 (sn2 θ log sn 2 θ sn 2 θ = 1 [ ] 8 (1 e r2 log(1 e r2 1. r G 2(e 2 1 r2 = r 2 log(1 e r2 + C r. (50 Hence by (49 an (50, ( G(e 2 1 r2 = 2mC 2 + r + 2m 1 ( r r 2 log(1 e r2 + C r = m log(1 e r2 + r2 e r2 1 + C. ( Par correlaton n menson 1. In menson one, the par correlaton form s the same as the par correlaton measure. We frst gve our unversal formula n the onemensonal case. Our formula agrees wth that of Hannay [Ha] for SU(2 polynomals. Theorem 4.1. Suppose m M = 1. Then K 2 ( z w ], K 2 [πδ (z, w = 0 (z w + H( 1 2 z w 2 where 2π z 2 2π w 2, H(t = (snh2 t + t 2 cosh t 2t snh t snh 3 t = t 2 9 t t5 + O(t 7. Proof. Makng the change of varables ζ = z w, we have by (37, (Ẑ E (z Ẑ (w 2π z 2 2π w 2 1 π 2 z z w w G(e 2 1 z w 2 = [ ] z z w w G(e 2 1 z w 2 2π z 2 [ ( 2 2 ] = G(e ζ ζ 2 1 ζ 2 2π z 2 [ = ] 4 2 G(e 2 1 r2 2π z 2 2π w 2. 2π w 2 2π w 2

14 784 P. Bleher, B. Shffman, S. Zeltch By (51 wth m = 1, we have ( 2 G(e r2 = r ] [log(1 e r2 + r2 r r e r2 1 = 4πδ 0 + 8(er r 2 e r2 (e r r 4 e r2 (e r (e r2 1 3 Fnally, [ 1+ 1 ] 4 2 G(e 2 1 r2 = πδ 0 + (er2 +1(e r r 2 e r2 (e r2 1+r 4 e r2 (e r2 +1 (e r2 1 3 = πδ 0 + (snh2 1 2 r r4 cosh 1 2 r2 r 2 snh 1 2 r2 snh r Par correlaton n hgher mensons. The lmt par correlaton measure s gven by K 2 (z, w = lm 2(m 1 K 2 ( z w, = K 2 (z, w 1 (m 1! ( m 1 2 z 2 ( 1 m 1 (m 1! 2 w 2. (The scalng 2(m 1 comes from the fact that ω( z = (τ ω 2 z 2. We now compute K 2 for the case of a manfol of general menson m>1. It s convenent to express ths measure n terms of the expecte ensty of zeros K 1 (z = lm m 1 K 1 ( z = m ( π V C m = 1 m π(m 1! 2 z 2. (52 We have the followng explct unversal formula for the lmt par correlaton measure. In partcular, t gves the scalng lmt par correlaton for the zeros of SU(m + 1- polynomals. Theorem 4.2. Suppose m M = m > 1. Then where K 2 (z, w = [γ m ( 1 2 z w 2 ] K 1 (z K 1 (w, γ m (t = = [ 12 (m 2 + m snh 2 t + t 2] cosh t (m + 1t snh t m 2 snh 3 t (m 1 2m t 1 + m 1 (m + 2(m m 6m 2 t (m + 4(m m 2 t m 1 2m (m + 6(m m 2 t 5 + O(t 7.

15 Poncaré Lelong Approach to Correlatons Between Zeros 785 Proof. By (37 an (51, agan wrtng ζ = z w (except ths tme ζ C m, K 2 (z, w = 1+ 4 m 2 2 m 2 G(e 1 2 z w 2 K 1 z j,k=1 j z j w k w (z K 1 (w k [ = 1+ 1 ] 4m 2 2 ζ G(e 1 2 ζ 2 K 1 (z K 1 (w (53 [ = 1+ 1 ( 2 4m 2 r 2 + 2m 1 ] (mlog(1 e r2 + r2 K r r e r2 1 1 (z K 1 (w. Computng the Laplacan n (53 leas to the state formula. ote that f we substtute m = 1 n the expresson for γ m (t, we obtan Hannay s functon H(t. However for the case m>1, the lmt measure s absolutely contnuous on C m C m, whereas n the one-mensonal case, there s a self-correlaton elta measure. Acknowlegement. The frst raft of ths paper was complete whle the thr author was vstng the Erwn Schrönger Insttute n July He wshes to thank that nsttuton for ts hosptalty an fnancal support. References [BD] Bleher, P. an D, X.: Correlatons between zeros of a ranom polynomal. J. Stat. Phys. 88, (1997 [BSZ] Bleher, P., Shffman, B. an Zeltch, S.: Unversalty an scalng of correlatons between zeros on complex manfols. Invent.Math. to appear [BBL] Bogomolny, E., Bohgas, O. an Leboeuf, P.: Quantum chaotc ynamcs an ranom polynomals. J. Stat. Phys. 85, (1996 [D] Deft, P.: Orthogonal polynomals an ranom matrces: A Remann Hlbert approach. Courant Lecture otes n Mathematcs, 3, ewyork: ewyork Unversty, Courant Insttute of Mathematcal Scences, 1999 [GH] Grffths, P. an Harrs,J.: Prncples of Algebrac Geometry. ew York: Wley-Interscence, 1978 [Ha] Hannay, J.H.: Chaotc analytc zero ponts: Exact statstcs for those of a ranom spn state. J. Phys. A: Math. Gen. 29, (1996 [SZ] Shffman, B. an Zeltch, S.: Dstrbuton of zeros of ranom an quantum chaotc sectons of postve lne bunles. Commun. Math. Phys. 200, (1999 [T] Tan, G.: On a set of polarze Kähler metrcs on algebrac manfols. J. Dff. Geometry 32, (1990 [Ze] Zeltch, S.: Szegö kernels an a theorem of Tan. Int. Math. Res. otces 6, (1998 Communcate by P. Sarnak