On uniform large genus asymptotics of Witten's intersection numbers

ON UNIF ORM LAR GE GENUS ASYMPTOTICS OF WITTEN’S INTERSECTION NUMBERS JINDONG GUO, DI Y ANG, DON ZA GIER Abstract. F ollo wing ideas from [14], we giv e a uniform large genus asymptotics for primitiv e psi-class intersection n um b ers on the mo duli space of stable algebraic curv es, and extend this result including insertions of zeros in a certain uniform wa y . Application to a particular formal solution of the Painlev ´ e I equation is given. W e also use a method from [14] to give a new pro of of the polynomiality conjecture on large gen us asymptotic expansions of psi-class intersection n um b ers. Contents 1. In tro duction and statemen ts of the main results 1 2. An explicit form ula for Witten’s in tersection num b ers 8 3. Uniform large gen us asymptotics of Witten’s intersection n um b ers 10 4. P olynomialit y in large genus 19 References 21 1. Introduction and st a tements of the main resul ts Let M g ,n b e the Deligne–Mumford mo duli space of stable algebraic curves of gen us g with n distinct mark ed p oints. In [28] Witten prop osed a striking conjecture on the relationship b etw een top ology of M g ,n and the celebrated Korteweg–de V ries (KdV) in tegrable hierarch y . T o b e precise, let Z ( t ; ϵ ) := exp  X g ,n ϵ 2 g − 2 n ! X d 1 ,...,d n t d 1 · · · t d n Z M g,n ψ d 1 1 · · · ψ d n n  , (1) b e the p artition function of psi-class interse ction numb ers , where t = ( t 0 , t 1 , t 2 , . . . ) is an infinite v ector of indeterminates, ϵ is an indeterminate often called the string coupling constan t, and ψ j denotes the first Chern class of the j th tautological line bundle on M g ,n . Here and b elo w, P i := P i ≥ 0 . Define u = u ( t ; ϵ ) := ϵ 2 ∂ 2 log Z ( t ; ϵ ) ∂ t 2 0 . (2) Then Witten’s conjecture claims that u satisfies the KdV hierarch y: ∂ u ∂ t k = 1 (2 k + 1)!! h  L 2 k +1 2  + , L i , k ≥ 0 , (3) 1 2 JINDONG GUO, DI Y ANG, DON ZA GIER where L := ϵ 2 ∂ 2 t 0 + 2 u is the Lax op erator (cf. e.g. [7]). The k = 1 equation in (3) reads ∂ u ∂ t 1 = u ∂ u ∂ t 0 + ϵ 2 12 ∂ 3 u ∂ t 3 0 , (4) whic h is the celebrated KdV equation. Another w a y of stating Witten’s conjecture is that Z ( t ; ϵ ) is a tau-function for the KdV hierarc h y; see e.g. [2, 3, 7, 10, 11] for the notion of a KdV tau-function. W e will refer to the psi-class in tersection num bers Z M g,n ψ d 1 1 · · · ψ d n n , g , n, d 1 , . . . , d n ≥ 0 , (5) as Witten ’s interse ction numb ers . They pla y imp ortan t roles in studying W eil–P etersson v olumes of M g ,n [19, 22, 25, 26, 27] and Masur–V eec h v olumes of mo duli space of quadratic differentials [1, 6]. They are called primitive if d 1 ≥ 2, . . . , d n ≥ 2. By the degree-dimension matching w e know that the n um b ers (5) v anish unless d 1 + · · · + d n = 3 g − 3 + n . (6) Witten’s conjecture was first prov ed by Kontsevic h [20]. This conjecture also can [8] b e equiv alen tly stated that the partition function Z = Z ( t ; ϵ ) satisfies the Virasoro constrain ts: L m ( Z ) = 0 , ∀ m ≥ − 1 , (7) where L m are linear operators defined by L m = − (2 m + 3)!! 2 m +1 ∂ ∂ t m +1 + X d (2 d + 2 m + 1)!! (2 d − 1)!! 2 m +1 t d ∂ ∂ t d + m + ϵ 2 2 m − 1 X d =0 (2 d + 1)!!(2 m − 2 d − 1)!! 2 m +1 ∂ 2 ∂ t d ∂ t m − 1 − d + t 2 0 2 ϵ 2 δ m, − 1 + 1 16 δ m, 0 , (8) whic h satisfy the Virasoro c omm utation relations: [ L m 1 , L m 2 ] = ( m 1 − m 2 ) L m 1 + m 2 , m 1 , m 2 ≥ − 1 . (9) It w as sho wn in [8] that the Virasoro constrain ts (7) are equiv alent to the following recursiv e relation: U ( d ) = n X j =2 (2 d j + 1) U ( d 2 , . . . , d j + d 1 − 1 , . . . , d n ) + 1 2 X a, b a + b = d 1 − 2  U ( a, b, d 2 , . . . , d n ) + X I ⊔ J = { 2 ,...,n } U ( a, d I ) U ( b, d J )  , (10) where d := ( d 1 , . . . , d n ) ∈ ( Z ≥ 0 ) n satisfying g ( d ) := 1 + 1 3 P n j =1 ( d j − 1) ∈ Z ≥ 0 and 2 g ( d ) − 2 + n ≥ 2, and U ( d ) is defined by U ( d ) := n Y j =1 (2 d j + 1)!! Z M g ( d ) ,n ψ d 1 1 · · · ψ d n n . (11) ON UNIFORM LARGE GENUS ASYMPTOTICS OF WITTEN’S INTERSECTION NUMBERS 3 Note that the integral on the righ t-hand side of (11) is understoo d as 0 if g ( d ) is not a non-negativ e integer. The relation (10) is no w widely known as the Dijkgr aaf–V erlinde– V erlinde (D VV) r elation . In [2] Bertola, Dubrovin and the second author of the present pap er deriv ed the follo wing explicit formula of generating series of n -p oint Witten’s in tersection num b ers: X d 1 ,...,d n U ( d ) Q n i =1 λ d i +1 i =        P g ≥ 1 (6 g − 3)!! 24 g g ! λ 3 g − 1 , n = 1 , − 1 n P σ ∈ S n tr Q n i =1 M ( λ σ ( i ) ) Q n i =1 ( λ σ ( i ) − λ σ ( i +1) ) − δ n, 2 ( λ 1 + λ 2 ) ( λ 1 − λ 2 ) 2 , n ≥ 2 , (12) where S n denotes the symmetric group, for an element σ ∈ S n the notation σ ( n + 1) is understo o d as σ (1), and M ( λ ) := 1 2 − P ∞ g =1 (6 g − 5)!! 24 g − 1 ( g − 1)! λ − 3 g +2 − 2 P ∞ g =0 (6 g − 1)!! 24 g g ! λ − 3 g 2 P ∞ g =0 6 g +1 6 g − 1 (6 g − 1)!! 24 g g ! λ − 3 g +1 P ∞ g =1 (6 g − 5)!! 24 g − 1 ( g − 1)! λ − 3 g +2 ! . (13) The n = 1 case of (12) is w ell known (see e.g. [28]). F ollo wing [14], an explicit form ula for Witten’s in tersection num bers will b e deduced from (12) in Section 2. In [23] Liu–Xu obtained the large gen us asymptotics for Witten’s in tersection n um- b ers with fixed n and fixed d 1 , . . . , d n − 1 ; the requirement that n should b e fixed is essen tial in [23], meaning that it cannot be impro v ed if one uses the normalizations considered in [23]. In [4, 6] Delecroix–Goujard–Zograf–Zoric h (DGZZ) introduced a remark able normalization of Witten’s in tersection num bers G ( d ) := 24 g ( d ) g ( d )! Q n j =1 (2 d j + 1)!! (6 g ( d ) + 2 n − 5)!! Z M g ( d ) , n ψ d 1 1 · · · ψ d n n , d ∈ ( Z ≥ 0 ) n , (14) and prop osed the following important conjecture: for an y C > 0, lim g → + ∞ max 1 ≤ n ≤ C log g max d ∈ ( Z ≥ 0 ) n | d | =3 g − 3+ n   G ( d ) − 1   = 0 , (15) or stated differently , the normalized in tersection n um b ers G ( d ) tend to 1 uniformly if n = O (log g ). Here | d | := d 1 + · · · + d n . The DGZZ conjecture was first pro v ed b y Aggarwal [1], and later indep enden tly b y t w o of the authors of the presen t pap er [15] by a differen t metho d. Actually , Aggarwal [1, Theorem 1.5] pro v ed the follo wing stronger v ersion of the DGZZ conjecture: lim ϵ → 0  lim g →∞ max n<ϵ √ g max d ∈ ( Z ≥ 0 ) n | d | =3 g − 3+ n    G ( d ) − 1     = 0 . (16) In other words, G ( d ) tends to 1 uniformly for n = o ( √ g ). According to [1, 4], o ( √ g ) cannot b e replaced by O ( √ g ) in this statement. Recen tly , Norbury and the three authors of the present pap er [14] discov ered and pro v ed a completely uniform large genus asymptotic formula for the Br´ ezin–Gross– Witten (BGW) n umbers (for the meaning of BGW n umbers see e.g. [14]). In this paper, w e aim to find and pro ve the corresp onding result for Witten’s in tersection num bers. 4 JINDONG GUO, DI Y ANG, DON ZA GIER F ollo wing the idea of [14], introduce a new normalization by C ( d ) := 2 2 g ( d ) Q n j =1 (2 d j + 1)!! 3 2 g ( d ) − 2+ n (2 g ( d ) − 3 + n )! Z M g ( d ) , n ψ d 1 1 · · · ψ d n n , d ∈ ( Z ≥ 0 ) n . (17) The num bers G ( d ) and C ( d ) are related by C (0 n − 1 , 3 g ( d ) − 3 + n ) G ( d ) = C ( d ) . (18) F rom the n = 1 case of (12) we kno w that C (3 g − 2) has the explicit expression: C (3 g − 2) = 3 (6 g − 3)!! 54 g g ! (2 g − 2)! , g ≥ 1 . (19) Explicit v alues of the n um b ers C ( d ) with g ( d ) = 2 , 3 are given in T able 1. g = 2 , D = 3888 (4) 35 144 0.24306 945 (2 , 3) 1015 3888 0.26106 1015 (2 , 2 , 2) 175 648 0.27006 1050 g = 3 , D = 7558272 (7) 25025 93312 0.26819 2027025 (2 , 6) 77077 279936 0.27534 2081079 (3 , 5) 38731 139968 0.27671 2091474 (4 , 4) 4249 15552 0.27321 2065014 (2 , 2 , 5) 6545 23328 0.28056 2120580 (2 , 3 , 4) 39235 139968 0.28031 2118690 (3 , 3 , 3) 714175 2519424 0.28347 2142525 (2 , 2 , 2 , 4) 6625 23328 0.28399 2146500 (2 , 2 , 3 , 3) 179375 629856 0.28479 2152500 (2 , 2 , 2 , 2 , 3) 120625 419904 0.28727 2171250 (2 , 2 , 2 , 2 , 2 , 2) 546875 1889568 0.28942 2187500 T able 1. Some normalized Witten’s intersection n um b ers C ( d ) Similar to the conjectural nesting prop ert y for BGW n umbers [14, Conjecture 1], w e observ e from T able 1 and further numerical data that the primitive normalized Witten’s in tersection num bers C ( d ) with a fixed genus g ≤ 13 lie b etw een the v alues C (3 g − 2) and C (2 3 g − 3 ). F or example, for g = 13 the num b ers C ( d ) with d 1 , . . . , d n ≥ 2 lie b etw een C (37) = 0 . 30674776 · · · and C (2 36 ) = 0 . 31263595 · · · . W e formulate the follo wing Conjecture 1. F or g ≥ 2 , n ≥ 1 and for d 1 , . . . , d n ≥ 1 satisfying | d | = 3 g − 3 + n , C (3 g − 2) ≤ C ( d ) ≤ C (2 3 g − 3 ) . (20) ON UNIFORM LARGE GENUS ASYMPTOTICS OF WITTEN’S INTERSECTION NUMBERS 5 Remark 1. The inequalit y (20) do es not hold if d j is allow ed to b e 0. E.g., for g = 2, C (0 6 , 10) = 1616615 6718464 ≈ 0 . 24062 < 0 . 24306 ≈ 35 144 = C (4) , (21) C (0 2 , 6) = 5005 15552 ≈ 0 . 32182 > 0 . 27006 ≈ 175 648 = C (2 3 ) . (22) Remark 2. A main conjecture for BGW num b ers in [14] was the monotonicit y of these n um b ers for fixed g in a suitable lexicographic ordering. How ever, this do es not hold for the normalized num b ers C ( d ), e.g., C (4 , 4) < C (3 , 5). Remark 3. There is also a weak monotonicit y conjecture [14] for BGW n umbers, sa ying that the in terv al I g ,n (the conv ex hull of normalized BGW n umbers of genus g and length n ) lies strictly to the left of I g ,n +1 . Even this weak monotonicity fails for normalized primitive Witten’s intersection n um b ers C ( d ), a coun terexample b eing C (2 5 , 8) = 727759375 2448880128 ≈ 0 . 29718 < 0 . 29746 ≈ 419588015525 1410554953728 = C (3 4 , 5) . (23) F ollowing [14], w e also observe from T able 1 and further numerical data that the normalized primitive Witten’s intersection n umbers C ( d ) for eac h fixed g are close to eac h other, e.g. the minim um and maxim um v alues C (37) and C (2 36 ) for g = 13 differ b y less than 2 p ercen t. In view of the nesting prop ert y , let us fo cus on the t wo v alues C (3 g − 2) and C (2 3 g − 3 ). On one hand, the v alue of C (3 g − 2) is given by (19), which b y Stirling’s form ula satisfies C (3 g − 2) ∼ 1 π  1 − 17 36 g + 1 2592 g 2 − 557 279936 g 3 + · · ·  , g → ∞ . (24) On the other hand, with the help of a deep result [17, 18] on a certain formal solution to the P ainlev ´ e I equation (cf. Section 3 for detail), one can obtain that C (2 3 g − 3 ) ∼ 1 π  1 − 2 9 g − 238 2025 g 2 − 198149 2733750 g 3 + · · ·  , g → ∞ . (25) Conjecture 1 together with formulas (24), (25) and (32) implies that for d ∈ ( Z ≥ 1 ) n with g ( d ) ∈ Z ≥ 1 , 1 π − 17 36 π g ( d ) + O  1 g ( d ) 2  ≤ C ( d ) ≤ 1 π − 2 9 π g ( d ) + O  1 g ( d ) 2  (26) as g ( d ) → ∞ . This motiv ates the following Theorem 1. F or n ≥ 1 and d ∈ ( Z ≥ 1 ) n satisfying g ( d ) ∈ Z ≥ 1 , we have C ( d ) = 1 π + O  1 g ( d )  (27) uniformly as g ( d ) → ∞ . Theorem 1 sa ys that there exists an absolute constant K 1 suc h that    C ( d ) − 1 π    ≤ K 1 g ( d ) (28) holds for all n ≥ 1 and d ∈ ( Z ≥ 1 ) n with sufficiently large g ( d ) ∈ Z . 6 JINDONG GUO, DI Y ANG, DON ZA GIER Our proof, which do es not use (25), is along the lines of [1, 14] and will b e giv en in Section 3. It mainly uses the D VV relation (10), which written in terms of C ( d ) reads C ( d ) = n X j =2 2 d j + 1 3 ( X ( d ) − 1) C ( d 2 , . . . , d j + d 1 − 1 , . . . , d n ) + X a, b a + b = d 1 − 2  2 3 ( X ( d ) − 1) C ( a, b, d 2 , . . . , d n ) + X I ⊔ J = { 2 ,...,n } ( X ( a, d I ) − 1)! ( X ( b, d J ) − 1)! 6 ( X ( d ) − 1)! C ( a, d I ) C ( b, d J )  , (29) where X ( d ) := 1 3 n X j =1 (2 d j + 1) = 2 g ( d ) − 2 + n . (30) In particular, C (0 , d ) = n X j =1 2 d j + 1 3( X (0 , d ) − 1) C ( d 1 , . . . , d j − 1 , . . . , d n ) , (31) C (1 , d ) = C ( d ) . (32) An application of Theorem 1 will b e giv en in Section 3. With the help of (31) we can improv e Theorem 1 as follows: Theorem 2. F or n ≥ 1 and d ∈ ( Z ≥ 0 ) n satisfying g ( d ) ∈ Z ≥ 1 , we have C ( d ) = 1 π p 0 ( d ) Y j =1  1 + 2 + j − p 0 ( d ) 3 X ( d ) − 3 p 1 ( d ) − 3 j  + O  1 g ( d )  , (33) uniformly as g ( d ) → ∞ , wher e p i ( d ) , i ≥ 0 , denotes the multiplicity of i in d . The pro of is given in Section 3. Theorem 2 says that there exists an absolute constan t K 2 > 0 such that     C ( d ) − 1 π p 0 ( d ) Y j =1  1 + 2 + j − p 0 ( d ) 3 X ( d ) − 3 p 1 ( d ) − 3 j      ≤ K 2 g ( d ) , (34) for all n ≥ 1 and d ∈ ( Z ≥ 0 ) n satisfying g ( d ) ∈ Z ≥ 1 . Alternativ ely , w e can write (33) as C ( d ) = 1 π  2 3  p 0 ( d )  3 X − 3 p 0 ( d ) − 3 p 1 ( d )+2 2  p 0 ( d ) ( X − p 1 ( d ) − p 0 ( d )) p 0 ( d ) + O  1 g ( d )  , (35) as g ( d ) → ∞ . Here, ( a ) b := a ( a + 1) · · · ( a + b − 1) denotes the Pochhammer symbol. As a particular example of Theorem 2, w e will prov e the following Corollary 1. As g → ∞ , for k = O ( √ g ) we have the uniform le ading asymptotics C (0 k , 2 3 g − 3+ k ) ∼ 1 π e − k 2 30 g , (36) and for k / √ g → ∞ we have C (0 k , 2 3 g − 3+ k ) → 0 . ON UNIFORM LARGE GENUS ASYMPTOTICS OF WITTEN’S INTERSECTION NUMBERS 7 In terms of the original Witten’s intersection n um b ers, (36) reads 2 2 g 5 3 g − 3+2 k 3 2 g − 2 (5 g − 6 + 2 k )! Z M g, 3 g − 3+2 k ψ 2 1 · · · ψ 2 3 g − 3+ k ∼ 1 π e − k 2 30 g . (37) Remark 4. Theorem 2 implies Aggarw al’s result (16). Actually , Corollary 1 cannot b e deduced from (16). Let us pro ceed to consider the full asymptotics of the normalized Witten’s in tersec- tion n um b ers. It w as sho wn by Liu and Xu [23] (cf. also [15]) that for fixed n ≥ 1 and fixed d ′ = ( d 1 , . . . , d n − 1 ) ∈ ( Z ≥ 0 ) n − 1 , the quotient C ( d ′ , 3 g − 3 + n − | d ′ | ) /C (3 g − 2) is a rational function of g , and therefore has a full asymptotic expansion which is a p o w er series of g − 1 . Then by using Stirling’s form ula and the 1-p oin t form ula (19) we know that C ( d ′ , 3 g − 3 + n − | d ′ | ) also has a full asymptotic expansion which is a p ow er series of g − 1 . The p olynomiality conjecture from [15] predicts that for eac h k the coefficient of g − k in the large gen us expansion of C ( d ′ , 3 g − 3 + n − | d ′ | ) is a p olynomial of n and the m ultiplicities of the argumen ts, and also that only the m ultiplicities of 0 , 1 , . . . , [3 k / 2] − 1 are in v olv ed. This conjecture w as pro ved b y Eynard et al [12] based on a formula pro ved in [29, 10] (cf. [2]). F ollowing [14], we will present and pro ve an impro v ed version (see Theorem 3 below) of the p olynomiality conjecture. F or n ≥ 1, d = ( d 1 , . . . , d n ) ∈ ( Z ≥ 0 ) n satisfying g ( d ) ∈ Z , we introduce a new normalization b C ( d ) of Witten’s in tersection num bers b y b C ( d ) := C ( d ) γ ( X ( d )) , (38) where γ ( X ) := 2 X √ π 3 3 X +1 2 Γ  3 X 2 + 1  Γ  X +3 2  Γ( X ) . (39) According to (19), we kno w that b C (3 g − 2) ≡ 1 for any g ≥ 1. Theorem 3. F or any fixe d n and fixe d d ′ ∈ ( Z ≥ 2 ) n − 1 , the numb ers b C ( d ) satisfy b C ( d ) ∼ X k b c k ( p 2 ( d ′ ) , p 3 ( d ′ ) , . . . ) X ( d ) k , d n → ∞ , (40) wher e d = ( d ′ , d n ) , b c k ar e universal p olynomials of p 2 , p 3 , . . . having r ational c o efficients, with b c 0 ≡ 1 and b c k | p b ≡ 0 = 0 ( k ≥ 1) . Mor e over, under the de gr e e assignments deg p d = 2 d + 1 ( d ≥ 1) , (41) the p olynomials b c k , k ≥ 1 , satisfy the de gr e e estimates deg b c k ≤ 3 k − 1 . (42) Sev eral explicit expressions for ˜ c k and b c k are given in the follo wing table. Organization of the pap er In Section 2 we give a closed form ula for Witten’s in- tersection num b ers. In Section 3 we pro v e Theorems 1 and 2. In Section 4 w e prov e Theorem 3. Ac kno wledgemen ts The work is supp orted b y NSFC 12371254 and CAS YSBR-032. P art of the w ork was done during visits of J.G. and D.Y. to MPIM, Bonn and visits of D.Z. to USTC; the authors thank b oth institutions for excellent w orking conditions. 8 JINDONG GUO, DI Y ANG, DON ZA GIER k ˜ c k ( p 1 , p 2 , . . . ) b c k ( p 1 , p 2 , . . . ) 0 1 1 1 − 17 18 0 2 613 648 − 5 72 p 2 − 5 72 p 2 3 65 648 p 2 + 35 216 p 3 − 33713 34992 5 144 p 2 + 35 216 p 3 4 1225 10368 p 2 2 + 130 729 p 2 + 9415 31104 p 3 1225 10368 p 2 2 + 1435 5184 p 2 + 175 384 p 3 − 1225 3456 p 4 − 385 3456 p 5 + 2424889 2519424 − 1225 3456 p 4 − 385 3456 p 5 T able 2. Explicit expressions for ˜ c k ( p 2 , p 3 , . . . ), b c k ( p 2 , p 3 , . . . ) with k = 0 , . . . , 4 2. An explicit formula for Witten’s intersection numbers In this section, based on (12) w e derive an explicit form ula for Witten’s intersection n um b ers. By taking the Laurent expansion on the righ t-hand side of (12) with n = 2, the follo wing formula w as obtained in [2]: Z M g, 2 ψ d 1 1 ψ d 2 2 = P d 1 l =0 ( d 1 + 1 − l ) ξ l − 1 , 3 g − l (2 d 1 + 1)!!(2 d 2 + 1)!! , (43) where g , d 1 , d 2 ≥ 0 satisfying d 1 + d 2 = 3 g − 1 and ξ k 1 ,k 2 =            (6 g 1 − 5)!! (6 g 2 − 5)!! 2 · 24 g 1 + g 2 − 2 ( g 1 − 1)! ( g 2 − 1)! , k 1 = 3 g 1 − 2 , k 2 = 3 g 2 − 2 , − (6 g 1 − 1)!! (6 g 2 − 1)!! 24 g 1 + g 2 g 1 ! g 2 ! 6 g 2 +1 6 g 2 − 1 , k 1 = 3 g 1 , k 2 = 3 g 2 − 1 , (6 g 1 − 1)!! (6 g 2 − 1)!! 2 · 24 g 1 + g 2 g 1 ! g 2 ! 6 g 1 +1 6 g 1 − 1 , k 1 = 3 g 1 − 1 , k 2 = 3 g 2 , 0 , otherwise . Zograf [30] deriv ed another formula for tw o-p oin t Witten’s in tersection num bers: C ( d 1 , 3 g − 1 − d 1 ) = 1 54 g (2 g − 1)! g d 1 − 1 X d = − 1 η g , d , (44) where η g ,d , d ≥ − 1, are defined by η g , d = (6 g − 3 − 2 d )!! (2 d + 1)!! ·        g − 2 j j ! ( g − j )! , d = 3 j − 1 , − 2 j ! ( g − 1 − j )! , d = 3 j , 2 j ! ( g − 1 − j )! , d = 3 j + 1 . (45) The equiv alence of (43) and (44) was sho wn in [13]. By taking the Lauren t expansion on the righ t-hand side of (12) with n ≥ 2, a form ula for Witten’s in tersection n um bers was deriv ed in [15], whose expression written in terms of C ( d ) is giv en in the follo wing prop osition. Before stating it, w e introduce a notation: a k 1 ,...,k n := 2 2 g ( k ) tr A k 1 · · · A k n 3 2 g ( k )+ n − 2 (2 g ( k ) + n − 3)! , (46) ON UNIFORM LARGE GENUS ASYMPTOTICS OF WITTEN’S INTERSECTION NUMBERS 9 for k = ( k 1 , . . . , k n ) ∈ Z n ≥− 1 , and a k 1 ,...,k n are defined as 0 if some of k j is less than or equal to − 2. Here M ( λ ) =: X k ≥− 1 A k λ − k . Prop osition 1 ([15]) . F or n ≥ 2 and d = ( d 1 , . . . , d n ) ∈ ( Z ≥ 0 ) n , we have C ( d ) = X σ ∈ S n σ ( n )= n ( − 1) | S − σ | +1 X J ∈ ( Z + 1 2 ) n { 1 ≤ q ≤ n | J q > 0 } = S + σ a d σ (1) + J 1 − J n ,...,d σ ( n ) + J n − J n − 1 . (47) The following prop osition is an analogue of [14, Prop osition 3]. Prop osition 2. F or n ≥ 2 and d = ( d 1 , . . . , d n ) ∈ ( Z ≥ 0 ) n , C ( d ) = X σ ∈ S n σ ( n )= n ( − 1) | S − σ | +1 X k 1 ,...,k n ≥− 1 k 1 + ··· + k n = d 1 + ··· + d n a k 1 ,...,k n ω d ,σ, k , (48) wher e the sets S + σ , S − σ ⊆ { 1 , . . . , n } ar e denote d as S + σ =  1 ≤ r ≤ n   σ ( r + 1) > σ ( r )  , S − σ =  1 ≤ r ≤ n   σ ( r + 1) < σ ( r )  , and the numb ers ω d ,σ, k have the fol lowing explicit expr ession ω d ,σ, k = max  0 , min r ∈ S + σ  r X q =1 ( d σ ( q ) − k q )  + min r ∈ S − σ  r X q =1 ( k q − d σ ( q ) )  . (49) The pro of is almost identical to that of [14, Proposition 3] and is therefore omitted. Let us give some examples for Prop osition 2. The n = 2 case of Prop osition 2 is equiv alent to (43). F or n ≥ 1, e ∈ Z n , introduce the notation M ( e ) := max  0 , min 1 ≤ i ≤ n { e i }  , (50) whic h has the generating function X e 1 ,...,e n ≥ 0 x e 1 − 1 1 · · · x e n − 1 n M ( e 1 , . . . , e n ) = 1 (1 − x 1 · · · x n ) Q n i =1 (1 − x i ) . (51) Using (50), the n = 3 case of Prop osition 2 can b e written as C ( d 1 , d 2 , d 3 ) = 2 X k 1 + k 2 + k 3 = d 1 + d 2 + d 3 a k 1 ,k 2 ,k 3 M ( d 1 − k 1 , d 1 + d 2 − k 1 − k 2 ) , (52) and the n = 4 case of Proposition 2 can be written as C ( d 1 , d 2 , d 3 , d 4 ) = 2 X k 1 + k 2 + k 3 + k 4 = d 1 + d 2 + d 3 + d 4 a k 1 ,k 2 ,k 3 ,k 4 ×  M  d 1 − k 1 , d 1 + d 2 − k 1 − k 2 , k 4 − d 4  − M  d 1 − k 2 , d 1 + d 2 − k 2 − k 3 , d 1 + d 3 − k 1 − k 2 , k 4 − d 4  − M  d 1 − k 1 , d 2 − k 3 , k 2 − d 3 , k 4 − d 4   , (53) where we ha v e used the fact that a k 1 ,k 2 ,k 3 ,k 4 = a k 2 ,k 3 ,k 4 ,k 1 = a k 1 ,k 4 ,k 3 ,k 2 . 10 JINDONG GUO, DI Y ANG, DON ZAGIER 3. Uniform large genus asymptotics of Witten’s intersection numbers In this section, w e prov e Theorem 1. Our pro of will mainly use the DVV relation (10), the tec hniques introduced by Aggarwal [1] (see also [14]), and the improv ement in [14]. 3.1. Lo w er Bound. In this subsection we giv e low er bounds for the normalized Wit- ten’s intersection n um b ers C ( d ). Lemma 1. F or n ≥ 1 and for d = ( d 1 , . . . , d n ) ∈ ( Z ≥ 0 ) n satisfying g ( d ) ∈ Z ≥ 0 and 2 g ( d ) − 2 + n > 0 , we have C ( d ) > 0 . (54) Pr o of. By using (29) and by recalling that C (0 3 ) = 1 / 3, C (1) = 1 / 6. □ The following lemma prov es one side of Conjecture 1. Lemma 2. F or g , n ≥ 1 and d ∈ ( Z ≥ 1 ) n satisfying | d | = 3 g − 3 + n , we have C ( d ) ≥ C (3 g − 2) . (55) Pr o of. F or n = 1, inequality (55) is trivial. F or n = 2 inequality (55) was obtained in [4, 5]. Now assume n ≥ 3, and let us pro v e (55) b y induction on X ( d ). F or X ( d ) ≤ 5, inequalit y (55) can b e chec ked directly from T able 1. F or d = ( d 1 , . . . , d n ) ∈ ( Z ≥ 1 ) n with X ( d ) ≥ 6, b ecause of the symmetry of C -v alues and form ula (32), w e can assume that 2 ≤ d 1 ≤ · · · ≤ d n . F or d 1 ≥ 3, by using (19), (29) and the inductive assumption w e hav e C ( d ) ≥ n X j =2 2 d j + 1 3( X ( d ) − 1) C (3 g − 2) + 2( d 1 − 1 + 4 3 ( X ( d ) − 2) ) 3 ( X ( d ) − 1) C (3 g − 5) = C (3 g − 2) − 2( d 1 − 1) 3( X ( d ) − 1)  C (3 g − 2) − C (3 g − 5)  + 8 C (3 g − 5) 9 ( X ( d ) − 1) ( X ( d ) − 2) = C (3 g − 2) + 2 C (3 g − 5) 3( X ( d ) − 1)  − ( d 1 − 1)(34 g − 35) 36 g (2 g − 3)( g − 1) + 4 3( X ( d ) − 2)  , (56) where in the inequality w e hav e omitted the quadratic-in- C term since they are non- negativ e and ha v e used the following estimate which is implied by equation (31) C (0 , d 1 − 2 , d 2 , . . . , d n ) ≥  1 + 2 3( X ( d ) − 2)  C (3 g − 5) . (57) The expression in the parenthesis of the righ t-hand side of (56) is nonnegative since − ( d 1 − 1)(34 g − 35) 36 g (2 g − 3)( g − 1) + 4 3( X ( d ) − 2) ≥ − (34 g − 35) 12 ng (2 g − 3) + 4 3(2 g − 4 + n ) ≥ 1 3 n  − (34 g − 35) 4 g (2 g − 3) + 12 (2 g − 4 + 3)  ≥ 0 , (58) where we hav e used the conditions d 1 − 1 ≤ 3 g − 3 n , 3 ≤ n ≤ 3 g − 3 and g ≥ 3 (since X ( d ) = 2 g − 2 + n ≥ 6). Inequalities (56) and (58) imply that C ( d ) ≥ C (3 g − 2) when d 1 ≥ 3. ON UNIFORM LARGE GENUS ASYMPTOTICS OF WITTEN’S INTERSECTION NUMBERS 11 F or d 1 = 2, we ha v e the estimate C (0 , 0 , d 2 , . . . , d n ) = n X j =2 2 d j + 1 3( X ( d ) − 2) C (0 , d 2 , . . . , d j − 1 , . . . , d n ) (59) ≥  1 + 1 3( X ( d ) − 2)   1 + 2 3( X ( d ) − 3)  C (3 g − 5) ≥ C (3 g − 2) . where for the equalit y w e hav e used (31), for the first inequalit y w e ha ve used the inductiv e assumption and an analogue of (57), and for the second inequality we hav e used the one p oint forumla (19) and the condition n ≤ 3 g − 3 and g ≥ 3. By using (29), (59), we obtain that C ( d 1 = 2 , . . . , d n ) ≥ n X j =2 2 d j + 1 3( X ( d ) − 1) C ( d 2 , . . . , d j + 1 , . . . , d n ) + 2 3( X ( d ) − 1) C (0 , 0 , d 2 , . . . , d n ) ≥ C (3 g − 2) . (60) This completes the pro of. □ Corollary 2. F or g , n ≥ 1 and d ∈ ( Z ≥ 0 ) n satisfying | d | = 3 g − 3 + n , we have C ( d ) ≥ C (3 g − 2) p 0 ( d ) Y j =1  1 + j + 2 − p 0 ( d ) 6 g − 6 + 3 n − 3 j  , (61) wher e p 0 ( d ) denotes the multiplicity of 0 in d . Pr o of. By (31) and Lemma 2 (some details can b e found in the pro of of Theorem 2 b elo w). □ W e mention that several other low er b ounds for Witten’s in tersection n um b ers are obtained in [1, Prop osition 4.1], [4, Theorem 5] and [21, Corollary 5.4]. 3.2. Upp er Bound. Analogous to [1, Lemma 3.1], we first pro v e the follo wing lemma. Lemma 3. F or n ≥ 1 , g ≥ 2 and for d ∈ ( Z ≥ 2 ) n satisfying | d | = 3 g − 3 + n , we have X a,b ≥ 0 a + b = d 1 − 2 X I ⊔ J = { 2 ,...,n } X ( a, d I ) ,X ( b, d J ) ∈ Z  X ( a, d I ) − 1  !  X ( b, d J ) − 1  ! 6  X ( d ) − 1  ! ≤ 2 ( X ( d ) − 1)( X ( d ) − 2) . (62) Pr o of. F or n = 1, the inequalit y (62) is trivial. Assume that n ≥ 2. F or each a, b, I , J satisfying a + b = d 1 − 2, I ⊔ J = { 2 , . . . , n } and X ( a, d I ) , X ( b, d J ) ∈ Z , we set n 1 = | I | , g 1 = g ( a, d I ) = ( a + | d I | + 2 − n 1 ) / 3 ∈ Z , n 2 = | J | , g 2 = g ( a, d J ) = ( b + | d J | + 2 − n 2 ) / 3 ∈ 12 JINDONG GUO, DI Y ANG, DON ZAGIER Z . By coun ting the num ber of 4-tuples ( a, b, I , J ) with given v alues of n i and g i , we find X a,b ≥ 0 a + b = d 1 − 2 X I ⊔ J = { 2 ,...,n } X ( a, d I ) ,X ( b, d J ) ∈ Z  X ( a, d I ) − 1  !  X ( b, d J ) − 1  ! 6  X ( d ) − 1  ! ≤ X n 1 + n 2 = n − 1 X g 1 ,g 2 ≥ 1 g 1 + g 2 = g  n − 1 n 1  (2 g 1 + n 1 − 2)! (2 g 1 + n 2 − 2)! 6 ( X ( d ) − 1)! = X n 1 + n 2 = n − 1  n − 1 n 1  n 1 ! ( X ( d ) − n 1 − 3)! 3 ( X ( d ) − 1)! + X g 1 ,g 2 ≥ 2 g 1 + g 2 = g (2 g 1 + n 1 − 2)! (2 g 1 + n 2 − 2)! 6 ( X ( d ) − 1)!  . (63) W e estimate the t wo terms on the righ t-hand side of (63) separately . F or the first term w e hav e X n 1 + n 2 = n − 1  n − 1 n 1  n 1 ! ( X ( d ) − n 1 − 3)! 3 ( X ( d ) − 1)! = 1 3 ( X ( d ) − 1) ( X ( d ) − 2) n − 1 X n 1 =0 n 1 Y j =1 n − j X ( d ) − 2 − j ≤ 1 3 ( X ( d ) − 1) ( X ( d ) − 2)  2 + ∞ X n 1 =2  3 5  n 1 − 2  = 3 2 ( X ( d ) − 1) ( X ( d ) − 2) , (64) where in the inequalit y w e ha v e used the fact that n ≤ 3 X ( d ) 5 (implied b y d ∈ ( Z ≥ 2 ) n ) and n ≤ X ( d ) − 2 (implied by g ≥ 2). F or the second term, we note that it v anishes if g ≤ 3, otherwise it has the upp er b ound X n 1 + n 2 = n − 1 X g 1 ,g 2 ≥ 2 g 1 + g 2 = g  n − 1 n 1  (2 g 1 + n 1 − 2)! (2 g 2 + n 2 − 2)! 6 ( X ( d ) − 1)! ≤ X g 1 ,g 2 ≥ 2 g 1 + g 2 = g n 1 + n 2 = n − 1  2 g − 4 2 g 1 − 2  − 1 6 ( X ( d ) − 1)( X ( d ) − 2) ≤ n ( g − 3)  2 g − 4 2  − 1 6 ( X ( d ) − 1)( X ( d ) − 2) ≤ 3 8 ( X ( d ) − 1) ( X ( d ) − 2) , (65) where for the first inequalit y we used the fact that X ( d ) − 3 = (2 g 1 + n 1 − 2) + (2 g 2 + n 2 − 2) and that  a 1 b 1  a 2 b 2  ≤  a 1 + a 2 b 1 + b 2  , (66) and for the last inequality we used g ≥ 4 and n ≤ 3 g − 3. Combining (63), (64), (65), w e obtain the lemma. □ F ollowing again the idea of Aggarwal [1] (cf. [14]), we no w define θ X , n = max d ∈ ( Z ≥ 1 ) n X ( d )= X C ( d ) , X , n ≥ 1 . (67) ON UNIFORM LARGE GENUS ASYMPTOTICS OF WITTEN’S INTERSECTION NUMBERS 13 According to (32) we ha v e θ X , n ≥ θ X − 1 , n − 1 , ∀ X , n ≥ 2 . (68) W e ha v e the follo wing Lemma 4. F or n ≥ 1 , d ∈ ( Z ≥ 1 ) n we have C (0 , d ) ≤ 3 X (0 , d ) − 3 p 1 ( d ) − 1 3 X (0 , d ) − 3 p 1 ( d ) − 3 θ X (0 , d ) − p 1 ( d ) − 1 ,n − p 1 ( d ) , (69) C (0 , 0 , d ) ≤ (3 X (0 2 , d ) − 3 p 1 ( d ) − 2)(3 X (0 2 , d ) − 3 p 1 ( d ) − 7) (3 X (0 2 , d ) − 3 p 1 ( d ) − 3)(3 X (0 2 , d ) − 3 p 1 ( d ) − 9) θ X (0 2 , d ) − p 1 ( d ) − 2 ,n − p 1 ( d ) . (70) Pr o of. Let us first pro v e (69). Since b oth sides of (69) are in v ariant if an y argumen t 1 in d is remov ed, it is sufficient to prov e (69) for the case when d ∈ ( Z ≥ 2 ) n . By using (31), (32) and (68) we ha v e C (0 , d ) = n X j =1 2 d j + 1 3 X (0 , d ) − 3 C ( d 1 , . . . , d j − 1 , . . . , d n ) ≤ 3 X (0 , d ) − 1 3 X (0 , d ) − 3 θ X (0 , d ) − 1 ,n , (71) whic h prov es inequalit y (69). T o prov e (70), for the same reason we can assume d ∈ ( Z ≥ 2 ) n . By using (31), (68) and (69) w e hav e C (0 2 , d ) = n X j =1 2 d j + 1 3 X (0 2 , d ) − 3 C (0 , d 1 , . . . , d j − 1 , . . . , d n ) ≤ 3 X (0 2 , d ) − 2 3 X (0 2 , d ) − 3 max  3 X (0 2 , d ) − 4 3 X (0 2 , d ) − 6 θ X (0 2 , d ) − 2 ,n , 3 X (0 2 , d ) − 7 3 X (0 2 , d ) − 9 θ X (0 2 , d ) − 3 ,n − 1  ≤ 3 X (0 2 , d ) − 2 3 X (0 2 , d ) − 3 3 X (0 2 , d ) − 7 3 X (0 2 , d ) − 9 θ X (0 2 , d ) − 2 ,n , (72) whic h finishes the pro of of (70). □ F ollowing [1] (see also [14]), in tro duce a n um b er-theoretic function f ( X , n ), defined through the recursion f ( X , n ) = 2 3 f ( X − 1 , n − 1) + 1 3 f ( X − 1 , n + 1) + 4 ( X − 1)( X − 2) , (73) for an y n ≥ 3, X ≥ 8 together with the initial data f ( X , n ) = 1 /π for 1 ≤ X ≤ 7 or n = 1 , 2. W e notice that the function f ( X , n ) is the same as the one in tro duced in [14]. The following prop osition is pro v ed in [14]. Lemma 5 ([14]) . The function f ( X , n ) satisfies the fol lowing pr op erties. (1) 1 /π ≤ f ( X , n ) ≤ 1 for any n ≥ 1 , X ≥ 1 . (2) f ( X , n ) is monotone incr e asing with r esp e ct to n . (3) F or 1 ≤ n ≤ X / 5 , as X → ∞ , f ( X , n ) = 1 /π + O (1 / X ) wher e the O -c onstant is uniform in n . The significance of the function f ( X , n ) is giv en by the following important lemma. 14 JINDONG GUO, DI Y ANG, DON ZAGIER Lemma 6. F or n ≥ 1 , X ≥ 1 , the numb ers θ X ,n have the upp er b ound θ X ,n ≤ f ( X , n ) . (74) Pr o of. F or 1 ≤ X ≤ 7, w e chec k individually that inequality (74) holds. F or n = 1, b y (19) inequality (74) is equiv alent to the inequalit y 3 (6 g − 3)!! 54 g g ! (2 g − 2)! ≤ 1 π , ∀ g ≥ 1 , (75) whic h is true b ecause the left-hand side is monotone increasing and asymptotic to 1 π b y Stirling’s formula. F or n = 2, according to [4] we ha v e for g ≥ 1, d 1 , d 2 ≥ 0 satisfying d 1 + d 2 = 3 g − 1, C ( d 1 , d 2 ) ≤ C (0 , 3 g − 1) . (76) Applying (31) and (19), w e hav e C ( d 1 , d 2 ) ≤ 6 g − 1 6 g − 3 3 (6 g − 3)!! 54 g g ! (2 g − 2)! ≤ 1 π , (77) where the second inequality holds for the similar reason as (75). No w consider the case that n ≥ 3 and X ≥ 8. Let us prov e inequality (74) by induction on X . F or X = 8, it is easy to c hec k the v alidit y of inequalit y (74). Consider X ≥ 9 and d = ( d 1 , . . . , d n ) ∈  Z ≥ 1  n satisfying X ( d ) = X . Without loss of generality , w e assume d 1 = min { d j } . F or d 1 ≥ 3, by using (29), Lemma 3 and (69) we can obtain C ( d ) ≤  1 − 2 d 1 − 2 3 X − 3  f ( X − 1 , n − 1) + 2 d 1 − 6 3 X − 3 f ( X − 1 , n + 1) + 4 3 X − 3 3 X − 4 3 X − 6 f ( X − 2 , n ) + 2 ( X − 1)( X − 2) . So C ( d ) ≤  1 − 2 d 1 − 2 3 X − 3  f ( X − 1 , n − 1) + 2 d 1 − 2 3 X − 3 f ( X − 1 , n + 1) + 26 9( X − 1)( X − 2) ≤ 2 3 f ( X − 1 , n − 1) + 1 3 f ( X − 1 , n + 1) + 26 9( X − 1)( X − 2) , (78) where for the first inequalit y we used the facts that f ( X − 2 , n ) ≤ f ( X − 1 , n + 1) and f ( X − 1 , n + 1) ≤ 1, and for the second inequalit y w e used the facts that d 1 ≤ 3 X − 3 2 n , n ≥ 3 and f ( X − 1 , n + 1) ≥ f ( X − 1 , n − 1). F or d 1 = 2, by using (29), Lemma 3 and (70) we obtain C ( d ) ≤  1 − 2 3 X − 3  f ( X − 1 , n − 1) + 2 ( X − 1)( X − 2) + 2 3 X − 3 3 X − 5 3 X − 6 3 X − 10 3 X − 12 f ( X − 3 , n − 1) . So C ( d ) ≤ 1 3 f ( X − 1 , n − 1) + 2 3 f ( X − 1 , n + 1) + 79 27( X − 1)( X − 2) , (79) ON UNIFORM LARGE GENUS ASYMPTOTICS OF WITTEN’S INTERSECTION NUMBERS 15 where w e used the facts that f ( X − 3 , n − 1) ≤ f ( X − 1 , n + 1), f ( X − 1 , n + 1) ≤ 1 and f ( X − 1 , n + 1) ≥ f ( X − 1 , n − 1). F or d 1 = 1, by using (32) w e hav e C ( d ) ≤ f ( X − 1 , n − 1) ≤ 1 3 f ( X − 1 , n − 1) + 2 3 f ( X − 1 , n + 1) . (80) F rom (78), (79), (80), w e get the v alidit y of (74). The lemma is prov ed. □ Lemma 7. F or X ≥ 4 , n ≥ 2 , we have θ X − 3 ,n − 1 ≤ θ X ,n + M X − 1 (81) for some c onstant M > 0 . Pr o of. T o prov e (81), it suffices to pro v e C (4 , d 1 , . . . , d n ) ≥ C ( d 1 , . . . , d n ) − M X (4 , d 1 , . . . , d n ) − 1 (82) for some constant M > 0. According to e.g. [2, 7, 10], for a KdV tau-function τ = τ ( t ) the following iden tities hold: ϵ 2 ∂ 2 log τ ∂ t 1 ∂ t 1 = u 4 144 + uu 2 6 + u 2 1 24 + u 3 3 , (83) ϵ 2 ∂ 2 log τ ∂ t 1 ∂ t 4 = u 10 2903040 + u u 8 60480 + 13 u 2 u 6 40320 + 7 u 3 u 4 2160 + 17 u 2 4 96768 + u u 2 3 448 + 5 u 4 u 2 288 + 41 u 2 u 2 2 2880 + 79 u 3 2 40320 + u 3 u 2 1 36 + u 4 1 384 + u 7 u 1 17280 + 41 u 2 u 6 241920 + 23 u 3 u 5 80640 + 17 u u 1 u 5 10080 + 13 u u 2 u 4 3360 + 251 u 2 1 u 4 120960 + 1 60 u 2 u 1 u 3 + 1 36 u u 2 1 u 2 + 151 u 1 u 2 u 3 20160 + u 6 144 , (84) where u = ∂ 2 x (log τ ), u k = ∂ k t 0 ( u ). Similar to [24], using (83) one can obtain the follo wing recursion for C ( d ): C ( d ) = C (0 6 , d ) + 1 6 X I ⊔ J = { 1 ,...,n } ( X (0 3 , d I ) − 1)!( X (0 3 , d J ) − 1)! ( X ( d ) + 1)! C (0 3 , d I ) C (0 3 , d J ) + 2 3 X I ⊔ J = { 1 ,...,n } ( X (0 2 , d I ) − 1)!( X (0 4 , d J ) − 1)! ( X ( d ) + 1)! C (0 2 , d I ) C (0 4 , d J ) + 1 3 X I ⊔ J ⊔ K = { 1 ,...,n } ( X (0 2 , d I ) − 1)!( X (0 2 , d J ) − 1)!( X (0 2 , d K ) − 1)! ( X ( d ) + 1)! C (0 2 , d I ) C (0 2 , d J ) C (0 2 , d K ) , (85) whic h can imply C (0 6 , d ) ≥ C ( d ) − M 1 X ( d ) + 1 (86) for some constan t M 1 > 0. Similarly , the identit y (84) leads to the inequalit y C (4 , d ) ≥ C (0 12 , d ) . (87) By applying (86) and (87), we obtain (82). The lemma is prov ed. □ W e are ready to pro ve Theorem 1. 16 JINDONG GUO, DI Y ANG, DON ZAGIER Pr o of of The or em 1. F or n ≥ 1, d ∈ ( Z ≥ 1 ) n satisfying g ( d ) ∈ Z , b y Lemma 2 and (67) w e hav e C (3 g ( d ) − 2) ≤ C ( d ) ≤ θ X ( d ) , n . (88) On one hand, we kno w from (24) that C (3 g ( d ) − 2) = 1 π + O ( 1 g ( d ) ) with an absolute O -constan t. On the other hand, according to Lemma 6, θ X ( d ) ,n ≤ f ( X ( d ) , n ), then b y Lemma 5 w e know that when n ≤ X 5 , θ X ,n = 1 π + O ( 1 X ) with an absolute O -constant. Consider d ∈ ( Z ≥ 1 ) n suc h that X ( d ) 5 < n < 3 X ( d ) 5 . Without loss of generalit y , assume d 1 = min { d j } . W e then conclude that d 1 ∈ { 1 , . . . , 6 } . Using the recursion (29) and using an estimate similar to (78), (79), w e hav e C ( d ) ≤  1 − 10 3 X ( d ) − 3  θ X ( d ) − 1 , n − 1 + 10 3 X ( d ) − 3 max { θ X ( d ) − 1 , n − 1 , θ X ( d ) − 3 , n − 1 } + M 2 ( X ( d ) − 1)( X ( d ) − 2) (89) for some M 2 > 0. By applying Lemma 7, we obtain θ X , n ≤ θ X − 1 , n − 1 + M 3 ( X − 1)( X − 2) (90) for some constant M 3 > 0. Iterating (90) t = [ 5 n −X +1 4 ] times and using Lemma 5 and Lemma 7 we obtain that θ X ,n is b ounded by 1 π + O ( 1 X ) uniformly when X 5 < n ≤ 3 X 5 . This together with (88) implies that form ula (27) holds for X ( d ) 5 ≤ n ≤ 3 X ( d ) 5 . F or n > 3 X ( d ) 5 , w e hav e θ X ( d ) , n = θ X ( d ) − 1 , n − 1 (indeed, since C ( d ) is unc hanged by removing an y 1 argumen t, we ha ve by definition θ X , n = θ X − 1 , n − 1 for n > 3 X 5 ), which implies that formula (27) still holds. Combining all three cases, w e obtain the statemen t of Theorem 1. □ Before pro ceeding, let us give an application of Theorem 1. Consider the Painlev ´ e I e quation : d 2 U dX 2 + 1 16 U 2 − 1 16 X = 0 . (91) This equation has (cf. e.g. [17, 18]) a unique formal solution U ( X ) of the form U ( X ) = ∞ X g =0 c g X 1 − 5 g 2 , c 0 = − 1 , c g ∈ C , (92) where c g , g ≥ 0, are determined by the recursion c g = 50 ( g − 1) 2 c g − 1 + 1 2 g − 2 X h =2 c h c g − h , g ≥ 3 , (93) with the initial data c 0 = − 1, c 1 = 2, c 2 = 98. Here w e use normalizations as in [9]. F rom (93) one can deduce the follo wing asymptotics for c g : c g ∼ A · 50 g ( g − 1)! 2  1 − 49 3750 g 3 − 49 1250 g 4 + · · ·  , g → ∞ , (94) where A is some constant whose determination requires deep analysis, which w as ob- tained in [17, 18] and w e will giv e a new pro of. ON UNIFORM LARGE GENUS ASYMPTOTICS OF WITTEN’S INTERSECTION NUMBERS 17 Theorem A ( [17, 18] ). Ther e holds that A = 1 2 π 2 r 3 5 . Pr o of. According to [16] (cf. also e.g. [9]), we know that c g can b e expressed in terms of Witten’s in tersection num bers as follows: c g = 2 g 3 3 g − 2 5 3 g − 3 (5 g − 5)! (5 g − 3) (3 g − 3)! C (2 3 g − 3 ) , g ≥ 2 . (95) The theorem is then prov ed by using Theorem 1 and Stirling’s form ula. □ Note that Theorem A cannot b e deduced from Aggarw al’s result (16), as n = 3 g − 3 whic h is certainly b eyond o ( √ g ). W e no w pro ceed and prov e Theorem 2. Pr o of of The or em 2. F or Y ≥ 1 and for α ≥ 0, define θ ( α ) Y := sup n ≥ 1 max d ∈ ( Z ≥ 1 ) n X (0 α , d ) − p 1 ( d )= Y C (0 α , d ) . (96) F rom Theorem 1 and Lemma 4 we kno w that as Y → ∞ , θ ( α ) Y = 1 /π + O ( Y − 1 ), α = 0 , 1 , 2, with an absolute O -constan t. Consider α ≥ 3. F or n, Y ≥ 1, and for any d ∈ ( Z ≥ 1 ) n satisfying X (0 α , d ) = p 1 ( d ) + Y , write d = (1 p 1 ( d ) , d p 1 ( d )+1 , . . . , d n ). By using (31) we ha v e C (0 α , d ) = C (0 α , d p 1 ( d )+1 , . . . , d n ) = n X j = p 1 ( d )+1 2 d j + 1 3( Y − 1) C (0 α − 1 , d p 1 ( d )+1 , . . . , d j − 1 , . . . , d n ) ≤  1 + 3 − α 3( Y − 1)  max { θ ( α − 1) Y − 1 , θ ( α − 1) Y − 2 } . (97) By taking maxim um we get θ ( α ) Y ≤  1 + 3 − α 3( Y − 1)  max { θ ( α − 1) Y − 1 , θ ( α − 1) Y − 2 } ≤ · · · ≤ max j =0 ,...,α − 2 { θ (2) Y +2 − α − j } α − 2 Y j =1  1 + 2 + j − α 3( Y − j )  . (98) Giv en sufficien tly small ϵ > 0, when α ≤ ϵY w e know from θ (2) Y = 1 /π + O ( Y − 1 ) that the maxim um in the right-hand side of (98) equals 1 /π + O ( Y − 1 ) with an absolute O -constan t, and when α > ϵY , the maximum is still b ounded b y some constan t but the pro duct in the right-hand side of (98) has the estimate α − 2 Y j =1  1 + 2 + j − α 3( Y − j )  ≤ exp  α − 2 X j =1 2 + j − α 3( Y − j )  ≤ exp  − ( α − 2)( α − 3) 6 Y  , (99) 18 JINDONG GUO, DI Y ANG, DON ZAGIER and hence is O ( Y − 1 ) with an absolute O -constant. Therefore, we conclude that θ ( α ) Y = 1 π α − 2 Y j =1  1 + 2 + j − α 3( Y − j )  + O ( Y − 1 ) , (100) with an absolute O -constant (independent of α ). Let us now consider the low er bound. F or Y ≥ 1 and for α ≥ 0, define ϕ ( α ) Y := inf n ≥ 1 min d ∈ ( Z ≥ 1 ) n X (0 α , d ) − p 1 ( d )= Y C (0 α , d ) . (101) F rom Theorem 1, inequalities (57) and (59) w e find that as Y → ∞ , θ ( α ) Y = 1 /π + O ( Y − 1 ), α = 0 , 1 , 2, with an absolute O -constant. By using a similar estimate of (97) and by taking minimum w e get ϕ ( α ) Y ≥  1 + 3 − α 3( Y − 1)  min { θ ( ϕ − 1) Y − 1 , ϕ ( α − 1) Y − 2 } ≥ · · · ≥ min j =0 ,...,α − 2 { ϕ (2) Y +2 − α − j } α − 2 Y j =1  1 + 2 + j − α 3( Y + 1 − 2 j )  . (102) Using similar argumen ts that prov es (100) w e can obtain that ϕ ( α ) Y = 1 π α − 2 Y j =1  1 + 2 + j − α 3( Y + 1 − 2 j )  + O ( Y − 1 ) , (103) with an absolute O -constan t (indep enden t of α ). Noticing that the upp er b ound (100) and the low er b ound (103) differ b y O ( Y − 1 ) with an absolute O -constan t indep enden t of α and that X ( d ) − p 1 ( d ) ≥ 2 g ( d ) − 2, we finish the proof of Theorem 2. □ Pr o of of Cor ol lary 1. F or k = k ( g ) = O ( √ g ), we know from Theorem 2 and Stirling’s form ula that C  0 k , 2 3 g − 3+ k  = 1 π  2 3  k  15 g +3 k − 13 2  k (5 g − 5 + k ) k + O  1 g  = 1 π  15 g +5 k − 13 3  k e − k 2 15 g +3 k − 13 (5 g − 5 + 2 k ) k e − k 2 2(5 g − 5+ k ) (1 + o (1)) + O  1 g  = 1 π e − k 2 30 g (1 + o (1)) + O  1 g  . (104) F or k / √ g → ∞ , w e hav e k Y j =1  1 + 2 + j − k 3(5 g − 5 + 2 k − j )  ≤ exp  k (5 − k ) 6(5 g − 5 + k )  → 0 , g → ∞ . (105) By using Theorem 2 w e know that C (0 k , 2 3 g − 3+ k ) → 0 as g → ∞ . This pro v es Corol- lary 1. □ ON UNIFORM LARGE GENUS ASYMPTOTICS OF WITTEN’S INTERSECTION NUMBERS 19 4. Pol ynomiality in large genus In this section, we pro v e Theorem 3 b y using the recursion (29). Pr o of of The or em 3. Fix d ′ = ( d 1 , . . . , d n − 1 ) ∈ ( Z ≥ 0 ) n − 1 . W rite d = ( d ′ , d n ) with d n ≥ 0. According to [23] (cf. [15]) and Stirling’s form ula we know that C ( d ) has the asymptotic expansion C ( d ) = 1 π X k C k ( d ′ ) X ( d ) k , as d n → ∞ , (106) where C k are functions of d ′ . By using the recursion (29) and by p erforming Laurent expansions, we obtain C k ( d, d ′ ) − C k ( d ′ ) = − k − 1 X l =1 ( − 1) k − l  k − 1 l − 1  C l ( d, d ′ ) + 1 3 n − 1 X j =1 (2 d j + 1)  C k − 1 ( d 1 , . . . , d j + d − 1 , . . . , d n − 1 ) − C k − 1 ( d ′ )  + X a,b a + b = d − 2 " 2 3  C k − 1 ( a, b, d ′ ) − C k − 1 ( d ′ )  + 1 3 X I ⊔ J = { 1 ,...,n − 1 } k − 2 X l =0 a ( a, d ′ I ) ,l,k C l ( b, d ′ J ) # , (107) where d ≥ 0 and a w ,l,k are certain n um bers defined via the follo wing generating function ( X ( w ) − 1)! C ( w ) ( X − X ( w ) − 1)! X ! ( X − X ( w )) l =: X k a w ,l,k X k . (108) W rite C k ( d ′ ) =: ˜ c k ( p 0 ( d ′ ) , p 1 ( d ′ ) , . . . ) , (109) where p r ( d ′ ) denotes the multiplicit y of r in d ′ . This defines functions ˜ c k ( p ), k ≥ 0, where p = ( p 0 , p 1 , p 2 , . . . ) is a infinite integer sequence satisfying P i p i < ∞ . Then form ula (107) b ecomes ˜ c k ( p + e d ) − ˜ c k ( p ) = − k − 1 X l =1 ( − 1) k − l  k − 1 l − 1  ˜ c l ( p + e d ) + 1 3 X i ≥ 0 (2 i + 1) p i  ˜ c k − 1 ( p − e i + e i + d − 1 ) − ˜ c k − 1 ( p )  + X a,b ≥ 0 a + b = d − 2 " 2 3  ˜ c k − 1 ( p + e a + e b ) − ˜ c k − 1 ( p )  + 1 3 X E ( t + e a ) ≤ k − 1 0 ≤ t r ≤ p r , r ≥ 0 k − 2 X l =0  ˜ c l ( p − t + e b ) α t + e a ,l,k Y i ≥ 0  p i t i  # , d ≥ 0 , (110) where E ( t ) := X (0 t 0 , 1 t 1 , . . . ), α t ,l,k = a (0 t 0 1 t 1 2 t 2 ··· ) ,l ,k , and e d denotes (0 , . . . , 0 , 1 , 0 , 0 , . . . ) with “1” appearing in the ( d + 1)th place. 20 JINDONG GUO, DI Y ANG, DON ZAGIER Let us now pro ve b y induction that ˜ c k ( p ), k ≥ 0, b elong to Q [ p 0 , p 1 , . . . ] and satisfy the degree estimates deg ˜ c k ( p ) ≤ 3 k − 1 , k ≥ 1 , (111) under the degree assignmen ts deg p d = 2 d + 1, d ≥ 0. F or k = 0, by using Theorem 1 w e kno w that ˜ c 0 ( p ) ≡ 1. Assume that for 1 ≤ l ≤ k − 1, ˜ c l ( p ) ∈ Q [ p 0 , p 1 , . . . ] are p olynomials satisfying deg ˜ c l ( p ) ≤ 3 l − 1. Then for k and for every d ≥ 0, the RHS of equation (110) are p olynomials in p 0 , . . . , p [(3 k − 5) / 2] . Moreo v er, b y the inductiv e assumption these polynomials are indep endent of d for every d ≥ 3 k − 1, i.e., ˜ c k ( p + e d ) − ˜ c k ( p ) = ( f d ( p 0 , . . . , p [(3 k − 5) / 2] ) , d ≤ 3 k − 2 , g ( p 0 , . . . , p [(3 k − 5) / 2] ) , d ≥ 3 k − 1 (112) for some f d ( d ≤ k ) and g in Q [ p 0 , . . . , p [(3 k − 5) / 2] ]. The compatibility of (112) implies that g ( p 0 , . . . , p [( k − 3) / 2] ) ≡ A is a constant. Solving (112) w e obtain that ˜ c k ha v e the form ˜ c k ( p ) = h ( p 0 , . . . , p 3 k − 2 ) + A n ′ ( p ) , (113) where h ∈ Q [ p 0 , . . . , p 3 k − 2 ] and n ′ ( p ) := P i ≥ 0 p i . W e aim to sho w that A = 0. Consider equation (110) with k replaced b y k + 1. Using a similar analysis and using (113), w e obtain that for every d ≥ 6 k − 1, ˜ c k +1 ( p + e d ) − ˜ c k +1 ( p ) = 4 3 A d + A ′ n ′ ( p ) + s ( p 0 , . . . , p 3 k − 2 ) , (114) where A ′ ∈ Q is a constant, and s ( p 0 , . . . , p 3 k − 2 ) is a polynomial in Q [ p 0 , . . . , p 3 k − 2 ] indep enden t of d . Now taking p = 0 , we find that equation (114) con tradicts with (44) unless A = 0. W e conclude from (113) that ˜ c k ( p ) ∈ Q [ p 0 , . . . , p 3 k − 2 ] . (115) No w taking d ≥ 3 k − 1 in equation (110) gives 0 = − k − 1 X l =1 ( − 1) k − l  k − 1 l − 1  ˜ c l ( p ) + 1 3 X i ≥ 0 (2 i + 1) p i  ˜ c k − 1 ( p − e i ) − ˜ c k − 1 ( p )  + [(3 k − 5) / 2] X a =0 4 3  ˜ c k − 1 ( p + e a ) − ˜ c k − 1 ( p )  + 1 3 X E ( t + e a ) ≤ k − 1 0 ≤ t r ≤ p r , r ≥ 0 k − 2 X l =0  ˜ c l ( p − t ) α t + e a ,l,k Y i ≥ 0  p i t i  ! . (116) ON UNIFORM LARGE GENUS ASYMPTOTICS OF WITTEN’S INTERSECTION NUMBERS 21 Using (116) and (110), w e obtain ∆ p d ˜ c k ( p ) = − k − 1 X l =1 ( − 1) k − l  k − 1 l − 1  ∆ p d ˜ c l ( p ) + 1 3 X i ≥ 0 (2 i + 1) p i ∆ p i + d ˜ c k − 1 ( p − e i ) + 2 3 d − 2 X a =0 ∆ p a ∆ p d − 2 − a ˜ c k − 1 ( p ) − 4 3 [(3 k − 5) / 2] X a = d − 1 ∆ p a ˜ c k − 1 ( p ) + 1 3 d − 2 X a =0 X E ( t + e a ) ≤ k − 1 0 ≤ t r ≤ p r ( r ≥ 0) k − 2 X l =0  ∆ p d − 2 − a ˜ c l ( p − t ) α t + e a ,l,k Y i ≥ 0  p i t i  + 1 3 [(3 k − 5) / 2] X a = d − 1 X E ( t + e a ) ≤ k − 1 0 ≤ t r ≤ p r ( r ≥ 0) k − 2 X l =0  ˜ c l ( p − t ) α t + e a ,l,k Y i ≥ 0  p i t i  , d ≥ 0 . (117) W e find by inductiv e assumption that each term of the right-hand side of (117) is of degree less than or equal to 3 k − 2 − 2 d for every d ≥ 0, whic h implies deg ˜ c k ≤ 3 k − 1. In particular, ˜ c k is a polynomial that only depends on p 0 , . . . , p [3 k/ 2] − 1 . No w restrict to the case when d ′ ∈ ( Z ≥ 2 ) n − 1 . F rom (106), (42) and Stirling’s form ula w e know that b C ( d ) ∼ X k b C k ( d ′ ) X ( d ) k , d n → ∞ , (118) where b C k are functions of d ′ with b C 0 ≡ 1. Define b c k ( p 2 , p 3 , . . . ) ∈ Q [ p 2 , p 3 , . . . ], k ≥ 0, via γ ( X ) X k b c k ( p 2 , p 3 , . . . ) X k = X k ˜ c k (0 , 0 , p 2 , p 3 , . . . ) X k , (119) where the left-hand side is understo o d as a pow er series in X − 1 . It then follows from deg ˜ c k ≤ 3 k − 1 that deg b c k ≤ 3 k − 1. Using (118), (106), (109), w e know that b C k ( d ′ ) = b c k ( p 1 ( d ′ ) , p 2 ( d ′ ) , . . . ) (120) for all d ′ . The statement that b c k (0 , 0 , . . . ) = 0 follows from the fact that C (3 g − 2) = γ (6 g − 3) for ev ery g ≥ 1. This finishes the pro of of Theorem 3. □ References [1] A. Aggarwal, Large genus asymptotics for intersection num b ers and principal strata volumes of quadratic differen tials. Inv en t. Math., 226 (2021), 897–1010. [2] M. Bertola, B. 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Surveys Diff. Geom., 1 (1991), 243–310. [29] J. Zhou, Emergent geometry and mirror symmetry of a p oint. [30] P . Zograf, An explicit formula for Witten’s 2-correlators. J. Math. Sci., 240 (2019), 535–538. Jindong Guo, School of Ma thema tical Sciences, University of Science and Technology of China, 230026 Hefei, P.R. China Email address : guojindong@mail.ustc.edu.cn Di Y ang, School of Ma thema tical Sciences, University of Science and Technology of China, 230026 Hefei, P.R. China Email address : diyang@ustc.edu.cn ON UNIFORM LARGE GENUS ASYMPTOTICS OF WITTEN’S INTERSECTION NUMBERS 23 Don Zagier, Max Planck Institute for Ma thema tics, 53111 Bonn, Germany, and Inter- na tional Centre f or Theoretical Physics, Trieste, It al y Email address : dbz@mpim-bonn.mpg.de