KP Hierarchy and the Tracy-Widom Law

Transcription

1 KP Hierarchy and the Tracy-Widom Law Yi Sun MIT May 7, 2012 Yi Sun (MIT) KP Hierarchy and the Tracy-Widom Law May 7, / 14

2 Introduction Theorem (Tracy-Widom Law) As n, the probabiity that a eigenvaues of an n n GUE matrix are at most u is given by ( ) det(i K [u, ) Airy ) = exp (α u)g 2 (α)dα, where g satisfies the Painevé II equation u g = xg + 2g 3 and has asymptotics g(x) exp( 2 3 x3/2 ) 2 πx 1/4 as x. Yi Sun (MIT) KP Hierarchy and the Tracy-Widom Law May 7, / 14

3 Gap Probabiities for GUE Let {φ n } n 0 be orthonorma functions given by φ n (x) = ψ n (x)e 1 2 x2, where {ψ n } n 0 are the Hermite poynomias. Eigenvaues of a N N GUE matrix have correation functions for the kerne Gap probabiities are ρ(x 1,..., x k ) = det(k N (x i, x j )) k i,j=1 K N (x, y) = where K E N (x, y) = K N(x, y)i E. N 1 i=0 φ i (x)φ i (y). P(no x i in E) = det(i K E N ), Yi Sun (MIT) KP Hierarchy and the Tracy-Widom Law May 7, / 14

4 Airy Kerne Theorem Taking convergence in the sup-norm, we have im n 1 2n 1/6 K n( 2n + x 2n 1/6, 2n + y 2n 1/6 ) = K Airy(x, y), where the Airy kerne is given by K Airy (x, y) = and the Airy function is given by 0 Ai(x + t) Ai(y + t)dt = Ai(x) Ai (y) Ai (x) Ai(y) x y Ai(x) = 1 π 0 ( ) u 3 cos 3 + xu du and satisfies the differentia equation Ai (x) x Ai(x) = 0. Yi Sun (MIT) KP Hierarchy and the Tracy-Widom Law May 7, / 14

5 Background on KP Consider a pseudo-differentia operator Want two conditions: L(t) = x + u 1 (x, t) 1 x + u 2 (x, t) 2 x + Continuous spectrum with wave function Ψ(x, t, z): L(t) Ψ(x, t, z) = zψ(x, t, z). Evoution of the wave function Ψ(x, t, z): i Ψ(x, t, z) = L i +Ψ(x, t, z), where L i + is the purey differentia part of L i and i = t i. KP hierarchy = conditions for this to be consistent Yi Sun (MIT) KP Hierarchy and the Tracy-Widom Law May 7, / 14

6 Background on KP (II) Sato Grassmannian Ω = Subspaces W C[[z, z 1 ]] with π : W C[[z]]. Points in Ω correspond to τ functions: ( τ(t) = det π : e ) j 1 t j z j W C[[z]], Theorem Soutions {u i } to KP with wave function Ψ(t, z) correspond to τ functions: where [z 1 ] = Ψ(t, z) = τ(t [z 1 ]) e i 1 t i z i, τ(t) ( z 1, z 2 2, z 3 3 ).,... Can characterize a τ(t) by a biinear reation. Yi Sun (MIT) KP Hierarchy and the Tracy-Widom Law May 7, / 14

7 KP and the Airy Kerne Exampe of KP: Take L(t) so that L 2 is purey differentia and L(0) 2 = x 2 x. Match wave function and Airy function: Ψ(x, 0, z) = 2 πz Ai(x + z 2 ) = e xz+ 2 z3( ) o(1). L(0) 2 Ψ(x, 0, z) = (x + z 2 )Ψ(x, 0, z) xψ(x, 0, z) = z 2 Ψ(x, 0, z) Extend Ψ(x, 0, z) to a times: Consider asymptotics: Ψ(x, 0, z) = e xz+ 2 3 z3 (1 + o(1)), Define subspace in Ω: { W = span i 0 i 1 Ψ(x, 0, z) }. Yi Sun (MIT) KP Hierarchy and the Tracy-Widom Law May 7, / 14

8 KP and the Airy kerne (II) Reca W = span i 0 { i 1 Ψ(x, 0, z) } Ω: Take τ function associated to W (Kontsevich integra): τ Airy (t) = im N ( exp Tr( 1 3 X 3 + X 2 Z) ) dx exp ( Tr(X 2, Z)) dx where X is drawn from N N GUE and Z = diag(z n ) with t n = 1 n i z n i δ n,3. By Theorem, get Ψ(x, t, z) corresponding to τ Airy (t) Check (abstracty) that Ψ(x, 0, z) = 2 πz Ai(x + z 2 ). Yi Sun (MIT) KP Hierarchy and the Tracy-Widom Law May 7, / 14

9 Vertex operator and Airy kerne KP vertex operator: X (t, y, z) := 1 z y exp i 1(z i y i )t i exp i 1 y i z i i i, For kernes of the form K E (t, y, z) = can write E K E (t, y, z) = Ψ(x, t, y)ψ (x, t, z)dx, X (t, y, z)τ(t). τ(t) Yi Sun (MIT) KP Hierarchy and the Tracy-Widom Law May 7, / 14

10 Fredhom determinants Theorem The Fredhom determinant of K E is given by: det(i λk E ) = 1 ( ) τ(t) exp λ X (t, z, z)dz τ(t). Proof idea: Consider discrete anaogue and take imit. X (t, y, z) 2 = 0, so exp(ax (t, y, z)) = 1 + ax (t, y, z), giving ( ) 1 τ(t) exp a i X (t, z i, z i ) τ(t) = 1 (1+a i X (t, z i, z i ))τ(t). τ(t) i Expand and use identity on product of vertex operators to get det(i + a j Ψ(x, t, z i )Ψ (x, t, z i )dx). E i Yi Sun (MIT) KP Hierarchy and the Tracy-Widom Law May 7, / 14

11 Virasoro constraints Consider the expansion: X (t, y, z) = 1 z y (z y) k k=0 k! = y k W (k). W (1) W (2) = reaization of Heisenberg agebra = reaization of Virasoro agebra Commutation reations among X (t, y, z) and X (t, y, z ) give: [ ] 1 ( ) 2 W (2), X (t, z, z) = z z +1 X (t, z, z). Integration by parts: [ 1 b ] 2 W (2), X (t, z, z)dz = b +1 X (t, b, b) a +1 X (t, a, a). a Yi Sun (MIT) KP Hierarchy and the Tracy-Widom Law May 7, / 14

12 Virasoro constraints (II) Reca: [ 1 b ] 2 W (2), X (t, z, z)dz = b +1 X (t, b, b) a +1 X (t, a, a) a and det(i λk [a,b] ) = 1 ( b ) τ(t) exp λ X (t, z, z)dz τ(t). a If had W (2) τ(t) = c τ(t) (obtained from biinear reations on τ), then combining gives ( b +1 b + a +1 a 1 2 W (2) + 1 ) ( ) 2 c exp λ X (t, z, z)dz τ(t) = 0, E so we see that ( b +1 b + a +1 a 1 2 W (2) + 1 ) 2 c τ(t) ker(i λk [a,b] ) = 0. Yi Sun (MIT) KP Hierarchy and the Tracy-Widom Law May 7, / 14

13 Virasoro constraints (III) ( ) Reca b +1 b + a +1 a 1 2 W (2) c τ(t) ker(i λk [a,b] ) = 0. Genera KP theory: ( ) τ(t) := τ(t) ker(i λk [a,b] ) = exp λ X (t, z, z)dz τ(t) E is a τ function. Use biinear reations on τ(t) in terms of t i to get reations on τ(t) in terms of a and b! Obtain constraints of form P(a, b, a, b ) og(τ(t) ker(i λk [a,b] )) = 0. Can remove τ(t) because differentia is independent of t. Yi Sun (MIT) KP Hierarchy and the Tracy-Widom Law May 7, / 14

14 References Mark Ader, T. Shiota, and P. van Moerbeke, Random matrices, vertex operators, and the Virasoro agebra, Physics Letters A 208 (1995), , Random matrices, Virasoro agebras, and noncommutative KP, Duke Mathematica Journa 94 (1998), no. 2, Mark Ader and P. van Moerbeke, Matrix integras, Toda symmetries, Virasoro constraints, and orthogona poynomias, Duke Mathematica Journa 80 (1995), no. 3, Maxim Kontsevich, Intersection theory on the modui space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), no. 1, (93e:32027) Yi Sun (MIT) KP Hierarchy and the Tracy-Widom Law May 7, / 14