Communications In Mathematical Physics - Volume 286 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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Comm. Pure & Appl. Math. 50, 241–290 (1997) 7. : Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems. Commun. Math. Phys. 207, 589–620 (1999) 8. : Integrals over classical groups, random permutations Toda and Toeplitz lattices. Comm. Pure Appl. Math. 54(2), 153–205 (2001) 9. : Recursion relations for Unitary integrals of Combinatorial nature and the Toeplitz lattice. Commun. Math. Phys. 237, 397–440 (2003) 10. : PDEs for the joint distributions of the Dyson, Airy and sine processes.

P 1 Q c ≤ K c p−1 − 21 . Similarly, 1 p−1 α L∞ ≤ K c . 45). For all (k, ) ∈ p, c Q kc (yc )Ak, (y) since Ak, ∈ Y and Bk, ∈ L ∞ , we have L2 c (Q kc ) (yc )Bk, (y) ≤ K c Q kc L2 1 L∞ ≤ K c (Q kc ) ≤ K c p−1 , 1 L2 ≤ K c p−1 + 14 . 48). 46). Since Rc (t) = Q c (x + (1 − c)t), we only have to prove that, for all t ∈ [−Tc , Tc ], inf R(t) − Q(. − y) y∈R 1 H1 ≤ K c p−1 . 48), taking c small enough so that |α (t)| < 21 , for all t ∈ [−Tc , Tc ], there exists a unique y(t) such that y(t) − α(y(t) + (1 − c)t) = 0.

X (x ( ) ) =1 e ( )2 − 21 x j +b ( ) x j +β ⎞ ( )2 xj − ∞ ( ) ( )i i=1 si x j ( ) dx j ⎠ j=1 ⎞ (µi(1) j )0≤i≤n 1 −1, 0≤ j≤N −1 ⎟ ⎜. ⎟, = det ⎜ ⎠ ⎝ .. ( p) (µi j )0≤i≤n p −1, 0≤ j≤N −1 ⎛ p 1 ci bi =0 p 1 ci βi =0 (67) x2 where F1 (x) := e− 2 in the inner-product below and where one only indicates the dependency on the auxiliary variables, ( ) µi j (t − s ( ) , β ) = E˜ x i+ j e− = x i e− for x2 2 +b ∞ ( ) k 1 sk x f |g 1 x+β x 2 eb = e x+β x 2 E˜ ( ) k ∞ 1 (tk −sk )x x je dx ∞ i 1 tk x 1 , f (x)g(x)F1 (x)d x.

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