Marc Raimondo - CMI

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Orthogonal Case P { max 1≤j≤N+1 xj ≤ x0} = � det(I − KN+1χ0), ⎛ ⎞ ⎛ ⎞ I KN+1(x,y) = ⎝ ǫ1 −∂2 0 0 ⎠SN+1,1(x,y) − ⎝ ⎠ T ǫ(x − y) 0 ∂2 ↔ ∂/∂y ǫ1 ↔ convolution in x with ǫ(x) = 1 2 sgn(x) T ↔ transpose x,y Key identity relates orthogonal at N + 1 to unitary at N : SN+1,1(x,y) = SN,2(x,y) + 1 2 φ(x)ǫψ(y). Luminy 12/08 – p.40
Orthogonal Case - Convergence P { max 1≤j≤N+1 xj ≤ x0} = � det(I − KN+1χ0), F1(s) = � ⎛ det(I − KGOE) ⎞ I Kτ(s,t) = ⎝ ǫ1 −∂2 ⎠(SN,2,τ(s,t) + T 1 2φτ(s)ǫψτ(t)) ⎛ ⎞ ⎛ ⎞ 0 0 − ⎝ ⎠ ǫ(x − y) 0 I KGOE(s,t) = ⎝ ǫ1 −∂2 ⎠(SA(s,t) + T 1 ⎛ ⎞ 2A(s)ǫA(t)) 0 0 − ⎝ ⎠. ǫ(x − y) 0 where O(N −1/3 ) terms cancel: Sτ(s,t) = SA(s,t) − ∆NA(s)A(t) + O(N −2/3 ), 1 2 φτ(s)ǫψτ(t) = 1 2 A(s)(ǫA)(t) + ∆NA(s)A(t) + O(N −2/3 ). Luminy 12/08 – p.41
Page 1 and 2: Marc Raimondo Luminy 12/08 - p.1
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Page 39: After rescaling: Sτ(s,t) = 1 2 Uni
Page 43 and 44: Tail p−values: s = 2 variables F

Marc Raimondo - CMI

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