Marc Raimondo - CMI

Rescaling and convergence to Airy function
Centred and scaled (weighted) polynomials
φτ(s) = (2N) 1/4 φN(uN + τNs),
ψτ(s) = (2N) 1/4 φN−1(uN−1 + τNs)
both satisfy differential equation
w ′′ N(s) = s(1 + O(N −1 ))wN(s).
Liouville-Green theory shows convergence to Airy A(s), solving
with O(N −2/3 ) rate:
A ′′ (s) = sA(s),
|φτ(s) − A(s)| ≤ CN −2/3 e −s/2 .
Luminy 12/08 – p.38