This paper analyses the periodic spectrum of Schr\"odinger's equation
−f′′+qf=λf when the potential is real, periodic, random and subject to
the invariant measure νNβ​ of the periodic KdV equation. This
νNβ​ is the modified canonical ensemble, as given by Bourgain ({Comm.
Math. Phys.} {166} (1994), 1--26), and νNβ​ satisfies a logarithmic
Sobolev inequality. Associated concentration inequalities control the
fluctuations of the periodic eigenvalues (λn​). For β,N>0 small,
there exists a set of positive νNβ​ measure such that (±2(λ2n​+λ2n−1​)​)n=0∞​ gives a sampling sequence
for Paley--Wiener space PW(Ï€) and the reproducing kernels give a Riesz
basis. Let (μj​)j=1∞​ be the tied spectrum; then
(2μj​​−j) belongs to a Hilbert cube in ℓ2 and is distributed
according to a measure that satisfies Gaussian concentration for Lipschitz
functions. The sampling sequence (μj​​)j=1∞​ arises from a
divisor on the spectral curve, which is hyperelliptic of infinite genus. The
linear statistics ∑j​g(λ2j​​) with test function g∈PW(π) satisfy Gaussian concentration inequalities.Comment: 34 page