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This paper analyses the periodic spectrum of Schr\"odinger's equation$-f''+qf=\lambda f$ when the potential is real, periodic, random and subject tothe invariant measure $\nu_N^\beta$ of the periodic KdV equation. This$\nu_N^\beta$ is the modified canonical ensemble, as given by Bourgain ({Comm.Math. Phys.} {166} (1994), 1--26), and $\nu_N^\beta$ satisfies a logarithmicSobolev inequality. Associated concentration inequalities control thefluctuations of the periodic eigenvalues $(\lambda_n)$. For $\beta, N>0$ small,there exists a set of positive $\nu_N^\beta$ measure such that $(\pm\sqrt{2(\lambda_{2n}+\lambda_{2n-1})})_{n=0}^\infty$ gives a sampling sequencefor Paley--Wiener space $PW(\pi )$ and the reproducing kernels give a Rieszbasis. Let $(\mu_j)_{j=1}^\infty$ be the tied spectrum; then$(2\sqrt{\mu_j}-j)$ belongs to a Hilbert cube in $\ell^2$ and is distributedaccording to a measure that satisfies Gaussian concentration for Lipschitzfunctions. The sampling sequence $(\sqrt{\mu_j})_{j=1}^\infty$ arises from adivisor on the spectral curve, which is hyperelliptic of infinite genus. Thelinear statistics $\sum_j g(\sqrt{\lambda_{2j}})$ with test function $g\inPW(\pi)$ satisfy Gaussian concentration inequalities.

Hill's Spectral Curves and the Invariant Measure of the Periodic KdV Equation