Spectral correlations in systems undergoing a transition from periodicity to disorder

We study the spectral statistics for extended yet finite quasi-one-dimensional systems, which undergo a transition from periodicity to disorder. In particular, we compute the spectral two-point form factor, and the resulting expression depends on the degree of disorder. It interpolates smoothly between the two extreme limits-the approach to Poissonian statistics in the (weakly) disordered case, and the universal expressions derived in T. Dittrich, B. Mehlig, H. Schanz, and U. Smilansky, Chaos Solitons Fractals 8, 1205 (1997); Phys. Rev. E 57, 359 (1998); B. D. Simons and B. L. Altshuler, Phys. Rev. Lett. 70, 4063 (1993); and N. Taniguchi and B. L. Altshuler, ibid. 71, 4031 (1993) for the periodic case. The theoretical results agree very well with the spectral statistics obtained numerically for chains of chaotic billiards and graphs.