Spectral correlations in systems undergoing a transition from periodicity to disorder

We study the spectral statistics for extended yet finite quasi-one-dimensio nal systems, which undergo a transition from periodicity to disorder. In pa rticular, we compute the spectral two-point form factor, and the resulting expression depends on the degree of disorder. It interpolates smoothly betw een the two extreme limits-the approach to Poissonian statistics in the (we akly) 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. I,. Altshuler, ibid. 7 1, 4031 (1993) for the periodic case. The theoretical results agree very we ll with the spectral statistics obtained numerically for chains of chaotic billiards and graphs. [S1063-651X(99)11005-5].