Stretched Exponential Relaxation and Independent Relaxation Modes

We discuss the origin of stretched exponential relaxation in disordered Ising spin systems by writing the master equation on the phase space, and the evolution of local and global spin autocorrelation functions, in terms of independent relaxation modes, which are eigenvectors of the time evolution operator. In this sense it is shown that when the relaxation modes are spatially delocalized, both local and global autocorrelation functions may present non-exponential relaxation. We also analyze results for random walks on the dilute hypercube, which may be associated with the phase space of a disordered Ising spin system. As expected, the results show a stretched exponential relaxation near the percolation transition, since it deals with random walks on a fractal percolating cluster de ned on a closed surface. We argue that the same type of topology is present in the available region of con guration space in Ising spin-glass systems near the glass transition, since these systems present very similar relaxation patterns in this temperature range.