RESTRICTED WALKS, STABILITY INSTABILITY TRANSITIONS, AND DYNAMIC SYMMETRIES

The principal results from applying a transition matrix approach to the problem of self-avoiding walks which Domb and Hioe obtained some years ago are recalled. Some results are then presented on two other different physical problems: one on a universal critical exponent for a class of stability-instability transitions in the classical Hamiltonian systems, and the other on the existence of a characteristic set of constants of evolution when a quantum system possesses a certain type of symmetry. The similarity in some of the key concepts and methods used in these three problems, which involve studies of the distribution of the appropriate eigenvalues and the utilization of the existing symmetry, and the dissimilarity in some of the details are noted. © 1990 Plenum Publishing Corporation.