Discrete Scale Invariance and Other Cooperative Phenomena in Spatially Extended Systems With Threshold Dynamics

Contents 1 Introduction 5 1.1 Imitation and Synchronisation . . . . . . . . . . . . . . . . . . . . 6 1.2 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Hierarchical systems . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 1=f-noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Scale invariance and discrete scale invariance . . . . . . . . . . . . 11 1.6 Exactly solvable models with discrete scale invariance . . . . . . 14 1.6.1 Potts model on a hierarchical lattice . . . . . . . . . . . . 14 1.6.2 Diffusion on a random lattice . . . . . . . . . . . . . . . . 19 1.6.3 An experimental hierarchical model . . . . . . . . . . . . . 21 1.7 Possible log-periodic signatures in cracking 1 . . . . . . . . . . . . 22 2 Hierarchical structures in growth models 27 2.1 Laplacian growth models . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.1 Numerical results . .