Dynamical localization of trajectories on a bond-disordered lattice

The chaotic dynamics of a random walker in a quenched environment is studied via the thermodynamic formalism of Ruelle, Sinai, and Bowen, in which chaotic properties are expressed in terms of a free energy-type function, ψ(β), of an inverse temperature-like parameter, β. Localization phenomena in this system are elucidated both analytically and numerically. The infinite system limit of the Ruelle pressure at β>1 and β<1 is shown to be controlled by rare configurations of the bond disorder, and this is related, respectively, to the extreme configurations associated to the minimum and maximum Lyapunov exponent in finite systems. These extreme values and the corresponding configurations are obtained numerically from Monte Carlo simulations based on the thermodynamic formalism.