Equilibrium of disordered systems: constructing the appropriate valleys in each sample via strong-disorder renormalization in configuration space

Abstract

To describe the equilibrium properties of disordered systems and the possible emergence of various ‘phases’ at low temperature, we adopt here the ‘broken-ergodicity’ point of view advocated in particular by Palmer (1982 Adv. Phys. 31 669): the aim is then to construct the valleys of configurations that become separated by diverging barriers and to study their relative weights, as well as their internal properties. To characterize the slow non-equilibrium dynamics of disordered systems, we have recently introduced (Monthus and Garel 2008 J. Phys. A 41 255002, Preprint arXiv:0804.1847)) a strong-disorder renormalization (RG) procedure in configuration space, based on the iterative elimination of the smallest barrier remaining in the system. In the present paper, we show how this renormalization procedure allows us to construct the longest-lived valleys in each disordered sample, and to obtain their free-energies, energies and entropies. This explicit RG formulation is very general since it can be defined for any master equation, and it gives new insights into the main ingredients of the droplet scaling picture. As an application, we have followed numerically the RG flow for the case of a directed polymer in a two-dimensional random medium to obtain histograms of the free-energy, entropy and energy differences between the two longest-lived valleys in each sample.