Eur. Phys. J. B (2013) 86: 323
https://doi.org/10.1140/epjb/e2013-40230-1

Regular Article

Typical and large-deviation properties of minimum-energy paths on disordered hierarchical lattices

Institut für Physik, Universität Oldenburg, Carl-von-Ossietzky Strasse, 26111 Oldenburg, Germany

^a e-mail: oliver.melchert@uni-oldenburg.de
^b e-mail: alexander.hartmann@uni-oldenburg.de

Received: 20 March 2013
Received in final form: 3 June 2013
Published online: 15 July 2013

Abstract

We perform numerical simulations to study the optimal path problem on disordered hierarchical graphs with effective dimension d_eff ≈ 2.32. Therein, edge energies are drawn from a disorder distribution that allows for positive and negative energies. This induces a behavior which is fundamentally different from the case where all energies are positive, only. Upon changing the subtleties of the distribution, the scaling of the minimum energy path length exhibits a transition from self-affine to self-similar. We analyze the precise scaling of the path length and the associated ground-state energy fluctuations in the vicinity of the disorder critical point, using a decimation procedure for huge graphs. Further, using an importance sampling procedure in the disorder we compute the negative-energy tails of the ground-state energy distribution up to 12 standard deviations away from its mean. We find that the asymptotic behavior of the negative-energy tail is in agreement with a Tracy-Widom distribution. Further, the characteristic scaling of the tail can be related to the ground-state energy flucutations, similar as for the directed polymer in a random medium.