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

TL;DR

This study investigates the properties of minimum-energy paths on disordered hierarchical lattices with mixed energy signs, revealing a transition in scaling behavior and the tail distribution of ground-state energies, including Tracy-Widom distribution characteristics.

Contribution

The paper introduces a detailed numerical analysis of energy path scaling and tail distributions in disordered lattices with mixed energies, highlighting a disorder-induced transition and tail behavior in the ground-state energy distribution.

Findings

Scaling transition from self-affine to self-similar paths.

Negative-energy tail follows Tracy-Widom distribution.

Ground-state energy fluctuations relate to tail scaling.

Abstract

We perform numerical simulations to study the optimal path problem on disordered hierarchical graphs with effective dimension d=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 vincinity 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.