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

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.

10 pages, 10 figures, 3 tables

References in corpus (12)

• Fractal dimension of domain walls in two-dimensional Ising spin glasses
• Probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension $d=1,2,3$ via a Monte-Carlo procedure in the disorder
• Probing tails of energy distributions using importance-sampling in the disorder with a guiding function
• Critical points of quadratic renormalizations of random variables and phase transitions of disordered polymer models on diamond lattices
• Domain walls and chaos in the disordered SOS model
• Negative-weight percolation
• Scaling behavior of domain walls at the T=0 ferromagnet to spin-glass transition
• Disordered Potts model on the diamond hierarchical lattice: Numerically exact treatment in the large-q limit
• Phase transitions in diluted negative-weight percolation models
• Structural crossover of polymers in disordered media
• Ising Spin Glasses on Wheatstone-Bridge Hierarchical Lattices
• Analysis of the loop length distribution for the negative weight percolation problem in dimensions d=2 through 6

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