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

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.Comment: 10 pages, 10 figures, 3 table

Configurational statistics of densely and fully packed loops in the negative-weight percolation model

By means of numerical simulations we investigate the configurational properties of densely and fully packed configurations of loops in the negative-weight percolation (NWP) model. In the presented study we consider 2d square, 2d honeycomb, 3d simple cubic and 4d hypercubic lattice graphs, where edge weights are drawn from a Gaussian distribution. For a given realization of the disorder we then compute a configuration of loops, such that the configurational energy, given by the sum of all individual loop weights, is minimized. For this purpose, we employ a mapping of the NWP model to the "minimum-weight perfect matching problem" that can be solved exactly by using sophisticated polynomial-time matching algorithms. We characterize the loops via observables similar to those used in percolation studies and perform finite-size scaling analyses, up to side length L=256 in 2d, L=48 in 3d and L=20 in 4d (for which we study only some observables), in order to estimate geometric exponents that characterize the configurations of densely and fully packed loops. One major result is that the loops behave like uncorrelated random walks from dimension d=3 on, in contrast to the previously studied behavior at the percolation threshold, where random-walk behavior is obtained for d>=6.Comment: 11 pages, 7 figure

A dedicated algorithm for calculating ground states for the triangular random bond Ising model

In the presented article we present an algorithm for the computation of ground state spin configurations for the 2d random bond Ising model on planar triangular lattice graphs. Therefore, it is explained how the respective ground state problem can be mapped to an auxiliary minimum-weight perfect matching problem, solvable in polynomial time. Consequently, the ground state properties as well as minimum-energy domain wall (MEDW) excitations for very large 2d systems, e.g. lattice graphs with up to N=384x384 spins, can be analyzed very fast. Here, we investigate the critical behavior of the corresponding T=0 ferromagnet to spin-glass transition, signaled by a breakdown of the magnetization, using finite-size scaling analyses of the magnetization and MEDW excitation energy and we contrast our numerical results with previous simulations and presumably exact results.Comment: 5 pages, 5 figure