Fixed-point distributions of short-range Ising spin glasses on hierarchical lattices

Fixed-point distributions for the couplings of Ising spin glasses with nearest-neighbor interactions on hierarchical lattices are investigated numerically. Hierarchical lattices within the Migdal-Kadanoff family with fractal dimensions in the range 2.58≤D≤7, as well as a lattice of the Wheatstone-Bridge family with fractal dimension D≈3.58 are considered. Three initial distributions for the couplings are analyzed, namely, the Gaussian, bimodal, and uniform ones. In all cases, after a few iterations of the renormalization-group procedure, the associated probability distributions approached universal fixed shapes. For hierarchical lattices of the Migdal-Kadanoff family, the fixed-point distributions were well fitted either by stretched exponentials, or by q-Gaussian distributions; both fittings recover the expected Gaussian limit as D→∞. In the case of the Wheatstone-Bridge lattice, the best fit was found by means of a stretched-exponential distribution.