Reduction of spin glasses applied to the Migdal-Kadanoff hierarchical lattice

A reduction procedure to obtain ground states of spin glasses on sparse graphs is developed and tested on the hierarchical lattice associated with the Migdal-Kadanoff approximation for low-dimensional lattices. While more generally applicable, these rules here lead to a complete reduction of the lattice. The stiffness exponent governing the scaling of the defect energy DeltaE with system size L, sigma(DeltaE) similar to L-y, is obtained as y(3) = 0.25546(3) by reducing the equivalent of lattices up to L = 2(100) in d = 3, and as y(4) = 0.76382(4) for up to L = 2(35) in d = 4. The reduction rules allow the exact determination of the ground state energy, entropy, and also provide an approximation to the overlap distribution. With these methods, some well-know and some new features of diluted hierarchical lattices are calculated.