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 ΔE with system size L, σ(ΔE) ~L <Superscript>y</Superscript>, is obtained as y <Subscript>3</Subscript>=0.25546(3) by reducing the equivalent of lattices up to L=2<Superscript>100</Superscript> in d=3, and as y <Subscript>4</Subscript>=0.76382(4) for up to L=2<Superscript>35</Superscript> 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. Copyright EDP Sciences, Springer-Verlag 2003