DOI: 10.1140/epjb/e2003-00184-5 THE EUROPEAN PHYSICAL JOURNAL B Reduction of spin glasses applied to the Migdal-Kadanoff

Abstract. 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 ∆E with system size L, σ(∆E) ∼ Ly, is obtained as y3 = 0.25546(3) by reducing the equivalent of lattices up to L = 2 100 in d = 3, and as y4 = 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. PACS. 05.50.+q Lattice theory and statistics (Ising, Potts, etc.) – 75.10.Nr Spin-glass and other random models – 02.60.Pn Numerical optimization