https://doi.org/10.1140/epjb/e2003-00184-5
Reduction of spin glasses applied to the Migdal-Kadanoff hierarchical lattice
Physics Department, Emory University, Atlanta, Georgia
30322, USA
Received:
20
February
2003
Revised:
4
April
2003
Published online:
3
July
2003
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 
with system size L,
, is obtained as 
by
reducing the equivalent of lattices up to 
in d=3, and as
for up to 
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
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2003
