3 research outputs found
A note on the Guerra and Talagrand theorems for Mean Field Spin Glasses: the simple case of spherical models
The aim of this paper is to discuss the main ideas of the Talagrand proof of
the Parisi Ansatz for the free-energy of Mean Field Spin Glasses with a
physicist's approach. We consider the case of the spherical -spin model,
which has the following advantages: 1) the Parisi Ansatz takes the simple ``one
step replica symmetry breaking form'', 2) the replica free-energy as a function
of the order parameters is simple enough to allow for numerical maximization
with arbitrary precision. We present the essential ideas of the proof, we
stress its connections with the theory of effective potentials for glassy
systems, and we reduce the technically more difficult part of the Talagrand's
analysis to an explicit evaluation of the solution of a variational problem.Comment: 20 pages, 5 figures. Added references and minor language correction
Replicated Transfer Matrix Analysis of Ising Spin Models on `Small World' Lattices
We calculate equilibrium solutions for Ising spin models on `small world'
lattices, which are constructed by super-imposing random and sparse Poissonian
graphs with finite average connectivity c onto a one-dimensional ring. The
nearest neighbour bonds along the ring are ferromagnetic, whereas those
corresponding to the Poisonnian graph are allowed to be random. Our models thus
generally contain quenched connectivity and bond disorder. Within the replica
formalism, calculating the disorder-averaged free energy requires the
diagonalization of replicated transfer matrices. In addition to developing the
general replica symmetric theory, we derive phase diagrams and calculate
effective field distributions for two specific cases: that of uniform sparse
long-range bonds (i.e. `small world' magnets), and that of (+J/-J) random
sparse long-range bonds (i.e. `small world' spin-glasses).Comment: 22 pages, LaTeX, IOP macros, eps figure