23,666 research outputs found
Spin Glass Computations and Ruelle's Probability Cascades
We study the Parisi functional, appearing in the Parisi formula for the
pressure of the SK model, as a functional on Ruelle's Probability Cascades
(RPC). Computation techniques for the RPC formulation of the functional are
developed. They are used to derive continuity and monotonicity properties of
the functional retrieving a theorem of Guerra. We also detail the connection
between the Aizenman-Sims-Starr variational principle and the Parisi formula.
As a final application of the techniques, we rederive the Almeida-Thouless line
in the spirit of Toninelli but relying on the RPC structure.Comment: 20 page
On differentiability of the Parisi formula
It was proved by Michel Talagrand in [10] that the Parisi formula for the
free energy in the Sherrington-Kirkpatrick model is differentiable with respect
to inverse temperature parameter. We present a simpler proof of this result by
using approximate solutions in the Parisi formula and give one example of
application of the differentiability to prove non self-averaging of the overlap
outside of the replica symmetric region
The Parisi formula for mixed -spin models
The Parisi formula for the free energy in the Sherrington-Kirkpatrick and
mixed -spin models for even was proved in the seminal work of
Michel Talagrand [Ann. of Math. (2) 163 (2006) 221-263]. In this paper we prove
the Parisi formula for general mixed -spin models which also include
-spin interactions for odd . Most of the ideas used in the paper are well
known and can now be combined following a recent proof of the Parisi
ultrametricity conjecture in [Ann. of Math. (2) 177 (2013) 383-393].Comment: Published in at http://dx.doi.org/10.1214/12-AOP800 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Un-inverting the Parisi formula
The free energy of any system can be written as the supremum of a functional
involving an energy term and an entropy term. Surprisingly, the limit free
energy of mean-field spin glasses is expressed as an infimum instead, a
phenomenon sometimes called an inverted variational principle. Using a
stochastic-control representation of the Parisi functional and convex duality
arguments, we rewrite this limit free energy as a supremum over martingales in
a Wiener space.Comment: 17 page
- …