The first approach was introduced by Mezard and
Parisi[3]. They employ the replica method to study
the 
moment of the partition function in the limit 
. I am sure that the details of replica methods will be
discussed here in other lectures, so I will only give a brief summary
of the main ideas and the consequent results. The basic idea is that
physical quantities are derived from the disorder-averaged free energy
It is difficult in practice to carry out such an average of a logarithm, but it may be accomplished by using an amusing mathematical identity:
Subtleties arise because one can really only calculate things for
integer n, and because one must interchange various limits (the
averaging and the limit, as well as the
thermodynamic limit). However, the nature of these difficulties are
fairly well understood, and certainly appear to be well under control
in this case. Many results can be re-obtained in replica-free ways
(e.g. via the cavity method described by Marc), and the answers
obtained are quite reasonable physically.
Technically, one proceeds by noticing that 
for integer n can
be written as a field theory involving n copies or replicas of the
original system, described by fields 
, with 
. Performing the disorder average then leads to an
interacting system,
The ``action'' is
where we have taken a Gaussian random potential with a two-point correlation function of the form
Various forms may be taken for R, depending upon whether one prefers to study short-range or long-range correlated disorder.
The replica action in Eq. 71 is highly non-trivial,
even for integer 
. To study it, a variational approach may
be employed. This method takes advantage of a bound on the path
integral employed first by Feynmann for the polaron problem. It
states that the effective action
is bounded below by the variational action
where 
is an arbitrary ``trial'' action,
is the trial partition function, and the 
denote an
average with respect to . A natural trial form is
The parameters K and 
may then be determined
variationally. It can also be shown that this approximation becomes
exact in the limit 
, where it becomes a
saddle-point calculation.
Replica symmetry breaking emerges in the solutions of the
self-consistent minimization equations of 
. The matrix
G may take, in the limit, either a replica
symmetric or replica symmetry breaking (RSB) form.
For this problem Mezard and Parisi have come up with a very compelling
physical picture of the meaning of RSB. As is apparent from the
Gaussian form of , the trial action decouples into a sum over
independent momenta. This implies that the distributions of Boltzmann
weight for each Fourier mode 
are statistically
independent. We can therefore concentrate on the distribution of a
particular mode 
.
For simplicity, let us consider the case of 1-step RSB, characterized
by two RSB parameters 
. Then the
trial distribution, in a particular disordered sample, can be
constructed as follows: First pick a random variable 
,
drawn from the distribution
Next, we pick an infinite sequence of other random variables,
, drawn independently from the distribution
For each i, we also pick a random ``free energy'' 
from an
exponential distribution,
Given this set of random variables, the distribution of 
is
where
and 
is a simple function of 
and 
.
This may seem somewhat complicated, but the net result, Eq. 80\
is very appealing. The thermal distribution for a single mode is simply a sum of Gaussians of magnitude specified by
exponentially distributed energies, each centered around some random
point in the space. This naturally produces a picture in
which the distribution function has many minima of different sizes,
distributed in a hierarchical way. The resulting ``states'', or
metastable configurations of thus form a tree-like
structure, suggestive of the tree-like form observed in the numerics
of the DP. Multi-step (and continuous) RSB simply involves an
iteration of this procedure, to produce more and more hierarchical
structures. We will come back to this general form a bit later when
we make a comparison with RG results.
What are the main results of the RSB treatment? In principle, any
correlation function may be calculated in an approximate way, becoming
exact as . First, the method correctly
produces the general phase diagram for random manifolds, i.e. the
existence of a thermal phase (with no RSB) for
N > 2d/(2-d) and d<2. Secondly, one may calculate the roughness
exponent in the low-temperature phase. For short-range correlated disorder,
The result for d>4 is generally believe to be act. For d<2, the
approximation appears to be quite poor for general N, though again
the large N result appears to be exact. It corresponds to
saturating the lower bound on 
for stability of the pinned
phase, 
. The intermediate result is more interesting. It
saturates DSF's proposed upper bound on 
and . It can
be reproduced by a simple ``Flory'' argument. We simply estimate the
two terms in the Hamiltonian by their naive power-counting scalings.
That is, for a delta-correlated V, we estimate
Equating the two terms in the free energy gives
which gives the result of Eq.82.
