The first approach was introduced by Mezard and Parisi. 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
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.