Let us now attack the problem directly at zero temperature. Provided our hypothesis of a T=0 fixed point is correct, this should provide an adequate description of the entire ordered (disorder-dominated) phase. Without thermal fluctuations, we are really faced with an optimization problem: What is the configuration of the manifold which best minimizes the sum of elastic and random energies?
In order to find such an optimal conformation, the manifold must distort itself (see Fig. 4). The extent to which this happens is the result of a competition between the elastic energy (which prefers a flat manifold) and the random potential (which encourages large excursions to take advantage of widely separated low energy regions). We expect that there should be a characteristic scaling for the result roughness of the manifold,
and
where 
is known as the roughness exponent. The simplest
hypothesis is that there is a single such scale for the transverse
distortions of the manifold on scale x, i.e. that the whole
probability distribution scales:
where f is a short-range order one function (normalized to one).
Figure 4: Illustration of a rough manifold in a particular sample.
This also means that all the moments of G scale:
There will also be large sample-to-sample variations in the minimum energy: in different random potentials, the best energy changes. We expect these variations to also obey scaling laws. In particular, we may consider the difference between the energy of the optimal configuration of a manifold of linear size L in a particular random potential and the mean ground-state energy of such a manifold (averaged over V),
This quantity is expected to be a random variable with a
distribution of width 
, i.e.
where again g is a normalized O(1) function. We can also consider another measure of energy variations, within a single sample. To do so, we compare the energy of an optimal manifold to that of one which has been constrained to be distorted an amount u over an (internal) area of linear size L. We can accomplish such a constraint, say, by fixing various points on the manifold and performing the remaining constrained optimization. The sample averaged energy difference is then expected to take a scaling form involving both exponents:
This last scaling result has an important consequence: the ``energy
exponent'' 
is actually just the negative of the RG eigenvalue
of temperature, i.e. 
. Why is that? Let's consider
a system at low but non-zero temperature T.
Eq. 48 tells us that typical excitations of size
L cost an energy of order . Since only those excitations
with energies less than T are activated, the typical scale for
thermal fluctuations around the ground state configuration (i.e. the
size of jumping regions) of the manifold is 
.
Now consider the RG prediction. For 
, temperature is an
irrelevant perturbation. As we coarse-grain, the renormalized
temperature 
shrinks until it eventually
becomes of the order of some microscopic energy scale 
. At that
point no further excitations can be activated. Setting 
, we find 
. Comparing these two forms
gives the desired result. From this point on, therefore, I will
discontinue the use of in favor of 
.
It would be nice at this point to be able to simply give some formulae for these exponents and scaling functions. Unfortunately, very few exact results are known for this problem, despite extensive efforts to obtain them. I will review some of these methods and their accomplishments in the next section, but first I would like to review what can be said in general about the scaling exponents. For further explanation of this scaling picture, see Ref. [6].
First, there is a rather simple and intuitive relation between the
roughness and energy exponents. A rather naive argument proceeds as
follows. The Hamiltonian of the system is just a sum of elastic and
random energies. It is their mutual competition which is responsible
for the non-trivial ground state, so we expect that the two terms
should be comparable in this optimal configuration. If so, we may
estimate the sample-to-sample energy variations simply by those of
either term. Choosing the elastic term for simplicity, we can obtain
a simple estimate by noting that the typical sample-to-sample
variations of u for a sample of size L are of order 
.
Integrating the squared gradient in d dimensions gives
or
It turns out that this somewhat naive argument is actually exact. Let me put off a discussion of why this is true until later, when it can be put into a more appropriate context.
This identity allows us to encapsulate the simple scaling in a single
unknown exponent. In fact, we can do a bit better by putting some
physical bounds on the allowed values of .
First we will obtain an upper bound. Suppose first we were to
constrain the manifold to be completely flat, i.e. 
. The
energy of the system would then be entirely given by the random
potential contribution. This is roughly a sum over 
independent random variables, where 
is the correlation length of
V in the internal coordinates. This trivial system thus has
sample-to-sample variations of order 
.
If we relax the constraint, the optimal manifold for each realization of the randomness will adjust itself to lower its energy. This means that it will choose to occupy regions of space in which the values of V lie as deep as possible into the (negative) tail of the distribution p(V). Provided this distribution is bounded (or exponentially narrow), this means that the sample-to-sample variations of the energy will be reduced. We therefore expect
which implies
This implies for that the roughness satisfies
The requirement of stability of the T=0 fixed point requires
or
A more rigid lower bound has been proposed by D. S. Fisher, which is
I haven't been able to come up with a satisfactory argument for the final bound, which is not to my knowledge explained in the literature, but it is consistent, as we will see, with the known results.
Lastly, let us close this section with a few values (see Table 1) for the scaling exponents, which are at least approximately known in the physical cases of interest.
Table 1: Scaling exponents for random manifolds
