Let us consider a fixed point of the RG flow of a generic system, and assume that it has two relevant directions corresponding to the coupling constants
[1], the temperature, and
, the external field. We suppose that
and
are transformed under the RG as:

where

and

are analytic functions given by the coarse graining procedure.
The fixed points

of the flow will be given by the solutions of:

Linearising the transformation around

, in terms of the reduced variables

and

we have:

where:

As previously stated we suppose

to be diagonalizable. We therefore write its eigenvalues as:

Note that we can always do that, it is just a simple definition. In other words, we are defining

and

as:

This way we can write:

After

iterations we will have:

and since ingeneral we know that

:

This is the scaling law of the correlation length.
From this we can determine the critical exponent

; in fact, setting

and choosing

so that

with

a positive real number
[2], we have:

Since in general

, we get:

This is an extremely important result! In fact, we see that once the RG transformation

is known,

is straightforward to compute and so we are actually able to calculate

and predict its value!
We can do even something more (including giving

a meaning) from the scaling law of the free energy density. After

iterations of the RG we have:

and choosing

so that

, then:

Comparing this to what we have seen in
An alternative expression for the scaling hypothesis we get:

- ↑ We have already stated that considering
as a coupling constant is equivalent to considering
as such.
- ↑ Remember that the value of
is not fixed, so we can choose the one we prefer; in this case we are making this choice because
does not necessarily have to be an integer.