Clas Blomberg - Physics of life-Elsevier Science (2007)

166 Part IV. Going further with thermodynamics
the calculation of on-intersecting chains on a square lattice, corresponding to the nonintersecting
random walk and also to chain molecule problems: these might be placed along
lattices but not allowed to intersect. On the other hand, the simple 1D Ising model leads only
to a kind of calculation method, while problems like the one below have further features and
turn out to be very useful for describing critical phenomena at phase transitions and also other
questions that turn out to be quite related to the phase transitions, such as the non-intersecting
one we next turn to.
Example 2:2D non-intersecting walks
This calculation is based upon calculations in a lattice, and, for simplicity, we use a
2D square lattice. We consider the problem to describe paths on a lattice that does not intersect.
The problem is also discussed in Section 19C, together with somewhat a simpler energy
arguments than we have here. The example here is taken from Stanley et al. (1982).
The main idea behind what is referred to as a lattice renormalization is to start with a particular
lattice, then make a transformation to a new lattice with fewer points. This means
that the points and bonds in the new lattice represent several points and several bonds in the
original lattice. The main idea is that the features of the original model in the original lattice
shall be represented, essentially the same way in the transformed lattice with the
important parameters transformed in a way that can be calculated. The relevant, power laws
follow from such transformations, as we will show. The philosophy behind this is that the
power laws are independent of the local, detailed features, and rather are related to how the
global features are “scaled” in these transformations.
We consider a square or cubic lattice and lines that go along the lattice sides and never
cross themself, that is, they never pass any lattice point more than once. The length of the
line is equal to the number of single steps, the number of passed lattice sides. A main question
concerns the average number of steps of a line that goes a certain distance in the lattice,
in particular, the relation between this number and the distance.
First, we need a general formalism. Introduce a quantity F(r, N) equal to the number of
lines of N steps that go a distance r in the lattice. The next step is to make a kind of transformation
by a relation:
Gru (, ) ∑ Fr (, Nu )
N
N
(16.33)
This is a sum over all non-intersecting lines that go a distance r where each step of the lines
is assigned a factor u. The number of lines grow with N, and the sum will not converge for
large values of u. There is a limit of convergence u 0 , and close to that, the function contains a
singular contribution (1 u/u 0 ) n . We put u u 0 (1 ), and u N u 0 N (1 ) N u 0 N e N .
It is reasonable that F(r, N) is proportional to a factor z N for large values of N, thus
F(r, N) F 1 (r, N) z N . If there is no further such factor in F 1 , z must be equal to 1/u 0 , the convergence
limit of u. We think of long lines and long distances, and it can be reasonable that
F 1 is a function essentially on one variable, r/R(N), where R is the typical extension of a