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The Potts model on the triangular lattice

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Introduction

Exact formulas associated with the statistical Potts model have been found
over both the square and triangular lattices in the plane. We described
relevant square lattice results (due to Temperley and Lieb) elsewhere.
Baxter, Temperley and Ashley (see [1]) derived the following generating
function for the Potts model on the triangular lattice:

Performing a logarithmic substitution, the integral can be rewritten
in the algebraic form:

The integral converges for . Here is a graph
of on an interval :

Although it isn't known whether the function
has a closed-form expression for all values of *y*, it can be evaluated
explicitly for any *y* which is a rational multiple of ;.
We consider several particular cases of in the
next section.

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Particular Cases

Let , where *p* and *q* are positive
integers. Then the formula (2) can be rewritten as

This integral belongs to the more common class of integrals

where and are positive
integers and is a cyclotomic polynomial. We assume
that the integral is convergent. The integral (4) can be envisaged in an
alternative form. Performing an integration by parts, we have

where and are positive
integers and is a cyclotomic polynomial. In the
last section we provide *Mathematica* code for evaluating integrals
(4) and (5). The reader must evaluate that cell in order to calculate all
the integrals below. Our results are consistent with a table presented
by Baxter, Temperley and Ashley in[1].

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First Derivative

We shall prove that:

Note that this formula can be rewritten in the following algebraic form:

As far as we know, the exact expression in (6) is new.

*Proof.* Differentiating the representation (2) relative
to *y* and performing trivial simplifications, we obtain:

The first two integrals are:

Although the third integral could be easily evaluated by *Mathematica*,
a more elegant way is to make use of:

so that the integrand can be rewritten as:

Running the integral in *Mathematica*, we obtain:

which completes the proof of formula (6). This is connected to the mean
cluster density for the bond percolation model on the triangular lattice.

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Second Derivative

Just as the first derivative has an interpretation as a mean value, the
second derivative can be regarded as a measure of "fluctuations" in the
number of clusters corresponding to the bond percolation model.
We shall prove that:

which, again, we believe is a new result. Note that this formula can
be rewritten in the following algebraic form:

*Proof*. Differentiating the integral on the right side
of representation (2) twice with respect to *y*, we obtain

Since

and, using differentiation by parameter, formula (5) can be rewritten
as:

Then running the integral in *Mathematica* we get:

Finally, we need to differentiate the above by
and find the limit as :

Therefore:

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*Mathematica* code

See the *Mathematica* notebook
for integration code.
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References

1.R. J. Baxter, H. N. V. Temperley and S. E. Ashley, *Triangular Potts
model at its transition temperature, and related models*, Proc. Royal
Soc. London A **358** (1978) 535-559.
2. S. Finch, *Random percolation constants*, HTML essay at World
Wide Web URL http://www.mathsoft.com/asolve/constant/rndprc/rndprc.html,
part of Favorite Mathematical Constants collection (1996).