The Potts model on the triangular lattice

Victor Adamchik, adamchik@cs.cmu.edu 
Steve Finch, sfinch@mathsoft.com 
Robert Ziff, rziff@engin.umich.edu

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.

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].

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.

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:

Mathematica code

See the Mathematica notebook for integration code.

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).