Perimeter generating functions for the mean-squared radius of gyration of convex polygons

Iwan Jensen

ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia

Email: I.Jensen@ms.unimelb.edu.au

Received 15 September 2005
Published 2 November 2005

PACS numbers: 05.50. + q, 05.70.Jk, 02.10.Ox

1. Introduction

A well-known long standing problem in combinatorics and statistical mechanics is to find the generating function for self-avoiding polygons (or walks) on a two-dimensional lattice. The models are of tremendous inherent interest as well as serving as simple models of polymers and vesicles [1-3]. Despite strenuous effort over the past 50 years or so, this problem has not been solved on any regular two-dimensional lattice. However, there are many simplifications of this problem that are solvable [4], but all the simpler models impose an effective directedness or other constraint that reduces the problem, in essence, to a one-dimensional problem.

One particular class of exactly solved polygon models are those with a convexity constraint (see figure 1). On the square lattice a polygon is said to be convex if it is convex with respect to both vertical and horizontal lines, i.e., any vertical line will intersect the polygon at zero or two horizontal edges while similarly any horizontal line will intersect the polygon at zero or two vertical edges. Alternatively, a convex polygon is a SAP of a length equal to the perimeter of its minimal bounding rectangle. If we further demand that the polygon must include the vertices in some of the corners of the minimal bounding rectangle, we can define a further five polygon models as illustrated in figure 1. The full perimeter and area generating functions are known for all these models [4]. Also of great interest is the mean-square radius of gyration, R²_n, which measures the typical size of a polygon with perimeter n. In this letter, we report on work leading to conjectured exact solutions for the generating functions associated with the mean-square radius of gyration for the class of convex polygons.

An n-step self-avoiding walk [b.omega] is a sequence of distinct vertices ω₀, ω₁,  ..., ω_n such that each vertex is a nearest neighbour of it predecessor. SAWs are considered distinct up to translations of the starting point ω₀. A self-avoiding polygon of length n is an n  -  1-step SAW such that ω₀ and ω_n - 1 are nearest neighbours and a closed loop can be formed by inserting a single additional step joining the two end points. We shall use the symbol [b.Omega]_n to mean the set of all SAPs of perimeter length n. Generally SAPs are considered distinct up to a translation, so if there are p_n SAPs of length n there are 2np_n walks (the factor of 2 arising since the walk can go in two directions). One expects in general that p_n ~ Aμⁿn^α - 3, where μ is the so-called connective constant while α is a critical exponent. In our cases μ and α are known from the exact solutions for the perimeter generating functions

where we took into account that polygons on the square lattice have even length. The generating functions thus have a singularity at the critical point x_c  =  1/μ² with critical exponent 2  -  α. The function A(x) is analytic at x  =  x_c. Note that both μ and α are model dependent.

The mean-square radius of gyration of n-step polygons is defined by

where we expect that R²_n ~ Bn^2ν. It is advantageous to look at the quantity r_n  =  n²p_nR²_n, which is an integer, and in particular we shall study the associated generating function

where we again used that r_n is non-zero only when n is even.

The values for the critical exponents are known exactly, though non-rigorously, for self-avoiding polygons due to the work by Nienhuis [5], α  =  1/2 and ν  =  3/4. As we shall demonstrate later, the exponent α takes on several different values for the convex polygons studied in this letter, but the exponent ν  =  1 in all cases.

In the next section, we briefly describe the algorithm used to calculate r_n and in the following section we list the various perimeter generating functions.

2. Computer enumeration

The first terms in the series for the polygon generating function are calculated using transfer matrix techniques to count the number of polygons spanning rectangles W  +  1 edges wide and L  +  1 edges long. The transfer matrix technique involves drawing a line through the rectangle intersecting a set of edges. For each configuration of occupied or empty edges along the intersection, we maintain a (perimeter) generating function for partial polygons cutting the intersection in that particular pattern. Due to the convexity constraint a vertical line will intersect the polygon exactly twice. The upper edge of the convex polygon performs a directed walk taking steps to the right and up until it reaches the top of the rectangle where it turns and then performs a directed walk with steps to the right and down. Likewise the lower edge performs a directed walk with right and down steps until it hits the bottom of the rectangle where it turns and takes only right and up steps. A convex polygon is formed once the two walks meet. In order to specify a configuration we just need to know the positions of the edges and whether or not the top and bottom of the rectangle have been touched. All the possible configurations can then be encoded by four (W  +  1) × (W  +  1)-matrices, one matrix for each possibility of touched borders. As the vertical boundary line is moved one step forward, the matrices are updated to allow for all the legal moves of the edge-walks (the walks must be directed as described above and never cross). The updating involves simple double sums over the indices. This approach was used by Guttmann and Enting [6] and is very efficient. However, in one iteration many steps can be inserted and this makes the calculation of the contributions to the radius of gyration somewhat cumbersome. We find it more convenient to use an algorithm in which the convex polygons in a given rectangle are enumerated by moving the intersection so as to add one vertex at a time. The method we used to enumerate convex polygons on the square lattice is a specialization of the method originally devised by Enting [7] for the enumeration of self-avoiding polygons. As noted earlier, convex polygons can be viewed as SAPs with a number of steps equal to the perimeter of the minimal bounding rectangle. So we could simply take our previous algorithm [8, 9], which we generalized in order to calculate the radius of gyration, and only extract the terms counting convex polygons. Due to the convexity constraint we were able to simplify the algorithm somewhat and make it more efficient. However, the algorithm is still quite similar to the SAP enumeration algorithm so we would not describe it further. Suffice to say that the method for calculating the radius of gyration coefficients r_n has been described in [9].

Using this algorithm we quickly (a few hours of CPU time) calculated the radius of gyration of the polygon models of figure 1 to length n  =  110, giving us 56 terms in the half-perimeter series. The first few terms p_n and r_n are listed in table 1. The full series for the generating functions studied in this letter can be obtained by sending a request to the author or via the web at http://www.ms.unimelb.edu.au/~iwan/.

Table 1. The number of polygons pn and their mean-squared radius of gyration rn  =  n2pnR2n.

3. The exact generating functions

In this section we use the series for r_n to find (numerically) the exact perimeter generating functions for the radius of gyration of convex polygons.

The perimeter generating function for convex polygons was first obtained by Delest and Viennot [10] using the method of algebraic languages and later by several other authors using different methods [6, 11, 12]:

From this we see that the critical point x_c  =  1/4 (and thus μ  =  2) while the critical exponent 2  -  α  =   - 2 (and thus α  =  4), corresponding to the dominant double pole at x  =  x_c. In addition there is a sub-dominant square root correction. Informed by this result it is natural to assume that the generation function for the mean-squared radius of gyration has a similar form. That is we assume that , where A(x) and B(x) are polynomials. Using the method of differential approximants [13] we easily established that γ  =  6. Next we wrote a simple Maple routine to find such a solution, that is we solve for the unknown coefficients a_i and b_i of A(x) and B(x). We simply form the series expansion for , match the series coefficients to those of and solve the resulting set of linear equations in the coefficients a_i and b_i. In this fashion we found a solution with polynomials of degree 10 requiring not more than 22 unknown coefficients. Since we have more than 50 known terms r_2n there are at least 30 unused series coefficients which serve as strong checks on the correctness of our solution. The generating function for the mean-squared radius of gyration of convex polygons is

From this we see that the critical exponent α  +  2ν  =  6 and thus ν  =  1. This should be compared to the result for self-avoiding polygons ν  =  3/4 [5]. Physically, there is a simple argument for ν  =  1. Convex polygons are relevant to the description of vesicles in the inflated regime, where they are space filling, and since the radius of gyration measures a typical size of a polygon R²_n is proportional to a typical area and hence ν  =  1 for convex polygons. The value ν  =  3/4 means that SAPs are much more ramified.

Directed and convex polygons were considered by Lin and Chang [11]. They calculated the full anisotropic generating function for directed and convex polygons. In the isotropic case which we consider here their result reduces to the very simple form

so we have x_c  =  1/4 while 2  -  α  =   - 1/2 and thus α  =  5/2. As for the convex case we start by looking for a solution to of the same form, that is , with γ  =  9/2 determined from differential approximants. However we were not successful at first, so next we tried a solution of the same form as for convex polygons and found that

So in this case we find the critical exponent α  +  2ν  =  9/2 and thus as before ν  =  1.

The model of staircase polygons is very well known and much studied, dating back at least to the work by Pólya [14] who showed that for n ≥ 2. This result was obtained by Delest and Viennot [10] in the more elegant form , where C_n are the famous and ubiquitous Catalan numbers. Consequently the generating function is

and x_c  =  1/4, while 2  -  α  =  1/2 and thus α  =  3/2. As per the previous cases we quite readily find the radius of gyration generation function

and we see that α  +  2ν  =  7/2 and once again ν  =  1.

Stack polygons were also considered by Lin and Chang [11] and their result for the generating function is

The critical point is now given by the zero of 1  -  3x  +  x² namely x_c  =  0.381 966 011 ... and the critical exponent is 2  -  α  =   - 1 or α  =  3. In this case the radius of gyration generation function is of the same form and again we have ν  =  1. Explicitly we find that

The generating function for Ferrers diagrams is trivial in that these polygons are simply formed from a directed walk with n  -  2 right or up steps, extended at the starting point with a horizontal step and at the end point with a vertical step, and then closed by straight lines to form a polygon with 2n steps. It immediately follows that the generating function is

and we have x_c  =  2 and α  =  3. The radius of gyration generation function is of the same form and with ν  =  1,

Rectangular polygons are obviously the simplest case and the generating function is simply

so that x_c  =  1 and α  =  4. The radius of gyration generating function is found to be

and again we have ν  =  1.

Now that these results for the radius of gyration of convex polygons are known from the numerical work presented here it should be easier to prove them rigorously.

Acknowledgment

We gratefully acknowledge financial support from the Australian Research Council.

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