# Self-avoiding walk enumeration via the lace expansion

Nathan Clisby, Richard Liang, and Gordon Slade

This website is an addendum to N. Clisby, R. Liang, G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor. 40:10973--11017, (2007), where we provide a method for enumeration of self-avoiding walks via enumeration of self-avoiding polygons and other so-called lace graphs. The Brydges-Spencer lace expansion gives a recurrence relation which expresses the number of self-avoiding walks in terms of the numbers of these lace graphs.

Here, we provide for download complete lists of our enumeration results of the number of self-avoiding walks and self-avoiding polygons, as well as the sum of squared displacements of self-avoiding walks, on the integer lattice Zd in dimensions 2 through 12. We also provide lists of our enumerations of restricted lace diagrams π(N)m,δ, which, given π(N)m,δ for all δ less than dimension d, are sufficient to calculate the lace expansion coefficients π(N)m, and therefore the coefficients πm, for dimension d.

References below to Section and Equation numbers refer to the above paper.

We list enumerations of the following quantities:

• cn -- the number of n-step self-avoiding walks -- is defined in Section 1.2;
• ρn -- the sum of squared displacements of n-step self-avoiding walks -- is defined in Section 1.2;
• pn -- the number of n-step self-avoiding polygons -- is defined in Section 1.2;
• πn -- the coefficient appearing in the lace expansion -- is defined in Section 3.2;
• θn -- the number of "theta" diagrams -- is defined in Equation (27), in Section 3.2;
• Rn -- the sum of squared displacements of the theta diagrams -- is defined in Equation (27), in Section 3.2;
• π(N)m,δ -- the number of N-loop, m-step lace graphs occupying exactly δ dimensions and taking steps in these dimensions in a rigid order -- is defined in Section 3.3.
• r(N)m,δ -- the sum of squared displacements of lace graphs, a modification of π(N)m,δ -- is defined in Section 3.3.
• &mu -- the connective constant -- is defined in Section 1.3.
• zc -- the critical point -- is the reciprocal of &mu .
• A -- the amplitude for cn -- is defined in Section 1.3.
• D -- the amplitude for the mean-square displacement -- is defined in Section 1.3.

For the 2-dimensional square lattice, the entries below for cn, pn, and ρn link to the webpage of Iwan Jensen, who has obtained the most extensive enumerations of these quantities. We have used his enumerations of cn to calculate πn up to n=71 via Equation (16).

The numbers provided below are in plain text files intended to be easily machine-readable, at the expense of human-readability. Column entries are separated by a single tab character, and rows are separated by newlines. Readers who wish to print or otherwise read the data themselves may prefer to consult N. Clisby, R. Liang, G. Slade, Self-avoiding walk enumeration via the lace expansion: tables (Unpublished, 2007).

## θn and Rn

d=2 d=2 d=2 d=2 d=2
d=3 d=3 d=3 d=3 d=3
d=4 d=4 d=4 d=4 d=4
d=5 d=5 d=5 d=5 d=5
d=6 d=6 d=6 d=6 d=6
d=7 d=7 d=7 d=7 d=7
d=8 d=8 d=8 d=8 d=8
d=9 d=9 d=9 d=9 d=9
d=10 d=10 d=10 d=10 d=10
d=11 d=11 d=11 d=11 d=11
d=12 d=12 d=12 d=12 d=12

## Coefficients in expansions in powers of (2d)-n

Updated 2007-05-21 15:04