Notebooks

Self-avoiding Random Walk (SAW)

A summary of some results on self-avoiding random walks. Please let me know of any errors.

Let $c_n$ be the number and $R_e$ and $R_g$ be the end-to-end distance and radius of gyration of a $n$-step self-avoiding random walk with vertices at $\omega_0,\omega_2,\cdots,\omega_n$. Then the radius of gyration and end-to-end distance, two relevant measures quantifying the "size" the SAW, are defined as follows:

$$ \begin{align*} R_{\rm g}^2 &= \frac{1}{2(n+1)^2} \sum_{i,j =0}^n(\omega_i-\omega_j)^2\\ R_\mathrm{ e}^2 &= |\omega_n-\omega_0|^2 \end{align*} $$

To get an idea about the size of $n$-step SAWs, one computes their values averaged over all $n$-step SAWs:

$$ \begin{align*} \langle R_{\rm g}^2\rangle &= \frac{1}{c_n}\sum_\omega R_\mathrm{ g}^2(\omega)\\ \langle R_{\rm e}^2\rangle &= \frac{1}{c_n}\sum_\omega R_\mathrm{ e}^2(\omega) \end{align*} $$

Numerical Results

There are no known method to calculate $c_n$ other than numerical enumeration and asymptotic analysis, as far as I know. The following table shows the length of largest SAWs that have been enumerated, with $c_n$ often becoming as large as $10^{30}$:

$d$ $\max(n)$ Symmetry Reference
2 71 Square Jensen (2004)
3 36 Simple cubic Lawler (2011)
3 28 BCC Schram (2017)
3 24 FCC Schram (2017)

Asymptotic results (large $n$)

Asymptotically, $\langle R_\mathrm{ g}^2\rangle$ and $\langle R_\mathrm{ e}^2\rangle$ are conjectured to behave as $\langle R_\mathrm{ e}^2\rangle \approx BN^{2\nu}$ and $\langle R_\mathrm{ g}^2\rangle \approx CN^{2\nu}$.
The non-universal amplitudes $B$ and $C$ depend on lattice symmetry and dimension.
$c_n$ is conjectured to behave as: $c_n \approx A \mu^n n^{\gamma -1}$ where $\mu$ is the connective constant (Sometimes it's called growth constant and $k=\log \mu$ is called connective constant).
The non-universal amplitude $A$ depends on both the dimension and lattice symmetry.
However, the $\nu$ and $\gamma$ only depend on the dimension—they are universal.
$d$ $\nu$ $\gamma$ $c_n\approx$ Comment Reference
2 $3/4$ $43/32 = 1.343 75$ $A \mu^n n^{11/32}$ Exact Nienhuis (1982), Nienhuis (1984)
3 0.58759700(40)$\approx 3/5$ $1.15695300(95) \approx 7/6$ $A \mu^n n^{1/6}$ Estimate Schram (2017), Slade (2019) (for summary)
4 $1/2$ with log correction $\langle R_\mathrm{ e}^2\rangle \approx B n(\log n)^{1/4}$ $\langle R_\mathrm{ g}^2\rangle \approx C n(\log n)^{1/4}$ 1 with log correction $A \mu^n (\log n)^{1/4}$ Upper critical dimension Slade (2019)
$\geq 5$ 1/2 1 $A \mu^n$ Probably exact Slade (2019), p.4

Recommended

Clisby, N., & Dünweg, B. (2016). High-precision estimate of the hydrodynamic radius for self-avoiding walks. Physical Review E, 94(5), 052102.
Jensen, I. (2004). Self-avoiding walks and polygons on the triangular lattice. Journal of Statistical Mechanics: Theory and Experiment, 2004(10), P10008.
Lawler, G. F. (2011). Scaling limits and the Schramm-Loewner evolution. Probability Surveys, 8.
Nienhuis, B. (1982). Exact Critical Point and Critical Exponents of $\mathrm{O}(n)$ Models in Two Dimensions. Physical Review Letters, 49(15), 1062–1065.
Nienhuis, B. (1984). Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. Journal of Statistical Physics, 34(5), 731–761.
Schram, R. D., Barkema, G. T., Bisseling, R. H., & Clisby, N. (2017). Exact enumeration of self-avoiding walks on BCC and FCC lattices. Journal of Statistical Mechanics: Theory and Experiment, 2017(8), 083208.
Slade, G. (2019). Self-avoiding walk, spin systems and renormalization. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 475(2221), 20180549.