Asymptotic expansion for the number of self-avoiding random walks

Question

This question is cross-posted from https://math.stackexchange.com/questions/4580314/asymptotic-expansion-for-the-number-of-self-avoiding-random-walks.

Let $c_n$ be the number of self-avoiding random walks on $\mathbb{Z}^2$ of length $n$ starting at $0$. It is conjectured that $$ c_n \ \sim \ A\mu^nn^{\frac{11}{32}}, $$ for suitable real numbers $A$ and $\mu$, where $f(n)\sim g(n)$ means $\lim_{n\to \infty} \frac{f(n)}{g(n)}=1$.

My question is whether anything is known or conjectured about additional terms of an asymptotic expansion for $c_n$. In other words, if we rewrite the conjecture above as stating
$$ c_n = A\mu^nn^{\frac{11}{32}} + o(\mu^nn^{\frac{11}{32}})$$ one could ask whether it would be reasonable to conjecture that $$ c_n = A\mu^nn^{\frac{11}{32}} + B\mu^nn^{\gamma_2-1} + o(\mu^nn^{\gamma_2-1})$$ for a suitable rational number $\gamma_2<\frac{11}{32}$, and if so whether a value for this number is known ? Maybe more complicated functions involing oscillatory behaviour needs to be introduced in the subleading terms ? Any thoughts or references on this problem would be greatly appreciated.

Answer 1

Allow me to answer my own question since the reference I have found might be of use to others.

It turns out that A. Guttmann and A.R. Conway have analyzed this problem in som detail in the following article

• Guttmann, A., Conway, A. "Square Lattice Self-Avoiding Walks and Polygons", Annals of Combinatorics 5, 319–345 (2001).

In this article the two authors are led to conjecture the asymtotic expansion

$$ c_nx_c^n \ \sim \ n^{\frac{11}{32}}(a_1+ a_2n^{-1} + a_3n^{-2} + ...) + (-1)^nn^{-\frac{3}{2}}(b_1 + b_3n^{-1} + ...)$$ where $x_c=1/\mu_c$. The conjecture is backed up by numerical evidence.