Two-dimensional self-avoiding walk asymptotics and critical exponent

Establish for two-dimensional lattices L (e.g., the square or honeycomb lattice) the precise asymptotic growth of the number σ_n(L) of n-step self-avoiding walks starting at a fixed vertex by proving that σ_n(L) ∼ A n^{γ−1} κ(L)^n for suitable constants A and γ, and ascertain that the exponent equals γ = 43/32. This includes confirming the predicted power-law correction in two dimensions in place of the exponential √n correction obtained by Hammersley and Welsh.

Background

Hammersley and Welsh proved upper bounds on the number of self-avoiding walks (SAWs) using bridge decompositions, yielding σ_n ≤ κ^n γ^{√n}. It is widely expected that in two dimensions the correction term is polynomial in n, with a specific critical exponent γ = 43/32 predicted by conformal field theory and SLE considerations.

The quoted passage frames this expectation as a believed but unproven asymptotic form, while noting that the result is established in high dimensions (d ≥ 5) by Hara and Slade with γ = 1. The two-dimensional case remains open and is believed to be linked to SLE_{8/3}.

This problem asks for a rigorous derivation of the asymptotic σ_n(L) ∼ A n^{γ−1} κ(L)^n and the value γ = 43/32 for two-dimensional lattices.