The usual methods fail when dealing with self avoiding walks. One approach I have pondered is to treat the walk as a parametric function of sorts in which the integral approximates the length of the walk and it's constrained by a integer value distances about $n$ number of $x$ & $y$ intercepts in such a way as it does not intersect with itself. So for a $n$ by $n$ lattice in which the walk starts at $0,0$, the functional would be of the form $(f,y)=y^n+a_1y^{n-1}...x^n...$ .

The graph would be inverted subject to the relevant constraints to find some sort of asymptotic behavior.

The usual methods fail when dealing with self avoiding walks. One approach I have pondered is to treat the walk as a parametric function of sorts in which the integral approximates the length of the walk and it's constrained by a integer value distances about $n$ number of $x$ & $y$ intercepts in such a way as it does not intersect with itself. So for a $n$ by $n$ lattice in which the walk starts at $0,0$, the functional would be of the form $f(x,y)=a_0y^n+a_1y^{n-1}...b_0x^n...$ .

The graph would be inverted subject to the relevant constraints to find some sort of asymptotic behavior.