A lattice point counting approach for the study of the number of self-avoiding walks on $\mathbb{Z}^{d}$

We reduce the problem of counting self-avoiding walks in the square lattice to a problem of counting the number of integral points in multidimensional domains. We obtain an asymptotic estimate of the number of self-avoiding walks of length $n$ in the square lattice. This new formalism gives a natural and unified setting in order to study the properties of the number of self-avoiding walks in the lattice $\mathbb{Z}^{d}$ of any dimension $d\geq 2$.