Consider the lattice $\mathbb{Z}^d$. Let $A_{n}$ be the set of returning self-avoiding paths (from $0$ to $0$) having length $n$. For any path $\omega \in A_{n}$, let $f(\omega)$ be the number of vertices in the path which have at least three neighbors belonging to the path as well. The function $f(\omega)$ provides an estimation of how much the path is ``tie''. Let $B_{n, k}$ be the set of elements in $\omega \in A_n$ such that $f(\omega)\leq k $.
How should one rescale $k$ with $n$ in order to have $\frac{|B_{n,k}|}{|A_n|}$ going to a constant $0 < C < 1$ as $n$ is large?
