We want to estimate, in $d$-dimensional space, the typical number of contacts of random walk of length $N$, as $N\to \infty$. Consider a sphere containing a random walk of length $N$, whose radius will scale with $\sqrt{N}$ and the volume will scale as $N^{d/2}$ in $d$ dimesions.
Now consider a polymer consisting of $N$ monomers modelled as a random walk. When we consider the monomers as independent particles confined in a volume $V_d$. What will be the probability that a given pair of monomers meet in the same point scale with $N$?