We know that a simple random walk (walk along one of the $2d$ lattices' directions with the same probability $\frac{1}{2d}$) on $\mathbb{Z}^{d}$ is transient when $d\geq 3$. And the probability $P_{0}(S_{2n}=0)$ can be computed: $P_{0}(S_{2n}=0)=\sum_{l_{1}+\dots+l_{d}=n}\frac{(2n)!}{(l_{1}!\cdots l_{d}!)^{2}}\frac{1}{(2d)^{2n}}$. Where $S_{2n}$ means that the random walk returns the origin after $2n$ times move.

It's easy to imagine that this probability with $n$ fixed is decreasing as $d$ is increased since if $d$ is larger, the random walk need return the origin at all $d$ directions, then the constraint on the random walk is more strict. But how can I prove the result rigorously?