  We give precise asymptotics to the number of random walks in the standard
orthogonal lattice in $\mathbb{R}^d$ that return to the starting point at step
$2n$, both for all such walks and for the ones that return for the first time.
The first set of asymptotics is obtained in an elementary way, by using a
combinatorial and geometric multiplication principle together with the
classical theory of Legendre polynomials. As an easy consequence we obtain a
unified proof of P\'{o}lya's theorem. By showing that the relevant generating
functions are $\Delta$-analytic, we use the deeper Tauberian theory of
singularity analysis to obtain the asymptotics for the first return paths.

Combinatorics;Probability