= Synchronous product of simple symmetric random walks
{title2=$p_{00}^{(2n)}=\left(\binom{2n}{n}2^{-2n}\right)^d\sim(\pi n)^{-d/2}$}
Take $d$ <independent> copies of the <simple random walk on the integer line>, each stepping left or right with equal <probability> at every integer time. Their <product Markov chain> moves all coordinates at once; each of its $2^d$ diagonal increments has <probability> $2^{-d}$. At odd times a return to the origin is impossible. At time $2n$, <independence> gives the displayed return probability, whose <asymptotic equivalence> follows from <Stirling's approximation>. The <recurrence criterion by return probabilities> and the <p-series> therefore imply <recurrence> for $d=1,2$ and a <transient Markov chain> for $d\geq3$.
For $d>1$ the chain from the origin visits only the <sublattice> whose coordinates all have the same parity. This is one <communicating class>; restricting to it gives an <irreducible Markov chain>. The <period of a state in a Markov chain> is two here. The conclusion applies to that class, without incorrectly claiming <Markov chain irreducibility> on all of $\mathbb Z^d$. This walk moves diagonally; the usual axis-step <simple random walk> has different transition probabilities, even though the same dimensional recurrence threshold holds.
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