= Solution
This is the <synchronous product of simple symmetric random walks>, rather than a walk that changes only one coordinate at a time. Independence of the three coordinate processes gives
$$
\mathbb P\bigl((X_{2m},Y_{2m},Z_{2m})=(0,0,0)\bigr)
=\left[\binom{2m}{m}2^{-2m}\right]^3
\sim\frac1{(\pi m)^{3/2}}.
$$
Odd-time return probabilities vanish. The <p-series> with exponent $3/2$ converges, so the sum of all return probabilities is finite. By the <recurrence criterion by return probabilities>, the origin is a <transient state>: \b[the three-dimensional process is transient]. From the origin its <communicating class> consists of integer triples whose coordinates have the same parity; translation gives the same classification for every other class.
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