An interesting question, known as the Gaussian moat problem, asks
whether it is possible to walk to infinity on Gaussian primes with
steps of bounded length. Our work examines a similar situation in the
real quadratic integer ring $\mathbb{Z}[\sqrt{2}]$ whose primes cluster
near the asymptotes $y = \pm x/\sqrt{2}$ as compared to Gaussian primes,
which cluster near the origin. We construct a probabilistic model of
primes in $\mathbb{Z}[\sqrt{2}]$ by applying the prime number theorem
and a combinatorial theorem for counting the number of lattice points
whose absolute values of their norms are at most $r^2$. We then prove
that it is impossible to walk to infinity if the walk remains within
some bounded distance from the asymptotes. Lastly, we perform a few
moat calculations to show that the longest walk is likely to stay
close to the asymptotes; hence, we conjecture that there is no walk to
infinity on $\mathbb{Z}[\sqrt{2}]$ primes with steps of bounded length.