This paper examines the question of whether there is an unbounded walk
of bounded step size along Gaussian primes. Percolation theory
predicts that for a low enough density of random Gaussian integers no
walk exists, which suggests that no such walk exists along prime
numbers, since they have arbitrarily small density over large enough
regions. In analogy with the Cramér conjecture, I construct a random
model of Gaussian primes and show that an unbounded walk of step size
$k@\sqrt{\log |z|}$ at $z$ exists with probability 1 if $k\More
\sqrt{2\ppi\lmbda_c}$, and does not exist with probability 1 if $k
\Less \sqrt{2\ppi\lmbda_c}$, where $\lmbda_c\Approx 0$.$35$ is a
constant in continuum percolation, and so conjecture that the critical
step size for Gaussian primes is also $\sqrt{2\ppi\lmbda_c\log|z|}$.