PRIME PERCOLATION

This paper examines the question of whether there is an unbounded walk of bounded step size along Gaussian primes. Percolation theory predic ts that for a low enough density of random Gaussian integers no walk e xists, which suggests that no such walk exists along prime numbers, si nce they have arbitrarily small density over large enough regions. In analogy with the Cramer conjecture, I construct a random model of Gaus sian primes and show that an unbounded walk of step size k root log\z\ at z exists with probability 1 if k > root 2 pi lambda(c), and does n ot exist with probability 1 if k < root 2 pi lambda(c), where lambda(c ) approximate to 0.35 is a constant in continuum percolation, and so c onjecture that the critical step size for Gaussian primes is also root 2 pi lambda(c)log\z\.