An infinite path across the minefield

Question

In $\mathbb{R}^2$, a mine is created at every integer point {$(z_1,z_2)\vert z_1,z_2\in \mathbb{Z}$} independently with probability $0\lt p\lt 1$. You choose a starting integer point, and can walk only along the integer lattice grid (up, down, left, right), without returning to previously walked path. What is the probability that there's a starting point from which you can safely walk on forever?

Answer 1

Elaborating on Ivan Neretin's comment, this particular model is known as site percolation on the square lattice.

Let $q = 1-p$ be the probability that a vertex does not have a mine. Then it is known that there exists some $q_c \in (0,1)$ for which the probability of an infintely long 'safe path' is $0$ for all $q \leq q_c$ and $1$ for $q > q_c$.

The exact value of $q_c$ is an open question and is in general difficult to compute. It has been estimated to be $q_c \approx 0.5924...$