Bond percolation on the square lattice: a union bound

On the square lattice $\mathbb{Z}^2$, every edge joining two vertices at Euclidean distance $1$ is independently open with probability $p$. Bounding the probability with a union bound (sub-additivity) over paths starting at the origin, one finds that the origin lies in an infinite open cluster with probability $0$ for all $p$ in an interval $(0,d)$. What is $d$?

$d=\tfrac13$. If the origin is in an infinite cluster, some self-avoiding open path of length $n$ starts there for every $n$. There are at most $4\cdot 3^{n-1}$ such paths (4 first steps, then no backtracking), each open with probability $p^n$, so the probability is at most $\tfrac43(3p)^n\to 0$ whenever $p<\tfrac13$.