Look at Bernoulli percolation on $\mathbb{Z}^2$ with $p> p_c$ ($p$ can be arbitrarily close to 1).

I am interested in the probability that there exists an infinite cluster starting at $(0,0)$ and it stays above the graph $y=x^n$ where $n$ is a large even number.

Is there a relation between $p$ and $n$ so that this probability is bounded from below?

I would also hope that a similar result holds for $\mathbb Z^d$. And we can relax the condition on the "cone" to be its horizontal width grows to infinity.