Define the planar percolation where each unit edge is open with probability $p$ very close to $1$.
Looking at the event where there exists a directed open path between $(0,0)$ and $(n,n)$. This event has probability close to $1$ provided for example: $p>0.9$ and $n$ is sufficiently large.
On this event, for sufficiently small $\epsilon> 0 $, I would like to give a lower bound $\delta_{\epsilon}$ for the probability of there existing a directed open path ${(0,0)\to (n,n)}$ where the number of corners is at least $\epsilon n$. The $\delta$ should depend only on $\epsilon$, and the lower bound should hold for all $n$ large and $p$ close to $1$.
Instead of corners, one could also consider there existing more than $\epsilon n$ "straight segments of length at most $K$" in a directed open path, where $K$ is some fixed large constant.
Is there a reference for this type of problem in percolation theory?
