I am interersted in continuum percolation with intensity $\lambda>0$. Formally, let $X$ be a Poisson point process in $\mathbb{R}^d$ with intensity $\lambda$ and $G$ the graph obtained by connecting all vertices at distance smaller than $1$ in $X\cup\{0\}$.

I am interested in the shortest path, conditionally that it exists, between $0$ and some close point, e.g. a neighbour in Delaunay triangulation. I actually am only looking for an upper bound on the length of this path. This kind of result might exist as a by-product in the percolation literature, in which I'm not a specialist, but does not seem to be especially highlighted.

In dimension 2, it might be possible to estimate it with some fine geometrical arguments not involving percolation, but this kind of question is likely to arise in the percolation framework. Anyway even an existing bound in dimension $2$ interests me.