Shortest loop containing 0 in continuum percolation

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.

• If the two points are Euclidean distance greater than 1 apart, there are configurations with arbitrarily long shortest paths between them. So, while it makes sense to ask for the probability of a shortest path of given length for given Euclidean distance and $\lambda$, I'm not sure what you mean by an upper bound. – user25199 Apr 30 '13 at 22:22
• Well this shortest pas has a random length, say $L$. The precise fact I want to prove is that $\mathbb{E}L^{2+\epsilon}<\infty$ for some $epsilon>0$. – kaleidoscop May 1 '13 at 9:02
• Thanks - makes sense now. Do you have a heuristic argument that it should hold at the percolation threshold? – user25199 May 1 '13 at 14:51
• Good point, it looks difficult. I actually don't need that and I removed the sentence "I need a bound not depending on $\lambda$". But if I have a bound depending on $\lambda$, I would like it not to increase too much when $\lambda$ is close to the threshold. – kaleidoscop May 1 '13 at 21:00
• Maybe of some use...arxiv.org/abs/1704.06400 – Alexander Kartun-Giles Feb 2 '18 at 16:27