Is the maximum nearest neighbor distance between points of the process, over all the infinitely many points of a stationary Poisson point process of intensity $\lambda$ in $\mathbb{R}^d$, almost surely finite? What is its distribution?


I am interested in the case $d=1$ and $d=2$. Also, if it is any easier, you can replace *between points of the process* by *between an arbitrary location in $\mathbb{R}^d$ and the closest point of the process*.



Maximum nearest neighbor distance for a Poisson point process
$\newcommand\la\lambda\newcommand\de\delta\newcommand\R{\mathbb R}$This maximum (or, rather, supremum) distance, say $M$, is $\infty$ almost surely (a.s.).


Indeed, recall that a (simple) Poisson point process of intensity $\la\in(0,\infty)$ on $\R^d$ is a random Borel measure $m$ over $\R^d$ such that, for any pairwise disjoint bounded Borel subsets $A\_1,\dots,A\_k$ of $\R^d$, the random variables (r.v.'s) $m(A\_1),\dots,m(A\_k)$ are independent Poisson r.v.'s with respective parameters $\la|A\_1|,\dots,\la|A\_k|$, where $|\cdot|$ is the Lebesgue measure. It is not hard to show (see e.g. [Proposition 9.1.III (ii, iii), p. 4](https://link.springer.com/book/10.1007/978-0-387-49835-5)) that $m=\sum\_{i=1}^\infty\de\_{X\_i}$, where $\de\_x$ is the Dirac delta measure supported on $\{x\}$, for $x\in\R^d$, and
the $X\_i$'s are random points in $\R^d$ that are a.s. pairwise distinct. So,
$$M=\sup\_i\min\{\|X\_k-X\_i\|\colon k\ne i\}$$
a.s., where $\|\cdot\|$ is the Euclidean norm.


Now take any real $a>0$ and any natural $n$. Take the hypercube $C\_{a,n}:=[0,3na)^d$ and partition it naturally into $n^d$ congruent smaller hypercubes each with edgelengths $3a$. In each of these $n^d$ hypercubes $C\_j$ ($j=1,\dots,n^d$) each with edgelengths $3a$, take the central sub-hypercube, say $B\_j$, with edgelengths $a$.
Note that  

$$p:=P\big(m(B\_j)=1,m(C\_j\setminus B\_j)=0\big) \\ 
=P\big(m(B\_1)=1\big)\,P\big(m(C\_1\setminus B\_1)=0\big)>0$$
for each $j=1,\dots,n^d$.


Then the probability $P(M\ge a)$ will be no less than the probability that at least one of the $n^d$ congruent smaller hypercubes $C\_j$ contains exactly one point of the random points $X\_i$ and that one point is in $B\_j$. The latter probability is
$$1-(1-p)^{n^d}\to1$$
as $n\to\infty$. So, $P(M\ge a)\ge1$ for all real $a>0$, and hence $P(M=\infty)=1$.


