For $1\leq i \leq n$, Let $X_i\sim \text{Uniform}(0,1)$, $Y_i \sim \text{Uniform}(0,1)$ be $n$ points chosen uniformly in the unit square. Denote the $k$-th smallest pairwise distances across the $n$ points as $d_k (1\leq k \leq {n\choose 2})$. I am interested in literature on:

1. $\mathbb{E}(d_k)$, or at least a good bound on $\mathbb{E}(d_k)$
2. The concentration of $d_k$ around $\mathbb{E}(d_k)$

For $1$, I was able to get the expectation for some $k$s such as $k=1,{n\choose 2}$, and some intermediate $k$ by complicated integrals. However, I can't seem to find a generalization of this.

For $2$, my "computational experiments" indicate that $d_k$ is extremely concentrated around its expectation, but I'm clueless on a tail bound that might be useful to prove this considering the dependence of $d_k$ variables (I played with Talagrand and Chernoff but both do not work).

Any ideas?

For $1\leq i \leq n$, let $X_i\sim \text{Uniform}(0,1)$, $Y_i \sim \text{Uniform}(0,1)$ be $n$ points chosen uniformly in the unit square. Denote the $k$-th smallest pairwise distances across the $n$ points as $d_k$ ($1\leq k \leq {n\choose 2}$). I am interested in literature on:

1. $\mathbb{E}(d_k)$, or at least a good bound on $\mathbb{E}(d_k)$.
2. The concentration of $d_k$ around $\mathbb{E}(d_k)$.

For 1, I was able to get the expectation for some $k$s such as $k=1,{n\choose 2}$, and some intermediate $k$ by complicated integrals. However, I can't seem to find a generalization of this.

For 2, my "computational experiments" indicate that $d_k$ is extremely concentrated around its expectation, but I'm clueless on a tail bound that might be useful to prove this considering the dependence of $d_k$ variables (I played with Talagrand and Chernoff but both do not work).

Any ideas?

For $1\leq i \leq n$, Let $X_i\sim Uniform(0,1), Y_i \sim Uniform(0,1)$ be $n$ points chosen uniformly in the unit square. Denote the $k-$th smallest pairwise distances across the $n$ points as $d_k (1\leq k \leq {n\choose 2})$. I am interested in literature on:

1. $\mathbb{E}(d_k)$, or at least a good bound on $\mathbb{E}(d_k)$
2. The concentration of $d_k$ around $\mathbb{E}(d_k)$

For $1$, I was able to get the expectation for some $k$s such as $k=1,{n\choose 2}$, and some intermediate $k$ by complicated integrals. However, I can't seem to find a generalization of this.

For $2$, my "computational experiments" indicate that $d_k$ is extremely concentrated around its expectation, but I'm clueless on a tail bound that might be useful to prove this considering the dependence of $d_k$ variables (I played with Talagrand and Chernoff but both do not work).

Any ideas?

For $1\leq i \leq n$, Let $X_i\sim \text{Uniform}(0,1)$, $Y_i \sim \text{Uniform}(0,1)$ be $n$ points chosen uniformly in the unit square. Denote the $k$-th smallest pairwise distances across the $n$ points as $d_k (1\leq k \leq {n\choose 2})$. I am interested in literature on:

1. $\mathbb{E}(d_k)$, or at least a good bound on $\mathbb{E}(d_k)$
2. The concentration of $d_k$ around $\mathbb{E}(d_k)$

For $1$, I was able to get the expectation for some $k$s such as $k=1,{n\choose 2}$, and some intermediate $k$ by complicated integrals. However, I can't seem to find a generalization of this.

For $2$, my "computational experiments" indicate that $d_k$ is extremely concentrated around its expectation, but I'm clueless on a tail bound that might be useful to prove this considering the dependence of $d_k$ variables (I played with Talagrand and Chernoff but both do not work).

Any ideas?