TITLE: Mean maximum distance for N random points on a unit square
QUESTION [9 upvotes]: Following up on Mean minimum distance for N random points on a one-dimensional line and Mean minimum distance for N random points on a unit square (plane), I have the following questions.
Given N random points in the unit square, what is the expected value of the Maximum distance between any two such points?
I know that the expected value of the distance is $\Big(2 + \sqrt{2} + 5 \log (1+ \sqrt{2}) \Big)$ and the (mysterious) distribution of the distances is given by the square line picking distribution (http://mathworld.wolfram.com/SquareLinePicking.html).
Thank you!

REPLY [2 votes]: I give a little more precised estimate from the arguments of N.T. and Matt in the case $N$ is large. 
I assume the number of point $N$ is not fixed but given by a Poisson law of mean $N$. As $N$ is large this problem sould be completely equivalent to the original one but calculations are simpler.
The extrems points are in the corners and as Matt we use a diamant to simplify the notation. We note $X_1=(1-x_1, y_1)$, $X_2=(-1+x_2, y_2)$, $X_3=(x_3, 1-y_3)$ and $X_4=(x_4, -1+y_4)$ where we expect $x_1,\cdots,y_4$ to be small. We have that 
$$\|X_1-X_2\| = \sqrt{(2 -x_1-x_2)^2+(y_1-y_2)^2}\approx (2 -x_1-x_2) $$So it is enough to consider $(1-x_1,x_2-1)$ and $(1-y_3,y_4-1)$ the extrema for the $x$ and $y$ coordinates. Because our point process is a Poisson point process of constant density $\frac{N}{2}$. Its projection on the $x$ axis is a Poisson point process of density $N(1-|x|)$. From this we deduce that for $1\geq a\geq 0$ $$\mathbb{P}(x_1\geq a) = \exp(-\int_0^a N u du) =\exp(-\frac{Na^2}{2}) $$
And we also have that $x_2, y_3,y_4$ follow this same law by symetry. Moreover $(x_1,x_2,y_3,y_4)$ are independent. We can now conclude writing $\tilde{x}=\sqrt{N}x$ that for large $N$ the maximal distance between two points behave like $$ D = 2-\frac{1}{\sqrt{N}}\min(\tilde{x}_1+\tilde{x}_2,\tilde{y}_3+\tilde{y}_4)$$ where $\tilde{x}_1,\tilde{x}_2,\tilde{y}_3,\tilde{y}_4$ are iid random variables with law $\mathbb{P}(\tilde{x}_1\geq a)=\exp{-\frac{a^2}{2}}$