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You are given a rectangular $n_1\times n_2$ grid with one light bulb $b_i$ at every node. Each bulb is on or off with probability $p$ and $1-p$, respectively, and furthermore you know that exactly $m$ bulbs are on. Equip the grid with the Manhattan metric $d$, and look at the set of distances between distinct bulbs which are on:

$$S:=\{d(b_i,b_j): i\neq j, b_i \textrm{ and } b_j \textrm{ are on}\}.$$

To understand how close two lit bulbs should be, I would like to compute the expected value of the random variable $X:=\min S$ (in terms of $n_1,n_2,p$ and $m$). Any hints on how to start thinking about this problem?

Thank you.

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If you demand that exactly $m$ bulbs are on, are you restricted to considering the case $p=m/(n_1n_2)$? – Chris Taylor Dec 6 '11 at 6:38
The value of $p$ is irrelevant. Since every configuration with $m$ bulbs on is equiprobable, one chooses one such configuration uniformly at random. – Did Dec 6 '11 at 6:44
Note that for each $n_1,n_2$ there is $m^*$ such that $\mathsf E X = 1$ for all $m\geq m^*$. – Ilya Dec 6 '11 at 7:48
It looks to me like a very hard problem. I would be surprised if there was a closed-form solution. – Robert Israel Dec 6 '11 at 7:50
@user17240: Expected values of a minimum, maximum or absolute value are often hard because if you write these out explicitly they have different analytic expressions for different cases that depend in a complicated manner on the relative magnitudes of the elementary variables; it's usually much easier to compute an expected value of a quantity that's defined by a single analytic expression in the elementary quantities. – joriki Jul 30 '12 at 7:30

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