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Math Help - expectation value of connected areas

  1. #1
    Newbie
    Joined
    May 2007
    Posts
    3

    expectation value of connected areas

    Hi!

    The problem is the following:
    We have an N-by-N square matrix of zeros. We turn M random elements of the matrix into 1-s. I need a closed formula for the expectation value of the number of connected 1-areas in the matrix (it is especially important, how it depends on M).
    An element is connected to another, when it is in its 8-neighborhood in the matrix. (If it is easier to calculate for a 4-neighborhood, then we can go with that one...). So eg.

    0 0 0
    0 1 0
    0 0 1

    These two 1-s are connected in an 8-neigborhood-way.
    In the following example, there are 2 connected areas:

    0 0 0 1 1 0
    0 0 0 0 1 0
    0 1 0 0 0 0
    0 0 1 1 0 0
    0 0 0 0 0 0

    I weren't able to solve this problem myself. Do you think there is a chance to obtain an analytic formula for this? I have already run simulations in MATLAB, here is the result:

    http://server6.pictiger.com/img/50412/other/expectation-value-of-connected-areas.php][IMG]http://images6.pictiger.com/thumbs/29/e4054ec690b745ccc2e6999648c7ba29.th.png

    I ask you for your help.

    Thank you very much.

    Kornel
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  2. #2
    Newbie
    Joined
    May 2007
    Posts
    3
    Sorry, didn't notice it is the high school section. Please close this thread, I have put my question into the college/university section. Thx
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