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meaning: Given a random line segment in the random diagram, what is the most \ likely size of the cluster it is in. For example, the last diagram above has \ two clusters of sizes 1 and 143. The averarge cluster size is (1+143)/2=72. \ But most line segments are in the bigger cluster, so given a random line \ segment, it is most likely to be in the bigger cluster. The correct way to \ calculate the \"most likely cluster size\" for this diagram is ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["1", "2"], "+", SuperscriptBox["143", "2"]}], ")"}], "/", RowBox[{"(", RowBox[{"1", "+", "143"}], ")"}]}], "="}], TraditionalForm]]], "142.014. Doing this calculation for the other two diagrams gives \"most \ likely cluster size\" 7.59 and 59.08, respectively. \n\nFor sufficiently big \ (or infinite) diagrams, there is a threshold ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "c"], ",", " ", RowBox[{"somewhere", " ", "near", " ", ".5"}], ","}], TraditionalForm]]], " so that if . ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"p", ">", SubscriptBox["p", "c"]}], ",", RowBox[{"there", " ", "is", " ", "only", " ", "one", " ", "cluster"}], ",", " ", "whereas", ",", " ", "if", " "}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"p", "<", SubscriptBox["p", "c"]}], ",", " "}], TraditionalForm]]], "there are many small clusters. The number ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "c"], " ", "has", " ", "been", " ", "studied", " ", "by", " ", "many", " ", "mathematicians", " ", "for", " ", "rectangular", " ", "grids"}], TraditionalForm]]], ", as above, and also for other grids in 2 and 3 dimensions.\n\nSee \ http://mathworld.wolfram.com/PercolationThreshold.html." }], "Text"], Cell[CellGroupData[{ Cell["The project:", "Subsection"], Cell[TextData[{ "Write ", StyleBox["Mathematica", FontSlant->"Italic"], " programs to generate diagrams and calculate the \"most likely cluster size\ \" as above. You could start with a function ", StyleBox["makediagram[n_,p_]", FontWeight->"Bold"], " that makes an n\[Times]n random diagram in which each line segment appears \ with probability p. A second function could slit up the diagram into \ clusters. You will also want to write functions to display these diagrams, \ and another to generate the most likely cluster size." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Extra:", "Subsection"], Cell[TextData[{ "Just for fun...here is a 100\[Times]100 random grid with ", StyleBox["p", FontSlant->"Italic"], "=.4. Since only three colors are used to distinguish different clusters, it \ is hard to see that there are 1060 clusters in this diagram. The largest \ cluster has size 244. 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