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Notebook[{

Cell[CellGroupData[{
Cell["The Gessel-Viennot Algorithm for size 2", "Subtitle"],

Cell[CellGroupData[{

Cell["\<\
Notebook 3:  Allow the Free Length to Grow
\
\>", "Section"],

Cell["\<\

Here I have created a plot of the results of the Gessel-Viennot algorithm for \
arbitrary starting points and angles and an arbitrary path length.\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Generalized Case-n Paths
\
\>", "Section"],

Cell["\<\

Try 1: Classic example. 2 paths, 0.5 alpha, one separation

This does appear to approach our limit, but rather slowly.
Note the strange \"sawtooth\" effect.  I wonder if this is just an artifact \
of incrementing the steps one at a time.\
\>", "Text"],

Cell[BoxData[{
    \(\(slopes\  = \ {0.5, \ 0.5};\)\), "\[IndentingNewLine]", 
    \(\(startPoints\  = \ {{0, \ 
            0}, \ {1, \ \(-1\)}};\)\), "\[IndentingNewLine]", 
    \(\(sizeMax\  = 5000;\)\)}], "Input"],

Cell["\<\
This algorithm takes [paths] paths, and assigns each a long-term slope shown. \
 It then finds how the entropy per unit changes with length.
\
\>", "Text"],

Cell[BoxData[{
    \(\(GetStartPoint[index_, \ 
          coord_]\  := \ \(startPoints[\([index]\)]\)[\([coord]\)];\)\), "\
\[IndentingNewLine]", 
    \(\(GetEndPoint[index_, \ coord_, length_]\  := \ 
        GetStartPoint[index, \ coord]\  + \ 
          If[coord \[Equal] 1, \ Floor[length\ slopes[\([index]\)]], \ 
            Floor[length\ \((1\  - \ 
                    slopes[\([index]\)])\)]];\)\), "\[IndentingNewLine]", 
    \(\(PathCount[length_, i_, \ j_]\  := 
        Binomial[
          GetEndPoint[j, \ 1, length] - GetStartPoint[i, \ 1] + 
            GetEndPoint[j, 2, length] - GetStartPoint[i, \ 2]\ , 
          GetEndPoint[j, \ 1, length] - 
            GetStartPoint[i, \ 1]\ ];\)\)}], "Input"],

Cell[BoxData[
    \(\(AllConfigs = 
        Array[PathCount, \ {sizeMax, Part[Dimensions[slopes], 1], 
            Part[Dimensions[slopes], 1]}];\)\)], "Input"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(dataGeneral = \ 
        Table[N[\(1\/sz\) Log[\ Det[\ \((AllConfigs[\([sz]\)]\ )\)]]], {sz, \ 
            1, \ sizeMax}];\)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(A := 
        ListPlot[dataGeneral, 
          PlotJoined \[Rule] True];\)\), "\[IndentingNewLine]", 
    \(\(B := 
        Plot[2\ Log[2], \ {x, \ 0, \ sizeMax}];\)\), "\[IndentingNewLine]", 
    \(Show[{A, \ B}]\)}], "Input"],

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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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