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Cell["Site-specific Hexagonal Walks 4", "Title",
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Cell["\<\
Nicholas Wheeler
16 January 2012\
\>", "Text",
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Cell["Introduction", "Subsection",
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Cell["\<\
In \"Hexagonal Walks\" (1 December 2011) I devised an elementary technique \
for constructing unbiased site-independent random walks on a hexagonal lattice\
\>", "Text",
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The problem is made challenging by the simple fact that the walker confronts \
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REMARK: Here I use arrow-length to signify not step-length but \
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while preserving site-independence\[Ellipsis]which proved to be easy, and \
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\[Bullet] String them together, to create walks:\
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I devised several relatively brief composite commands that are capable of \
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accommodating site-specificity, nor any of the other recursive procedures I \
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\[LongDash]ignorant as I presently am of the ",
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\"By hand\" construction of hexagonal walks with site-specific weights\
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which are displayed as red \[ScriptCapitalR]-vectors and blue \
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To construct a 100-step walk we have first to describe the site-specific \
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Cell["\<\
and then to execute (repeatedly,  to gain a sense of the situation) the \
series of 100 commands that are presented below.

REMARK: I have been unable to devise a master command that generates those \
sequential commands automatically, was forced to write them out by hand.\
\>", "Text",
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Cell["\<\
To generate this simplest of all hexagonal walks (which we have been able to \
construct by a variety of alternative means) define\
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To produce biased walks of the type illustrated in a previous figure, set\
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of which this is a selected (but\[LongDash]so diverse are \
they\[LongDash]hardly \"typical) example:\
\>", "Text",
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to indicate that all red nodes present the same site-independent set of \
unbalanced weights, ditto all blue nodes, but the red/blue nodes are \
distinctly unbalanced. \
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REMARK: Most of those figures show a similar trend. I should be possible to \
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{1, 4, 4}.\
\>", "Text",
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Assuming that the corrugations run parallel to the y-axis we expect the \
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Cell["we get densely over-written walks of the form", "Text",
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To reduce the overwrite we introduce some probability of drift up the y-axis:\
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We find that such walks tend to be bounded on left and right by \[MinusPlus]5.\
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One could extend my 100-command sequence to (say) 200 without much \
difficulty, and even to 1000 with an hour or so of tedious work. But one \
would need to find a way to automate that procedure if one were interested in \
really long walks. Relatedly, there is a limit to what one can learn from one \
or a few illustrative walks; one needs to devise a way to automate the \
construction of many walks if one wants to use a Monte Carlo technique to \
extract persistent general features of such walks.

Physically, I can imagine writing something like\
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to set up the quantum theory of electron (exiton?) motion on graphene, but do \
not see how one gets from there to a picture of electrons \"hopping\" from \
sites to nearest-neighbor sites. The methods used to study Markov processes \
(matrix-driven evolution of a stochastic vector) would seem more apt\
\[Ellipsis]the problem there being that such matrices/vectors would \
necessarily be very high dimensional. And it is not easy so see how matrix \
elements in rectangular  array are most naturally to be associated with \
lattice nodes in hexagonal array.

Finally, one needs a physically-motivated rationale for assigning structure \
to the weight functions wr[x, y] and wb[x, y].

In short, I remain doubtful that study of random walks on currugated graphene \
provides a good way to study the quantum physics of such a system.\
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