Branched Polymers
From the article by Richard Kenyon and Peter Winkler in the August/September
2009 issue of American Mathematical Monthly
A {\em branched polymer} is a connected configuration of non-overlapping unit balls in space.
Building on and from the work of David Brydges and John Imbrie, this article presents an
elementary calculation of the volume of the space of branched polymers of order n in the plane
and in 3-space. Our development reveals some more general identities, and allows exact random
sampling. In particular we show that a random 3-dimensional branched polymer of order n has
diameter of order . Along the way, we give the first elementary proof of Rayleigh's notorious
"random flight" theorem, which says that the probability that an n-step unit-vector random walk
in the plane ends within distance one of its starting point is 1/(n+1).
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