Branched
Polymers
Peter Winkler, Dartmouth
joint work with
Rick Kenyon, Brown
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Statistical physics  Combinatorics
hard-core model
monomer-dimer
Potts model
percolation
linear polymers
branched polymers
random independent sets
random matchings
random colorings
random subgraphs
self-avoiding
random walks
random lattice trees
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Grid versus Space
Some of these models were originally intended for
Euclidean space, but were moved to the grid to:
•permit simulation;
•prove theorems;
•entice combinatorialists!?
But: combinatorics can help even in space!
E.g. Bollobas-Riordan, Randall-W., Bowen-Lyons-Radin-W.…
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Definition
A branched polymer is a connected set of
labeled, non-overlapping unit balls in space.
This one is order 11, dimension 2.
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Branched polymers in modern science
artificial blood
catalyst recovery
artificial
photosynthesis
Q: what do random branched polymers look like?
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Parametrization
To understand what a random (branched) polymer is,
we must parametrize the configuration space
(separately, for each combinatorial tree.)
Fortunately, there is a natural way to do this: anchor ball
#1 at the origin, and consider the (spherical) angle made
by each ball with the ball it touches on the way to ball #1.
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Volume of configuration space, n=3,4
For order 3 in the plane:
3(2p)(4p/3) = 8p 2
8p 3
40p 3
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Volume of configuration space, n=5
80p 4/27
3680p 4/27
6608p 4/27
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Results of Brydges and Imbrie
Using methods such as localization and equivariant
cohomology, Brydges and Imbrie [’03] proved a
deep connection between branched polymers in
dimension D+2 and the hard-core model in dimension D.
They get exact formulas for the volume of the space
of branched polymers in dimensions 2 and 3.
On the plane: vol. of order-n polymers = (n-1)!(2p)
In 3-space: vol. of order-n polymers = n n-1 (2p)
n-1
n-1
.
.
Our objectives: find elementary proof; generalize;
try to construct and understand random polymers.
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Invariance principle
Theorem: The volume of the space of n-polymers
in the plane is independent of their radii !
Proof: Calculus. The boundaries between treepolymers are polymers with cycles; as radii change,
volume moves across these cycles and is preserved.
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Calculating the volume using invariance
Let the radius of the ith ball be e i , for e small.
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Calculating the volume by taking a limit
Thus, as e -> 0 , the “inductive trees” (in which
balls 2, 3 etc. are added one by one) score full
n-1
volume (2p) while the rest of the trees lose
a dimension and disappear.
Consequently, the total vol. of order-n polymers,
regardless of radii, is (n-1)!(2p) n-1 as claimed.
We are also now in a position to “grow” uniformly
random plane polymers one disk at a time, by
adding a tiny new disk and growing it, breaking
cycles according to the volume formula.
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Growing a random 5-polymer from a random 4
When a cycle forms, a volume-gaining tree is
selected proportionately and the corresponding
edge deleted; the disk continues to grow.
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Generating
random
polymers
This random
polymer was
grown in
accordance
with the
stated scheme.
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Generalization to graphs
Definition: Let G be a graph with edge-lengths e.
A G-polymer is an embedding of V(G) in the plane
such that for every edge u,v, d(u,v) is at least
e(u,v), with equality over some spanning subgraph.
5
2
1
4
Theorem: The volume of the space of G-polymers
n-1
is |T(1,0)|(2p) , where T is the Tutte polynomial
of G, and does not depend on the edge-lengths.
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Polymers in dimension 3
Volume invariance does not hold, but Archimedes’
principle allows reparametrizing by projections
onto x-axis and yz-plane.
x1
x2 x 3 x4
x5
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Distribution of projections on x-axis
x1
x2 x 3 x4
x5
“unit-interval” graph
Probability proportional to Pg(i) where g(i) is the
number of points to the left of x i within distance 1
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A construction with the same distribution
edge-lengths chosen
uniformly from [0,1]
uniformly random
rooted, labeled tree
x1
x2 x 3 x4
x5
tree laid out sideways
and projected to x-axis
This tree is “imaginary”---not the polymer tree!
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Conclusions from the random tree construction
depth of uniformly random
labeled tree is order n 1/2
(Szekeres’ Theorem)
number of rooted,
labeled trees is n n-1
(Cayley’s Theorem)
thus diameter of uniformly
random n-polymer in 3-space
is order n 1/2 as well.
thus total volume of n-polymers
in 3-space is n n-1 (2p)
n-1
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Spitzer’s “random flight” problem
Theorem: Suppose you take a unit-step random walk
in the plane (n steps, each a uniformly random unit
vector. Then the probability that you end within
distance 1 of your starting point is exactly 1/(n+1).
Problem: Proving this is a notoriously difficult;
Spitzer suggests developing a theory of
Fourier transforms of spherically symmetric
functions. Is there a combinatorial proof?
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Spitzer’s Problem: solution.
Let G be an n+1-cycle; then T(1,0) = -1+(n+1)
n
and thus the volume of G-polymers is n(2p) .
Of these, 1 out of n+1 will break between vertex
1 and vertex n+1; these represent the walks that
end at distance at least 1 from the start point.
It follows that the probability that an n-step walk
does end within distance 1 of the start point is
n
n
n
((2p) – n(2p) /(n+1))/(2p) = 1/(n+1). Done!
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Conclusions & open questions
Combinatorics can play a useful role in statistical
physics, even when model is not moved to a grid.
What is diameter of random n-polymer in the plane?
In dimensions 4 and higher?
What about other features, such as number of
leaves, or scaling limit of polymer shape?
Thank you for your attention!
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