Slide 1

Branched Polymers

joint work with Rick Kenyon, Brown

Peter Winkler, Dartmouth

Slide 2

Statistical physics Combinatorics

hard-core modelrandom independent

sets

monomer-dimer random matchings

branched polymers random lattice trees

Potts model random colorings

linear polymers self-avoiding random walks

percolation random subgraphs

Slide 3

Grid versus Space

Some of these models were originally intended forEuclidean 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.…

Slide 4

Definition

A branched polymer is a connected set of labeled, non-overlapping unit balls in space.

This one is order 11, dimension 2.

Slide 5

Branched polymers in modern science

artificial blood catalyst recoveryartificial

photosynthesis

Q: what do random branched polymers look like?

Slide 6

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.

ParametrizationTo understand what a random (branched) polymer

is, we must parametrize the configuration space

(separately, for each combinatorial tree.)

Slide 7

Volume of configuration space, n=3,4

40

38 3

For order 3 in the plane:

3(2)(4/3) = 82

Slide 8

Volume of configuration space, n=5

6608/2743680/27

480/274

Slide 9

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.

In 3-space: vol. of order-n polymers = n (2) . n-1n-1

On the plane: vol. of order-n polymers = (n-1)!(2) . n-1

Our objectives: find elementary proof; generalize;

try to construct and understand random polymers.

Slide 10

Invariance principle

Theorem: The volume of the space of n-polymersin the plane is independent of their radii !

Proof: Calculus. The boundaries between tree-polymers are polymers with cycles; as radii change,

volume moves across these cycles and is preserved.

Slide 12

Calculating the volume using invariance

Let the radius of the ith ball be , for small. i

Slide 13

Calculating the volume by taking a limit

Thus, as -> 0 , the “inductive trees” (in whichballs 2, 3 etc. are added one by one) score full

volume (2) while the rest of the trees losea dimension and disappear.

n-1

Consequently, the total vol. of order-n polymers,

regardless of radii, is (n-1)!(2) as claimed. n-1

We are also now in a position to “grow” uniformlyrandom plane polymers one disk at a time, by

adding a tiny new disk and growing it, breakingcycles according to the volume formula.

Slide 14

Growing a random 5-polymer from a random 4

When a cycle forms, a volume-gaining tree isselected proportionately and the corresponding

edge deleted; the disk continues to grow.

Slide 15

This random polymer was

grown inaccordance

with the stated

scheme.

Generating

randompolymers

Slide 16

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 leaste(u,v), with equality over some spanning subgraph.

1

5

2 4

Theorem: The volume of the space of G-polymers

is |T(1,0)|(2) , where T is the Tutte polynomialof G, and does not depend on the edge-lengths.

n-1

Slide 17

x xxx

Polymers in dimension 3

Volume invariance does not hold, but Archimedes’principle allows reparametrizing by projections

onto x-axis and yz-plane.

321 x54

Slide 18

Distribution of projections on x-axis

Probability proportional to (i) where (i) is the number of points to the left of x within

distance 1 i

“unit-interval” graph

x xxx 321 x54

Slide 19

A construction with the same distribution

uniformly randomrooted, labeled tree

This tree is “imaginary”---not the polymer tree!

edge-lengths chosen

uniformly from [0,1]

x xxx 321 x54

tree laid out sidewaysand projected to x-

axis

Slide 20

Conclusions from the random tree construction

number of rooted, labeled trees is n (Cayley’s Theorem)

n-1

thus total volume of n-polymers

in 3-space is n (2)n-1n-1

depth of uniformly random labeled tree is order n (Szekeres’

Theorem)

1/2

thus diameter of uniformly

random n-polymer in 3-space is order n as

well.

1/2

Slide 21

Spitzer’s “random flight” problem

Theorem: Suppose you take a unit-step random walkin the plane (n steps, each a uniformly random unit

vector. Then the probability that you end withindistance 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 symmetricfunctions. Is there a combinatorial proof?

Slide 22

Spitzer’s Problem: solution.

Of these, 1 out of n+1 will break between vertex1 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 walkdoes end within distance 1 of the start point is

((2) – n(2) /(n+1))/(2) = 1/(n+1). Done! n n n

Let G be an n+1-cycle; then T(1,0) = -1+(n+1)

and thus the volume of G-polymers is n(2) . n

Slide 23

Conclusions & open questions

Combinatorics can play a useful role in statistical

physics, even when model is not moved to a grid.

Thank you for your attention!

What about other features, such as number ofleaves, or scaling limit of polymer shape?

What is diameter of random n-polymer in the plane?

In dimensions 4 and higher?