1. Slide 1 Branched Polymers joint work with Rick Kenyon, Brown Peter Winkler, Dartmouth

2. Slide 2 Statistical physics  Combinatorics hard-core model random independent sets monomer-dimer random matchingsbranched polymers random lattice trees Potts model random colorings linear polymers self-avoiding random walks percolation random subgraphs

3. Slide 3 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.…

4. 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.

5. Slide 5 Branched polymers in modern science artificial bloodcatalyst recovery artificial photosynthesis Q: what do random branched polymers look like?

6. 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. Parametrization To understand what a random (branched) polymer is, we must parametrize the configuration space (separately, for each combinatorial tree.)

7. Slide 7 Volume of configuration space, n=3,4 40  3 88 3 For order 3 in the plane: 3(2  )(4  /3) = 8  2

8. Slide 8 Volume of configuration space, n=5 6608  /27 4 3680  /27 4 80  /27 4

9. 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-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.

10. Slide 10 Invariance principle Theorem: The volume of the space of n-polymers in 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.

11. Slide 11 Invariance principle, quantitatively In fact: The local volume change in the ith tree due to increasing the radius of ball j is proportional to the projection of the ith edge onto a line whose angle is the mean of the two angles at ball j.    j i

12. Slide 12 Calculating the volume using invariance Let the radius of the ith ball be , for  small. i

13. Slide 13 Calculating the volume by taking a limit Thus, as  -> 0, the “inductive trees” (in which balls 2, 3 etc. are added one by one) score full volume (2  ) while the rest of the trees lose a 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” 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.

14. Slide 14 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.

15. Slide 15 This random polymer was grown in accordance with the stated scheme. Generating random polymers

16. 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 least e(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 polynomial of G, and does not depend on the edge-lengths. n-1

17. Slide 17 xxxx Polymers in dimension 3 Volume invariance does not hold, but Archimedes’ principle allows reparametrizing by projections onto x-axis and yz-plane. 321 x 54

18. 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 xxxx 321 x 54

19. Slide 19 A construction with the same distribution uniformly random rooted, labeled tree This tree is “imaginary”---not the polymer tree! edge-lengths chosen uniformly from [0,1] xxxx 321 x 54 tree laid out sideways and projected to x-axis

20. 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-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

21. Slide 21 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?

22. Slide 22 Spitzer’s Problem: solution. 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 ((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

23. 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 of leaves, or scaling limit of polymer shape? What is diameter of random n-polymer in the plane? In dimensions 4 and higher?