Branched Polymers

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

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

3. Some of these models were originally intended for Euclidean space, but were moved to the grid to: E.g.Bollobas-Riordan, Randall-W., Bowen-Lyons-Radin-W.… But: combinatorics can help even in space! • entice combinatorialists!? • permit simulation; • prove theorems; Grid versus Space

4. A branched polymer is a connected set of labeled, non-overlapping unit balls in space. This one is order 11, dimension 2. Definition

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

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

7. For order3in the plane: 3(2p)(4p/3) = 8p 40p 8p 3 3 2 Volume of configuration space, n=3,4

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

9. n-1 n-1 n-1 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. Our objectives: find elementary proof; generalize; try to construct and understand random polymers. On the plane:vol. of order-n polymers = (n-1)!(2p) . In 3-space:vol. of order-n polymers = n (2p) . Results of Brydges and Imbrie

10. Proof: Calculus. The boundaries between tree- polymers are polymers with cycles; as radii change, volume moves across these cycles and is preserved. Theorem: The volume of the space of n-polymers in the plane is independent of their radii ! Invariance principle

11. i Let the radius of the ith ball be e , for e small. Calculating the volume using invariance

12. n-1 n-1 Thus, as e-> 0 , the “inductive trees” (in which balls 2, 3 etc. are added one by one) score full volume (2p) while the rest of the trees lose a dimension and disappear. 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. Consequently, the total vol. of order-n polymers, regardless of radii,is(n-1)!(2p) as claimed. Calculating the volume by taking a limit

13. When a cycle forms, a volume-gaining tree is selected proportionately and the corresponding edge deleted; the disk continues to grow. Growing a random 5-polymer from a random 4

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

15. 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. Theorem: The volume of the space of G-polymers is |T(1,0)|(2p) , where T is the Tutte polynomial of G, and does not depend on the edge-lengths. 1 5 2 4 n-1 Generalization to graphs

16. x x x x x 3 2 1 5 4 Volume invariance does not hold, but Archimedes’ principle allows reparametrizing by projections onto x-axis and yz-plane. Polymers in dimension 3

17. Probability proportional to Pg(i) where g(i) is the number of points to the left of x within distance 1 “unit-interval” graph x x x x x i 3 2 1 5 4 Distribution of projections on x-axis

18. uniformly random rooted, labeled tree edge-lengths chosen uniformly from [0,1] tree laid out sideways and projected to x-axis This tree is “imaginary”---not the polymer tree! x x x x x 3 2 1 5 4 A construction with the same distribution

19. number of rooted, labeled trees is n (Cayley’s Theorem) depth of uniformly random labeled tree is order n (Szekeres’ Theorem) thus diameter of uniformly random n-polymer in 3-spaceis order n as well. thus total volume of n-polymers in 3-space is n (2p) n-1 n-1 n-1 1/2 1/2 Conclusions from the random tree construction

20. 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? Spitzer’s “random flight” problem

21. Let G be an n+1-cycle; then T(1,0) = -1+(n+1) and thus the volume of G-polymers is n(2p) . It follows that the probability that an n-step walk does end within distance 1 of the start point is ((2p) – n(2p) /(n+1))/(2p) = 1/(n+1). Done! 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. n n n n Spitzer’s Problem: solution.

22. Combinatorics can play a useful role in statistical physics, even when model is not moved to a grid. 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? Thank you for your attention! Conclusions & open questions