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Branched Polymers. joint work with Rick Kenyon, Brown. Peter Winkler, Dartmouth. 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.
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Branched Polymers joint work with Rick Kenyon, Brown Peter Winkler, Dartmouth
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
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
A branched polymer is a connected set of labeled, non-overlapping unit balls in space. This one is order 11, dimension 2. Definition
artificial photosynthesis artificial blood catalyst recovery Q: what do random branched polymers look like? Branched polymers in modern science
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
For order3in the plane: 3(2p)(4p/3) = 8p 40p 8p 3 3 2 Volume of configuration space, n=3,4
6608p /27 3680p /27 80p /27 4 4 4 Volume of configuration space, n=5
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
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
i Let the radius of the ith ball be e , for e small. Calculating the volume using invariance
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
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
This random polymer was grown in accordance with the stated scheme. Generating random polymers
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
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
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
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
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
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
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.
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