|
SPEAKER: |
|
|
TITLE: |
"Random Polymers and Weakly Interacting Lattice Trees" |
|
DAY: |
Wednesday, September 30, 1998 |
|
TIME: |
3:30 p.m. [Coffee in BSB-202 at 3:00 p.m.] |
|
PLACE: |
BSB-108 |
Polymers are long molecules consisting of many building blocks. Polymers have two characteristic properties. The first is that they are irregular, because there are different possibilities for the angles between the building blocks. The second is that they try to avoid self-intersections because of polarization of the building blocks or the so-called excluded-volume-effect. An important property of a polymer is its functionality, the number of connections the building blocks can make. If this functionality is two, we speak of a linear polymer, if it is higher than two, we speak of a branched polymer.
Probabilistic models for polymers are based on lattice random walks or lattice branching random walks with a self-repellent interaction. The paths of these processes model the configuration of the polymer in space. The (branching) random walk models the irregularity, the self-repellence penalizes self-intersections. We study the Domb-Joyce model based on simple random walk as a model for a linear polymer. The Domb-Joyce model is a generalization of the self-avoiding walk where self-intersections are unlikely, but not impossible. Furthermore, we investigate a model of weakly interacting lattice trees as a model for a branched polymer. A lattice tree is a finite connected set of bonds that does not contain any cycle.
We are interested in the behaviour of the end-to-end distance of the polymer as the number of building blocks increases. The end-to-end distance gives an indication of what the spatial extent of the polymer is as a function of the number of building blocks.
Dr Remco van der Hofstad received his Master's and PhD degrees from the University of Utrecht (The Netherlands). He spent time as a post-doctoral fellow at McMaster University, where he worked with Dr Gordon Slade. He also worked at the Theory Group of Microsoft Research. He has just started at the Delft University of Technology as an assistant professor, and is currently visiting the Fields Institute working in the Probability program. His research interests are in statistical physical models, like polymer models.
The following articles have been provided by Dr van der Hofstad to be used as background for his talk. At least two of them will be placed on reserve at Thode Library (STATS 770: Statistics Seminar).
[1] R. van der Hofstad and F. den Hollander (1995), "Scaling for a Random Polymer," COMMUNICATIONS OF MATHEMATICAL PHYSICS 169, pp. 397-440.
[2] R. van der Hofstad, F. den Hollander and G. Slade (1998), "A New Inductive Approach to the Lace Expansion," PROBABILITY THEORY AND RELATED FIELDS 111, pp. 253-286.
[3] C. Borgs, J. Chayes, R. van der Hofstad and G. Slade (1998), "Mean-Field Lattice Trees," submitted to ANNALS OF COMBINATORICS.
[4] R. van der Hofstad (1998), "One-Dimensional Random Polymers," Chapter 1 of PhD Thesis.
[5] N. Madras and G. Slade (1993), THE SELF-AVOIDING WALK, Birkhauser, Boston.