Dresden 2009 – wissenschaftliches Programm

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DY: Fachverband Dynamik und Statistische Physik

DY 29: Statistical physics of complex networks

DY 29.8: Vortrag

Freitag, 27. März 2009, 12:15–12:30, ZEU 255

Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions — •Marc Timme¹, Frank van Bussel¹, Denny Fliegner², Sebastian Stolzenberg³, and Christoph Ehrlich⁴ — ¹Network Dynamics Group, MPIDS Göttingen — ²Dept. of Nonlinear Dynamics, MPIDS Göttingen — ³Dept. of Physics, Cornell University, USA — ⁴Dept. of Physics, TU Dresden

Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in mathematical graph theory, and in computer science. Counting problems, however, are among the hardest problems to access computationally. Here we suggest a novel method to access a benchmark counting problem, finding chromatic polynomials of graphs. We develop a vertex-oriented symbolic pattern matching algorithm that exploits the equivalence between the chromatic polynomial and the zero-temperature partition function of the Potts antiferromagnet on the same graph. Implementing this bottom-up algorithm using appropriate computer algebra, the new method outperforms standard top-down methods by several orders of magnitude, already for moderately sized graphs. As a first application we compute chromatic polynomials of samples of the simple cubic lattice, for the first time computationally accessing three-dimensional lattices of physical relevance. The method offers straightforward generalizations to several other counting problems.