Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions

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.Comment: 7 pages, 4 figure

Chromatic roots are dense in the whole complex plane

I show that the zeros of the chromatic polynomials P_G(q) for the generalized theta graphs \Theta^{(s,p)} are, taken together, dense in the whole complex plane with the possible exception of the disc |q-1| < 1. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z_G(q,v) outside the disc |q+v| < |v|. An immediate corollary is that the chromatic zeros of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof.Comment: LaTeX2e, 53 pages. Version 2 includes a new Appendix B. Version 3 adds a new Theorem 1.4 and a new Section 5, and makes several small improvements. To appear in Combinatorics, Probability & Computin

Chromatic roots are dense in the whole complex plane

I show that the zeros of the chromatic polynomials P-G(q) for the generalized theta graphs Theta((s.p)) are taken together, dense in the whole complex plane with the possible exception of the disc \q - l\ < l. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z(G)(q,upsilon) outside the disc \q + upsilon\ < \upsilon\. An immediate corollary is that the chromatic roots of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem oil the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof

Partition Function Zeros of an Ising Spin Glass

We study the pattern of zeros emerging from exact partition function evaluations of Ising spin glasses on conventional finite lattices of varying sizes. A large number of random bond configurations are probed in the framework of quenched averages. This study is motivated by the relationship between hierarchical lattice models whose partition function zeros fall on Julia sets and chaotic renormalization flows in such models with frustration, and by the possible connection of the latter with spin glass behaviour. In any finite volume, the simultaneous distribution of the zeros of all partition functions can be viewed as part of the more general problem of finding the location of all the zeros of a certain class of random polynomials with positive integer coefficients. Some aspects of this problem have been studied in various branches of mathematics, and we show how polynomial mappings which are used in graph theory to classify graphs, may help in characterizing the distribution of zeros. We finally discuss the possible limiting set as the volume is sent to infinity.Comment: LaTeX, 18 pages, hardcopies of 15 figures by request to [email protected], CERN--TH-7383/94 (a note and a reference added

Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions

I show that there exist universal constants $C(r) < \infty$ such that, for all loopless graphs $G$ of maximum degree $\le r$, the zeros (real or complex) of the chromatic polynomial $P_G(q)$ lie in the disc $|q| < C(r)$. Furthermore, $C(r) \le 7.963906... r$. This result is a corollary of a more general result on the zeros of the Potts-model partition function $Z_G(q, {v_e})$ in the complex antiferromagnetic regime $|1 + v_e| \le 1$. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of $Z_G(q, {v_e})$ to a polymer gas, followed by verification of the Dobrushin-Koteck\'y-Preiss condition for nonvanishing of a polymer-model partition function. I also show that, for all loopless graphs $G$ of second-largest degree $\le r$, the zeros of $P_G(q)$ lie in the disc $|q| < C(r) + 1$. Along the way, I give a simple proof of a generalized (multivariate) Brown-Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.Comment: 47 pages (LaTeX). Revised version contains slightly simplified proofs of Propositions 4.2 and 4.5. Version 3 fixes a silly error in my proof of Proposition 4.1, and adds related discussion. To appear in Combinatorics, Probability & Computin

Chromatic Zeros On Hierarchical Lattices and Equidistribution on Parameter Space

Associated to any finite simple graph $\Gamma$ is the chromatic polynomial $P_\Gamma(q)$ whose complex zeroes are called the chromatic zeros of $\Gamma$. A hierarchical lattice is a sequence of finite simple graphs $\{\Gamma_n\}_{n=0}^\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\mu_n$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\mu_n$ converges to some measure $\mu$ as $n$ tends to infinity. We call $\mu$ the limiting measure of chromatic zeros associated to $\{\Gamma_n\}_{n=0}^\infty$. In the case of the Diamond Hierarchical Lattice we prove that the support of $\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications.Comment: To appear in Annales de l'Institut Henri Poincar\'e D. We have added considerably more background on activity currents and especially on the Dujardin-Favre classification of the passive locus. Exposition in the proof of the main theorem was improved. Comments welcome