On the Zeros of Plane Partition Polynomials

Chromatic Polynomials, Potts Models and All That

The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these...

Chromatic Roots are Dense in the Whole Complex Plane

I show that the zeros of the chromatic polynomials PG(q) for the generalized theta graphs Θ(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) ZG(q, v) outside the disc |q + v| < |v|. An immediate corollary is that the chr...

Some compact generalization of inequalities for polynomials with prescribed zeros

‎Let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial‎ ‎of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$‎. ‎In this paper we obtain some new results about the dependence of $|p(Rz)|$ on $|p(rz)| $ for $r^2leq rRleq k^2$‎, ‎$k^2 leq rRleq R^2$ and for $Rleq r leq k$‎. ‎Our results refine and generalize certain well-known polynomial inequalities‎.

Yang–Lee zeros of the Yang–Lee model

To understand the distribution of the Yang–Lee zeros in quantum integrable field theories we analyse the simplest of these systems given by the 2D Yang– Lee model. The grand-canonical partition function of this quantum field theory, as a function of the fugacity z and the inverse temperature β, can be computed in terms of the thermodynamics Bethe Ansatz based on its exact S-matrix. We extract t...

On the average number of sharp crossings of certain Gaussian random polynomials