Negative Correlation Properties for Matroids

In pursuit of negatively associated measures, this thesis focuses on certain negative correlation properties in matroids. In particular, the results presented contribute to the search for matroids which satisfy $$P(\{X:e,f\in X\}) \leq P(\{X:e\in X\})P(\{X:f\in X\})$$ for certain measures, $P$, on the ground set. Let $\mathcal M$ be a matroid. Let $(y_g:g\in E)$ be a weighting of the ground set and let $${Z = \sum_{X}\left( \prod_{x\in X} y_x\right) }$$ be the polynomial which generates Z-sets, were Z $\in \{$ B,I,S $\}$. For each of these, the sum is over bases, independent sets and spanning sets, respectively. Let $e$ and $f$ be distinct elements of $E$ and let $Z_e$ indicate partial derivative. Then $\mathcal M$ is Z-Rayleigh if $Z_eZ_f-ZZ_{ef}\geq 0$ for every positive evaluation of the $y_g$s. The known elementary results for the B, I and S-Rayleigh properties and two special cases called negative correlation and balance are proved. Furthermore, several new results are discussed. In particular, if a matroid is binary on at most nine elements or paving or rank three, then it is I-Rayleigh if it is B-Rayleigh. Sparse paving matroids are B-Rayleigh. The I-Rayleigh difference for graphs on at most seven vertices is a sum of monomials times squares of polynomials and this same special form holds for all series parallel graphs

The multivariate Tutte polynomial (alias Potts model) for graphs and matroids

The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M, and encodes much important combinatorial information about the graph (indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar two-variable Tutte polynomial -- and therefore also its one-variable specializations such as the chromatic polynomial, the flow polynomial and the reliability polynomial -- but is considerably more flexible. I begin by giving an introduction to all these problems, stressing the advantages of working with the multivariate version. I then discuss some questions concerning the complex zeros of the multivariate Tutte polynomial, along with their physical interpretations in statistical mechanics (in connection with the Yang--Lee approach to phase transitions) and electrical circuit theory. Along the way I mention numerous open problems. This survey is intended to be understandable to mathematicians with no prior knowledge of physics

Enumerative Combinatorics

Enumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds

The ZZ-invariant massive Laplacian on isoradial graphs

We introduce a one-parameter family of massive Laplacian operators $(\Delta^{m(k)})_{k\in[0,1)}$ defined on isoradial graphs, involving elliptic functions. We prove an explicit formula for the inverse of $\Delta^{m(k)}$, the massive Green function, which has the remarkable property of only depending on the local geometry of the graph, and compute its asymptotics. We study the corresponding statistical mechanics model of random rooted spanning forests. We prove an explicit local formula for an infinite volume Boltzmann measure, and for the free energy of the model. We show that the model undergoes a second order phase transition at $k=0$, thus proving that spanning trees corresponding to the Laplacian introduced by Kenyon are critical. We prove that the massive Laplacian operators $(\Delta^{m(k)})_{k\in(0,1)}$ provide a one-parameter family of $Z$-invariant rooted spanning forest models. When the isoradial graph is moreover $\mathbb{Z}^2$-periodic, we consider the spectral curve of the characteristic polynomial of the massive Laplacian. We provide an explicit parametrization of the curve and prove that it is Harnack and has genus $1$. We further show that every Harnack curve of genus $1$ with $(z,w)\leftrightarrow(z^{-1},w^{-1})$ symmetry arises from such a massive Laplacian.Comment: 71 pages, 13 figures, to appear in Inventiones mathematica

Negative correlation and log-concavity

We give counterexamples and a few positive results related to several conjectures of R. Pemantle and D. Wagner concerning negative correlation and log-concavity properties for probability measures and relations between them. Most of the negative results have also been obtained, independently but somewhat earlier, by Borcea et al. We also give short proofs of a pair of results due to Pemantle and Borcea et al.; prove that "almost exchangeable" measures satisfy the "Feder-Mihail" property, thus providing a "non-obvious" example of a class of measures for which this important property can be shown to hold; and mention some further questions.Comment: 21 pages; only minor changes since previous version; accepted for publication in Random Structures and Algorithm

Hepp's bound for Feynman graphs and matroids

We study a rational matroid invariant, obtained as the tropicalization of the Feynman period integral. It equals the volume of the polar of the matroid polytope and we give efficient formulas for its computation. This invariant is proven to respect all known identities of Feynman integrals for graphs. We observe a strong correlation between the tropical and transcendental integrals, which yields a method to approximate unknown Feynman periods.Comment: 26 figures, comments very welcom