Kac-Ward formula and its extension to order-disorder correlators through a graph zeta function

A streamlined derivation of the Kac-Ward formula for the planar Ising model's partition function is presented and applied in relating the kernel of the Kac-Ward matrices' inverse with the correlation functions of the Ising model's order-disorder correlation functions. A shortcut for both is facilitated by the Bowen-Lanford graph zeta function relation. The Kac-Ward relation is also extended here to produce a family of non planar interactions on $\mathbb{Z}^2$ for which the partition function and the order-disorder correlators are solvable at special values of the coupling parameters/temperature.Comment: An extension of the Kac-Ward determinantal formula beyond planarity was added (Section 5). To appear in Journal of Statistical Physic

Spanning trees of 3-uniform hypergraphs

Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and related to the class of Pfaffian graphs. We prove a complexity result for recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian 3-graphs -- one of these is given by a forbidden subgraph characterization analogous to Little's for bipartite Pfaffian graphs, and the other consists of a class of partial Steiner triple systems for which the property of being 3-Pfaffian can be reduced to the property of an associated graph being Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are not 3-Pfaffian, none of which can be reduced to any other by deletion or contraction of triples. We also find some necessary or sufficient conditions for the existence of a spanning tree of a 3-graph (much more succinct than can be obtained by the currently fastest polynomial-time algorithm of Gabow and Stallmann for finding a spanning tree) and a superexponential lower bound on the number of spanning trees of a Steiner triple system.Comment: 34 pages, 9 figure

Topological graph polynomials and quantum field theory, Part II: Mehler kernel theories

We define a new topological polynomial extending the Bollobas-Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behavior under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials, occurring in the parametric representation of the non-commutative Grosse-Wulkenhaar quantum field theory. An explicit solution of the parametric representation for commutative field theories based on the Mehler kernel is also provided.Comment: 58 pages, 23 figures, correction in the references and addition of preprint number

On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings

We study two variants of the well-known orthogonal drawing model: (i) the smooth orthogonal, and (ii) the octilinear. Both models form an extension of the orthogonal, by supporting one additional type of edge segments (circular arcs and diagonal segments, respectively). For planar graphs of max-degree 4, we analyze relationships between the graph classes that can be drawn bendless in the two models and we also prove NP-hardness for a restricted version of the bendless drawing problem for both models. For planar graphs of higher degree, we present an algorithm that produces bi-monotone smooth orthogonal drawings with at most two segments per edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

On the breakdown of perturbative integrability in large N matrix models

We study the perturbative integrability of the planar sector of a massive SU(N) matrix quantum mechanical theory with global SO(6) invariance and Yang-Mills-like interaction. This model arises as a consistent truncation of maximally supersymmetric Yang-Mills theory on a three-sphere to the lowest modes of the scalar fields. In fact, our studies mimic the current investigations concerning the integrability properties of this gauge theory. Like in the field theory we can prove the planar integrability of the SO(6) model at first perturbative order. At higher orders we restrict ourselves to the widely studied SU(2) subsector spanned by two complexified scalar fields of the theory. We show that our toy model satisfies all commonly studied integrability requirements such as degeneracies in the spectrum, existence of conserved charges and factorized scattering up to third perturbative order. These are the same qualitative features as the ones found in super Yang-Mills theory, which were enough to conjecture the all-loop integrability of that theory. For the SO(6) model, however, we show that these properties are not sufficient to predict higher loop integrability. In fact, we explicitly demonstrate the breakdown of perturbative integrability at fourth order.Comment: 27 page