A Combinatorial Interpretation of the Free Fermion Condition of the Six-Vertex Model

The free fermion condition of the six-vertex model provides a 5 parameter sub-manifold on which the Bethe Ansatz equations for the wavenumbers that enter into the eigenfunctions of the transfer matrices of the model decouple, hence allowing explicit solutions. Such conditions arose originally in early field-theoretic S-matrix approaches. Here we provide a combinatorial explanation for the condition in terms of a generalised Gessel-Viennot involution. By doing so we extend the use of the Gessel-Viennot theorem, originally devised for non-intersecting walks only, to a special weighted type of \emph{intersecting} walk, and hence express the partition function of $N$ such walks starting and finishing at fixed endpoints in terms of the single walk partition functions

Two operators on sandpile configurations, the sandpile model on the complete bipartite graph, and a Cyclic Lemma

We introduce two operators on stable configurations of the sandpile model that provide an algorithmic bijection between recurrent and parking configurations. This bijection preserves their equivalence classes with respect to the sandpile group. The study of these operators in the special case of the complete bipartite graph ${K}_{m,n}$ naturally leads to a generalization of the well known Cyclic Lemma of Dvoretsky and Motzkin, via pairs of periodic bi-infinite paths in the plane having slightly different slopes. We achieve our results by interpreting the action of these operators as an action on a point in the grid $\mathbb{Z}^2$ which is pointed to by one of these pairs of paths. Our Cyclic lemma allows us to enumerate several classes of polyominoes, and therefore builds on the work of Irving and Rattan (2009), Chapman et al. (2009), and Bonin et al. (2003).Comment: 28 page

Higher connectivity of fiber graphs of Gr\"obner bases

Fiber graphs of Gr\"obner bases from contingency tables are important in statistical hypothesis testing, where one studies random walks on these graphs using the Metropolis-Hastings algorithm. The connectivity of the graphs has implications on how fast the algorithm converges. In this paper, we study a class of fiber graphs with elementary combinatorial techniques and provide results that support a recent conjecture of Engstr\"om: the connectivity is given by the minimum vertex degree.Comment: 18 pages. Minor revision

Spacetime Approach to Phase Transitions

In these notes, the application of Feynman's sum-over-paths approach to thermal phase transitions is discussed. The paradigm of such a spacetime approach to critical phenomena is provided by the high-temperature expansion of spin models. This expansion, known as the hopping expansion in the context of lattice field theory, yields a geometric description of the phase transition in these models, with the thermal critical exponents being determined by the fractal structure of the high-temperature graphs. The graphs percolate at the thermal critical point and can be studied using purely geometrical observables known from percolation theory. Besides the phase transition in spin models and in the closely related $\phi^4$ theory, other transitions discussed from this perspective include Bose-Einstein condensation, and the transitions in the Higgs model and the pure U(1) gauge theory.Comment: 59 pages, 18 figures. Write-up of Ising Lectures presented at the National Academy of Sciences, Lviv, Ukraine, 2004. 2nd version: corrected typo

Monopole currents and Dirac sheets in U(1) lattice gauge theory

We show that the phases of the 4-dimensional compact U(1) lattice gauge theory are unambiguously characterized by the topological properties of minimal Dirac sheets as well as of monopole currents lines. We obtain the minimal sheets by a simulated-annealing procedure. Our results indicate that the equivalence classes of sheet structures are the physical relevant quantities and that intersections are not important. In conclusion we get a percolation-type view of the phases which holds beyond the particular boundary conditions used.Comment: 13 pages, latex, 5 figures include