On the asymptotics of dimers on tori

We study asymptotics of the dimer model on large toric graphs. Let $\mathbb L$ be a weighted $\mathbb{Z}^2$-periodic planar graph, and let $\mathbb{Z}^2 E$ be a large-index sublattice of $\mathbb{Z}^2$. For $\mathbb L$ bipartite we show that the dimer partition function on the quotient $\mathbb{L}/(\mathbb{Z}^2 E)$ has the asymptotic expansion $\exp[A f_0 + \text{fsc} + o(1)]$, where $A$ is the area of $\mathbb{L}/(\mathbb{Z}^2 E)$, $f_0$ is the free energy density in the bulk, and $\text{fsc}$ is a finite-size correction term depending only on the conformal shape of the domain together with some parity-type information. Assuming a conjectural condition on the zero locus of the dimer characteristic polynomial, we show that an analogous expansion holds for $\mathbb{L}$ non-bipartite. The functional form of the finite-size correction differs between the two classes, but is universal within each class. Our calculations yield new information concerning the distribution of the number of loops winding around the torus in the associated double-dimer models.Comment: 48 pages, 18 figure

Few Long Lists for Edge Choosability of Planar Cubic Graphs

It is known that every loopless cubic graph is 4-edge choosable. We prove the following strengthened result. Let G be a planar cubic graph having b cut-edges. There exists a set F of at most 5b/2 edges of G with the following property. For any function L which assigns to each edge of F a set of 4 colours and which assigns to each edge in E(G)-F a set of 3 colours, the graph G has a proper edge colouring where the colour of each edge e belongs to L(e).Comment: 14 pages, 1 figur

A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in R^2, there is a line l such that in both line sets, for both halfplanes delimited by l, there are n^{1/2} lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are (n/3)^{1/2} of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erdos-Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to Graph Drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labelled lines that are universal for all n-vertex labelled planar graphs. As a side note, we prove that every set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar graphs

Quantum 2+1 evolution model

A quantum evolution model in 2+1 discrete space - time, connected with 3D fundamental map R, is investigated. Map R is derived as a map providing a zero curvature of a two dimensional lattice system called "the current system". In a special case of the local Weyl algebra for dynamical variables the map appears to be canonical one and it corresponds to known operator-valued R-matrix. The current system is a kind of the linear problem for 2+1 evolution model. A generating function for the integrals of motion for the evolution is derived with a help of the current system. The subject of the paper is rather new, and so the perspectives of further investigations are widely discussed.Comment: LaTeX, 37page

The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space \l^3

We show that a complete embedded maximal surface in the 3-dimensional Lorentz-Minkowski space $L^3$ with a finite number of singularities is, up to a Lorentzian isometry, an entire graph over any spacelike plane asymptotic to a vertical half catenoid or a horizontal plane and with conelike singular points. We study the space $G_n$ of entire maximal graphs over $\{x_3=0\}$ in $L^3$ with $n+1 \geq 2$ conelike singularities and vertical limit normal vector at infinity. We show that $G_n$ is a real analytic manifold of dimension $3n+4,$ and the coordinates are given by the position of the singular points in $R^3$ and the logarithmic growth at the end. We also introduce the moduli space $M_n$ of {\em marked} graphs with $n+1$ singular points (a mark in a graph is an ordering of its singularities), which is a $(n+1)$-sheeted covering of $G_n.$ We prove that identifying marked graphs differing by translations, rotations about a vertical axis, homotheties or symmetries about a horizontal plane, the corresponding quotient space $M_n$ is an analytic manifold of dimension $3n-1.$Comment: 32 pages, 4 figures, corrected typos, former Theorem 3.3 (now Theorem 2.2) modifie

Expected length of the longest common subsequence for large alphabets

We consider the length L of the longest common subsequence of two randomly uniformly and independently chosen n character words over a k-ary alphabet. Subadditivity arguments yield that the expected value of L, when normalized by n, converges to a constant C_k. We prove a conjecture of Sankoff and Mainville from the early 80's claiming that C_k\sqrt{k} goes to 2 as k goes to infinity.Comment: 14 pages, 1 figure, LaTe