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

Spectral problem on graphs and L-functions

The scattering process on multiloop infinite p+1-valent graphs (generalized trees) is studied. These graphs are discrete spaces being quotients of the uniform tree over free acting discrete subgroups of the projective group $PGL(2, {\bf Q}_p)$. As the homogeneous spaces, they are, in fact, identical to p-adic multiloop surfaces. The Ihara-Selberg L-function is associated with the finite subgraph-the reduced graph containing all loops of the generalized tree. We study the spectral problem on these graphs, for which we introduce the notion of spherical functions-eigenfunctions of a discrete Laplace operator acting on the graph. We define the S-matrix and prove its unitarity. We present a proof of the Hashimoto-Bass theorem expressing L-function of any finite (reduced) graph via determinant of a local operator $\Delta(u)$ acting on this graph and relate the S-matrix determinant to this L-function thus obtaining the analogue of the Selberg trace formula. The discrete spectrum points are also determined and classified by the L-function. Numerous examples of L-function calculations are presented.Comment: 39 pages, LaTeX, to appear in Russ. Math. Sur

Families of noncongruent numbers

Let E_k denote the elliptic curve defined by y^2 = x(x^2 - k^2). We consider the curves with k = pl, p = l = 1 mod 8 primes, and show that the density of rank-0 curves among them is at least 1/2 by explicitly constructing nontrivial elements in the 2-part of the Tate-Shafarevich group of E_k

Modular transformations of admissible N = 2 and Affine sl(2|1;C) characters

This thesis is a study of the affine super-algebra sl(2|l; C) and N = 2 superconformal algebra at fractional levels. In the first chapter we review background material on Conformal Field Theory, and how it appears in the context of string theory and the Wess - Zumino – Novikov - Witten model. We also discuss integrable and admissible representations of infinite dimensional algebras and their modular transformations. In Chapter 2 we elaborate some more on modular transformations and we derive them in the case of non - unitary minimal N = 2 characters. Some very explicit formulas are presented. In Chapter 3 we discuss character formulas for the affine sl(2|l;C) algebra and some of their general properties are given, in particular their behaviour under spectral flow. In Chapter 4 we turn to the study of sumrules for sl(2|l;C) at level k. These involve the product of sl(2) characters at level k, k', and 1 with {k + l){k' + !) = 1. We consider k + 1 = for = 1, p e Z*, u eN and show that the sumruleswe have obtained agree with the literature when the parameter p is restricted to p = 1. We use the integral form of the sumrules to study the modular properties of sl(2|l) characters at fractional level in the last section of Chapter 4.The advisor for this work has been Dr. Anne Taormina

The decomposition of the hypermetric cone into L-domains

The hypermetric cone \HYP_{n+1} is the parameter space of basic Delaunay polytopes in n-dimensional lattice. The cone \HYP_{n+1} is polyhedral; one way of seeing this is that modulo image by the covariance map \HYP_{n+1} is a finite union of L-domains, i.e., of parameter space of full Delaunay tessellations. In this paper, we study this partition of the hypermetric cone into L-domains. In particular, it is proved that the cone \HYP_{n+1} of hypermetrics on n+1 points contains exactly {1/2}n! principal L-domains. We give a detailed description of the decomposition of \HYP_{n+1} for n=2,3,4 and a computer result for n=5 (see Table \ref{TableDataHYPn}). Remarkable properties of the root system $\mathsf{D}_4$ are key for the decomposition of \HYP_5.Comment: 20 pages 2 figures, 2 table

Random Matrix Theory and the Attraction of Zeros of L-Functions From the Central Point

We explore the attraction of zeros near the central point of L-functions associated with elliptic curves and modular forms. Specifically, we consider families of twists of elliptic curves, the family of weight 2 modular forms, and the family of level 1 modular forms. We observe experimentally an attraction of the zeros near the central point, and that the attraction decreases with the rank r of the L-function. However, for each set of L-functions of rank r within a particular family we observe a statistically significant increase in the attraction as the conductors of the L-functions increase. This indicates a correspondence with the random matrix theory result about the vanishing of the distance between eigenangles near 1 as the size of the matrix increases, but also that this correspondence only exists in the limit, since we observe less attraction otherwise. Additionally, we begin preliminary investigation on a new statistic, the relationship between the value of the first zero above the central point and the value of the L-function at s = 1/2

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