Loop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model

We use the nested loop approach to investigate loop models on random planar maps where the domains delimited by the loops are given two alternating colors, which can be assigned different local weights, hence allowing for an explicit Z_2 domain symmetry breaking. Each loop receives a non local weight n, as well as a local bending energy which controls loop turns. By a standard cluster construction that we review, the Q = n^2 Potts model on general random maps is mapped to a particular instance of this problem with domain-non-symmetric weights. We derive in full generality a set of coupled functional relations for a pair of generating series which encode the enumeration of loop configurations on maps with a boundary of a given color, and solve it by extending well-known complex analytic techniques. In the case where loops are fully-packed, we analyze in details the phase diagram of the model and derive exact equations for the position of its non-generic critical points. In particular, we underline that the critical Potts model on general random maps is not self-dual whenever Q \neq 1. In a model with domain-symmetric weights, we also show the possibility of a spontaneous domain symmetry breaking driven by the bending energy.Comment: 44 pages, 13 figure

Scalar fields: from domain walls to nanotubes and fulerenes

In this work we review some features of topological defects in field theory models for real scalar fields. We investigate topological defects in models involving one and two or more real scalar fields. In models involving a single field we examine two different subclasses of models, which support one or more topological defects. In models involving two or more real scalar fields, we explore the presence of defects that live inside topological defects, and junctions and networks of defects. In the case of junctions of defects we investigte structures that simulate nanotubes and fulerenes. Our investigations may also be used to describe nonlinear properties of polymers, Langmuir films and optical solitons in fibers.Comment: Revtex, 10 pages, 5 figures. Talk presented at XXII Encontro Nacional de Fisica de Particulas e Campos, Sao Lourenco, MG, Brazil, October 2001; v2, 2 references adde

The Lie h-Invariant Conformal Field Theories and the Lie h-Invariant Graphs

We use the Virasoro master equation to study the space of Lie h-invariant conformal field theories, which includes the standard rational conformal field theories as a small subspace. In a detailed example, we apply the general theory to characterize and study the Lie h-invariant graphs, which classify the Lie h-invariant conformal field theories of the diagonal ansatz on SO(n). The Lie characterization of these graphs is another aspect of the recently observed Lie group-theoretic structure of graph theory.Comment: 38p

The flexible coefficient multinomial logit (FC-MNL) model of demand for differentiated products

We show FC-MNL is flexible in the sense of Diewert (1974), thus its parameters can be chosen to match a well-defined class of possible own- and cross-price elasticities of demand. In contrast to models such as Probit and Random Coefficient-MNL models, FC-MNL does not require estimation via simulation; it is fully analytic. Under well-defined and testable parameter restrictions, FC-MNL is shown to be an unexplored member of McFadden’s class of Multivariate Extreme Value discrete-choice models. Therefore, FC-MNL is fully consistent with an underlying structural model of heterogeneous, utility-maximizing consumers. We provide a Monte-Carlo study to establish its properties and we illustrate the use by estimating the demand for new automobiles in Italy

Legendre-Gauss-Lobatto grids and associated nested dyadic grids

Legendre-Gauss-Lobatto (LGL) grids play a pivotal role in nodal spectral methods for the numerical solution of partial differential equations. They not only provide efficient high-order quadrature rules, but give also rise to norm equivalences that could eventually lead to efficient preconditioning techniques in high-order methods. Unfortunately, a serious obstruction to fully exploiting the potential of such concepts is the fact that LGL grids of different degree are not nested. This affects, on the one hand, the choice and analysis of suitable auxiliary spaces, when applying the auxiliary space method as a principal preconditioning paradigm, and, on the other hand, the efficient solution of the auxiliary problems. As a central remedy, we consider certain nested hierarchies of dyadic grids of locally comparable mesh size, that are in a certain sense properly associated with the LGL grids. Their actual suitability requires a subtle analysis of such grids which, in turn, relies on a number of refined properties of LGL grids. The central objective of this paper is to derive just these properties. This requires first revisiting properties of close relatives to LGL grids which are subsequently used to develop a refined analysis of LGL grids. These results allow us then to derive the relevant properties of the associated dyadic grids.Comment: 35 pages, 7 figures, 2 tables, 2 algorithms; Keywords: Legendre-Gauss-Lobatto grid, dyadic grid, graded grid, nested grid