Two-Dimensional Scaling Limits via Marked Nonsimple Loops

We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE(6) and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We explain how these marked loops should yield continuum versions of near-critical percolation, dynamical percolation, minimal spanning trees and related plane filling curves, and invasion percolation. We show that this yields for some of the continuum objects a conformal covariance property that generalizes the conformal invariance of critical systems. It is an open problem to rigorously construct the continuum objects and to prove that they are indeed the scaling limits of the corresponding lattice objects.Comment: 25 pages, 5 figure

Scaling limit for a drainage network model

We consider the two dimensional version of a drainage network model introduced by Gangopadhyay, Roy and Sarkar, and show that the appropriately rescaled family of its paths converges in distribution to the Brownian web. We do so by verifying the convergence criteria proposed by Fontes, Isopi, Newman and Ravishankar.Comment: 15 page

Income distribution and the evaluation of user benefits from changes in transport systems

Transport policies usually affect specific social groups in different ways. Nevertheless, distributional issues have not formed part of the mainstream of research on modelling in transport planning and evaluation. Traditionally economists have thought in terms of redistribution between income groups, often with the implicit assumption that the marginal utility of income is greater for the poor than for the rich. Nevertheless, the assumption of constancy or near constancy for the marginal utility of income has often served as a basis for using Marshallian consumer's surplus as a measure of user benefits from transport systems. In this study, a framework is developed in order to estimate the benefits provided to specific income groups under alternative investment scenarios. The commonly accepted assumptions concerning household and personal incomes in transport demand modelling are reconsidered, and alternative measures of user benefits based on the trade-off between goods and leisure are examined. One of these, the compensating variation, is compared with the traditional measure of consumer's surplus for each income category. Hypothetical but realistic scenarios are constructed for analysis. These are based on matrices of journeys and travel costs by different transport modes between over two hundred districts covering Greater London. The matrices resulted from modelling exercises relating to alternative investment policies considered in a set of studies that had been carried out in the late 1980s. When applied to these scenarios, the framework demonstrated clear differences between the policies in terms of distribution of benefits across the income spectrum. Differences were also found between the estimates of travellers' monetary valuation of changes in utility level as given by the compensating variation and consumer surplus. The results are presented separately for each income category and are discussed in terms of their distributional implications in the evaluation of transport changes in large urban areas

Growth, Inequality and Poverty: Some Empirical Evidence from Minas Gerais State, Brazil

This chapter is motivated by the fact that the Brazilian economy has one of the highest income inequality index in the world. According to Paes de Barros et al(2000), average income of the 10% richest people in Brazil is 28 times higher than the average income of the 40% poorest people. In Argentina, it is 10 times, 13 times in Costa Rica and 5 times in France. Brazilian growth did not benefit all classes and inequality is increasing since the 60´s. While the 10% richest people get 48% of total income, the 10% poorest people get 0,8% of total income. The inequality problem also arises in the Brazilian regional income analysis. Minas Gerais is a rich and dynamic state with 300.000 km2 divided into 10 different regions, 66 microregions and 853 towns. It is located in the Southeast developed part of the country and is responsible for 10% of Brazilian GDP. As the rest of Brazil, it has a dual economy with prosperity and poverty and social and economic heterogeneity. This chapter empirically analyses the economic growth and income inequality behavior in Minas Gerais towns and microregions from 1970 to 2000, using the income convergence hypothesis. Convergence tests such as Barro and Sala-i-Martin(1992), σ- convergence, Drennan & Lobo(1999) and Quah(1993) are performed. The role of human capital in growth is analysed for Minas Gerais 66 microregions. A comparison is also made between very rich regions and very poor regions of this state to see the relationship between regional inequality and poverty.

Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation

Consider a cellular automaton with state space $\{0,1 \}^{{\mathbb Z}^2}$ where the initial configuration $\omega_0$ is chosen according to a Bernoulli product measure, 1's are stable, and 0's become 1's if they are surrounded by at least three neighboring 1's. In this paper we show that the configuration $\omega_n$ at time n converges exponentially fast to a final configuration $\bar\omega$, and that the limiting measure corresponding to $\bar\omega$ is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents $\beta$, $\eta$, $\nu$ and $\gamma$, and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of ${\mathbb Z}^2$ (i.e., for independent $*$-percolation on ${\mathbb Z}^2$), we prove that the bootstrapped percolation model has the same scaling limit and critical exponents. This type of bootstrap percolation can be seen as a paradigm for a class of cellular automata whose evolution is given, at each time step, by a monotonic and nonessential enhancement.Comment: 15 page

Large deviations principle for Curie-Weiss models with random fields

In this article we consider an extension of the classical Curie-Weiss model in which the global and deterministic external magnetic field is replaced by local and random external fields which interact with each spin of the system. We prove a Large Deviations Principle for the so-called {\it magnetization per spin} $S_n/n$ with respect to the associated Gibbs measure, where $S_n/n$ is the scaled partial sum of spins. In particular, we obtain an explicit expression for the LDP rate function, which enables an extensive study of the phase diagram in some examples. It is worth mentioning that the model considered in this article covers, in particular, both the case of i.\,i.\,d.\ random external fields (also known under the name of random field Curie-Weiss models) and the case of dependent random external fields generated by e.\,g.\ Markov chains or dynamical systems.Comment: 11 page

GMO Guidelines project.

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