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Doctoral Thesis
DOI
Document
Author
Full name
Eric Ossami Endo
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2018
Supervisor
Committee
Proença, Rodrigo Bissacot (President)
Enter, Aernout Coert Daniel van
Fontes, Luiz Renato Goncalves
Kuelske, Christof
Ny, Arnaud Le
Title in English
Gibbs Measures for Models on Lines and Trees
Keywords in English
Cayley trees
Dyson model
g-measures
Gibbs measures
Ising model
Abstract in English
In this thesis we study various properties of the spins models, in particular, Ising and Dyson models. We study the stability of the phase transition of the nearest-neighbor ferromagnetic Ising model when we add a perturbation to the critical external field that becomes weaker far from the root of the Cayley tree. We also study the relation between g-measures and Gibbs measures, showing that the Dyson model at sufficiently low temperature is not a g-measure. Counting contours on trees is also studied, showing the characterization of the trees that have infinite number of contours, and comparisons between various definitions of contours. We also study the measures of the spatial Gibbs random graphs, and their local convergence.
Title in Portuguese
Medidas de Gibbs para modelos em retas e árvores
Keywords in Portuguese
Árvores de Cayley
g-medidas
Medidas de Gibbs
Modelo de Dyson
Modelo de Ising
Abstract in Portuguese
Nesta tese estudamos diversas propriedades dos modelos de spins, em particular, os modelos de Ising e Dyson. Estudamos a estabilidade da transição de fase no modelo de Ising ferromagnético de primeiros vizinhos quando adicionamos uma perturbação no campo externo crítico pela qual se torna mais fraca ao estar distante da raiz da árvore de Cayley. Estudamos a relação entre g-medidas e medidas de Gibbs, mostrando que a medida de Gibbs do modelo de Dyson a temperaturas suficientemente baixas não é uma g-medida. Também estudamos contagem de contornos em árvores, mostramos uma caracterização das árvores que possuem um número infinito de contornos de um tamanho fixo envolvendo um vértice, e comparamos entre diversas definições de contornos. Estudamos também as medidas de grafos aleatórios spatial Gibbs, e suas convergências locais.
 
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doutoradoEndo.pdf (1.58 Mbytes)
Publishing Date
2019-05-06
 
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