Indiana University-Purdue University Indianapolis (IUPUI) / Associated to any finite simple graph $\Gamma$ is the {\em chromatic polynomial} $\P_\Gamma(q)$ whose complex zeros are called the {\em chromatic zeros} of $\Gamma$. A hierarchical lattice is a sequence of finite simple graphs $\{\Gamma_n\}_{n=0}^\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\mu_n$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\mu_n$ converges to some measure $\mu$ as $n$ tends to infinity. We call $\mu$ the {\em limiting measure of chromatic zeros} associated to $\{\Gamma_n\}_{n=0}^\infty$. In the case of the Diamond Hierarchical Lattice we prove that the support of $\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.

Neste trabalho estudamos o comportamento crítico do modelo de Potts p-estados na árvore de Cayley, através das propriedades do conjunto de zeros de Yang-Lee da função de partição. Tratando a transformação do grupo de renormalização como um mapeamento racional na esfera de Riemann utiliza-se alguns resultados da teoria de Julia e Fatou para obter-se uma descrição geométrica do comportamento crítico do modelo. Mostra-se de que forma o conjunto de zeros de Yang-Lee se relaciona com o conjunto de Julia do mapa do grupo de renormalização, e calculam-se alguns parâmetros geométricos desse conjunto que descrevem o comportamento não universal do modelo. / We study the critical behavior of the p-state Potts model on a Cayley tree, looking for the properties of the Yang-Lee zeros set of the partition function. We treated the renormalization group transformation as a rational mapping on the Riemann sphere, and use some results from the Julia and Fatou theory to obtain a geometrical description of the critical properties of the model. We show how the Yang-Lee zeros set is associated with the Julia set of the renormalization group map, and we also calculate some geometrical parameters of this set which describes the non-universal behavior of the model.

Neste trabalho estudamos o comportamento crítico do modelo de Potts p-estados na árvore de Cayley, através das propriedades do conjunto de zeros de Yang-Lee da função de partição. Tratando a transformação do grupo de renormalização como um mapeamento racional na esfera de Riemann utiliza-se alguns resultados da teoria de Julia e Fatou para obter-se uma descrição geométrica do comportamento crítico do modelo. Mostra-se de que forma o conjunto de zeros de Yang-Lee se relaciona com o conjunto de Julia do mapa do grupo de renormalização, e calculam-se alguns parâmetros geométricos desse conjunto que descrevem o comportamento não universal do modelo. / We study the critical behavior of the p-state Potts model on a Cayley tree, looking for the properties of the Yang-Lee zeros set of the partition function. We treated the renormalization group transformation as a rational mapping on the Riemann sphere, and use some results from the Julia and Fatou theory to obtain a geometrical description of the critical properties of the model. We show how the Yang-Lee zeros set is associated with the Julia set of the renormalization group map, and we also calculate some geometrical parameters of this set which describes the non-universal behavior of the model.

Made available in DSpace on 2014-12-17T15:14:49Z (GMT). No. of bitstreams: 1 FranciscoAPP.pdf: 1034371 bytes, checksum: b3ff17842c3ee8ab8282b0e829786698 (MD5) Previous issue date: 2007-12-14 / The usual Ashkin-Teller (AT) model is obtained as a superposition of two Ising models coupled through a four-spin interaction term. In two dimension the AT model displays a line of fixed points along which the exponents vary continuously. On this line the model becomes soluble via a mapping onto the Baxter model. Such richness of multicritical behavior led Grest and Widom to introduce the N-color Ashkin-Teller model (N-AT). Those authors made an extensive analysis of the model thus introduced both in the isotropic as well as in the anisotropic cases by several analytical and computational methods. In the present work we define a more general version of the 3-color Ashkin-Teller model by introducing a 6-spin interaction term. We investigate the corresponding symmetry structure presented by our model in conjunction with an analysis of possible phase diagrams obtained by real space renormalization group techniques. The phase diagram are obtained at finite temperature in the region where the ferromagnetic behavior is predominant. Through the use of the transmissivities concepts we obtain the recursion relations in some periodical as well as aperiodic hierarchical lattices. In a first analysis we initially consider the two-color Ashkin-Teller model in order to obtain some results with could be used as a guide to our main purpose. In the anisotropic case the model was previously studied on the Wheatstone bridge by Claudionor Bezerra in his Master Degree dissertation. By using more appropriated computational resources we obtained isomorphic critical surfaces described in Bezerra's work but not properly identified. Besides, we also analyzed the isotropic version in an aperiodic hierarchical lattice, and we showed how the geometric fluctuations are affected by such aperiodicity and its consequences in the corresponding critical behavior. Those analysis were carried out by the use of appropriated definitions of transmissivities. Finally, we considered the modified 3-AT model with a 6-spin couplings. With the inclusion of such term the model becomes more attractive from the symmetry point of view. For some hierarchical lattices we derived general recursion relations in the anisotropic version of the model (3-AAT), from which case we can obtain the corresponding equations for the isotropic version (3-IAT). The 3-IAT was studied extensively in the whole region where the ferromagnetic couplings are dominant. The fixed points and the respective critical exponents were determined. By analyzing the attraction basins of such fixed points we were able to find the three-parameter phase diagram (temperature ? 4-spin coupling ? 6-spin coupling). We could identify fixed points corresponding to the universality class of Ising and 4- and 8-state Potts model. We also obtained a fixed point which seems to be a sort of reminiscence of a 6-state Potts fixed point as well as a possible indication of the existence of a Baxter line. Some unstable fixed points which do not belong to any aforementioned q-state Potts universality class was also found / O modelo Ashkin-Teller (AT) usual consiste na superposi??o de dois modelos de Ising acoplados por um termo de intera??o de quatro spins. Em duas dimens?es o modelo AT apresenta uma linha de pontos fixos com expoentes cr?ticos variando continuamente, sobre a qual ele se torna sol?vel atrav?s de um mapeamento no modelo Baxter. Motivado por esta riqueza de comportamento multicr?tico em duas dimens?es, Grest e Widom introduziram e estudaram o modelo Ashkin-Teller de N cores (AT-N), nas vers?es anisotr?pica (AAT-N) e isotr?pica (IAT-N), atrav?s de v?rios m?todos anal?ticos e computacionais. Neste trabalho apresentamos uma vers?o mais geral do modelo Ashkin-Teller de 3 cores (AT-3) onde e introduzido um acoplamento de 6 spins. Estudamos o modelo atrav?s da an?lise da estrutura de suas simetrias, seguido de an?lises de poss?veis diagramas de fases determinados por t?cnicas de grupo de renormaliza??o no espa?o real. Esses diagramas s?o obtidos em temperatura finita na regi?o onde predomina o comportamento ferromagn?tico. Com o aux?lio do conceito de transmissividade obtemos as rela??es de recorr?ncia em redes hier?rquicas com liga??es peri?dicas e quasi-peri?dicas. Numa an?lise preliminar, consideramos inicialmente o modelo Ashkin-Teller de duas cores, a fim de obter resultados que possam servir de guia ao nosso objetivo principal. No caso anisotr?pico (AAT-2), o modelo foi tratado na Ponte de Wheatstone, conforme j? havia sido estudado por Claudionor Bezerra na sua disserta??o de mestrado. Usando ferramentas computacionais mais adequadas, encontramos superf?cies cr?ticas isomorfas previstas no trabalho citado, mas ainda n?o identificadas explicitamente. Al?m disso, analisamos a vers?o isotr?pica (IAT-2), em uma rede hier?rquica aperi?dica. Mostramos,neste caso, como a aperiodicidade da rede afeta as flutua??es geom?tricas, causando mudan?as no comportamento cr?tico do modelo. Essas an?lises foram feitas utilizando defini??es apropriadas de transmissividade. Em seguida passamos ao estudo do modelo Ashkin-Teller de 3 cores onde, al?m do acoplamento de 4 spins, introduzimos um acoplamento de 6 spins, que torna o modelo mais atraente do ponto de vista das simetrias que ele passa a apresentar. Calculamos rela??es de recorr?ncias gerais para o modelo na vers?o anisotr?pica (AAT-3), de onde podemos obter o caso particular do sistema isotr?pico (IAT-3), em certas redes hier?rquicas. A vers?o IAT-3 do modelo foi estudada detalhadamente na regi?o onde predominam as intera??es ferromagn?ticas. Determinamos os pontos fixos e respectivos expoentes cr?ticos. Analisando as bacias de atra??o desses pontos fixos, conseguimos obter o diagrama de fases tri-dimensional (temperatura ? acoplamento de quatro spins ? acoplamento de seis spins). Identificamos pontos fixos do tipo Ising e de Potts de 4 e de 8 estados, al?m de ind?cios de um ponto fixo reminiscente do Potts de 6 estados e uma possibilidade de uma linha de Baxter. Identificamos tamb?m pontos fixos cr?ticos inst?veis que n?o pertencem a nenhuma classe de universalidade identificada com o modelo de Potts q estados