Neste trabalho estabelecemos o Teorema Central do Limite para o modelo O(N) de Heisenberg hierárquico na criticalidade via equação a derivadas parciais no limite N -> infinito. Por simplicidade consideramos apenas o caso d = 4, sendo o teorema também válido para d > 4. Pelo estudo de uma dada equação a derivadas parciais (EDP) determinamos a temperatura inversa crítica do modelo esférico hierárquico contínuo para um d > 2 qualquer, havendo conexão entre criticalidade e o ponto fixo da EDP. Por meio de uma análise geométrica da trajetória crítica obtemos informações sobre a dinâmica e distribuição dos zeros de Lee-Yang. / In this work we stablish the Central Limit Theorem for the hierarchical O(N) Heisenberg model at criticality via partial differential equation in the limit N -> infinity. For simplicity we only treat the d = 4 case but the theorem is still valid for d > 4. By studying a given partial differential equation (PDE) we determine for any d > 2 the critical inverse temperature of the continuum hierarchical spherical model, and we show a connection between criticality and the fixed point of PDE. By means of a geometric analysis of the critical trajectory we obtain some informations about Lee-Yang zeros´s dynamics and distribution.

Neste trabalho estabelecemos o Teorema Central do Limite para o modelo O(N) de Heisenberg hierárquico na criticalidade via equação a derivadas parciais no limite N -> infinito. Por simplicidade consideramos apenas o caso d = 4, sendo o teorema também válido para d > 4. Pelo estudo de uma dada equação a derivadas parciais (EDP) determinamos a temperatura inversa crítica do modelo esférico hierárquico contínuo para um d > 2 qualquer, havendo conexão entre criticalidade e o ponto fixo da EDP. Por meio de uma análise geométrica da trajetória crítica obtemos informações sobre a dinâmica e distribuição dos zeros de Lee-Yang. / In this work we stablish the Central Limit Theorem for the hierarchical O(N) Heisenberg model at criticality via partial differential equation in the limit N -> infinity. For simplicity we only treat the d = 4 case but the theorem is still valid for d > 4. By studying a given partial differential equation (PDE) we determine for any d > 2 the critical inverse temperature of the continuum hierarchical spherical model, and we show a connection between criticality and the fixed point of PDE. By means of a geometric analysis of the critical trajectory we obtain some informations about Lee-Yang zeros´s dynamics and distribution.

<p dir="ltr">The Ising model is one of the most important theoretical models in statistical physics, which was originally developed to describe ferromagnetism. A system of magnetic particles, for example, can be modeled as a linear chain in one dimension or a lattice in two dimensions, with one particle at each lattice point. Then each particle is assigned a spin $\sigma_i\in \{\pm 1\}$. The $q$-state Potts model is a generalization of the Ising model, where each spin $\sigma_i$ may take on $q\geq 3$ a number of states $\{0,\cdots, q-1\}$. Both models have temperature $T$ and an externally applied magnetic field $h$ as parameters. Many statistical and physical properties of the $q$-~state Potts model can be derived by studying its partition function. This includes phase transitions as $T$ and/or $h$ are varied.</p><p><br></p><p dir="ltr">The celebrated \textit{Lee-Yang Theorem} characterizes such phase transitions of the $2$-state Potts model (the Ising model). This theorem does not hold for $q>2$. Thus, phase transitions for the Potts model as $h$ is varied are more complicated and mysterious. We give some results that characterize the phase transitions of the $3$-state Potts model as $h$ is varied for constant $T$ on the binary rooted Cayley tree. Similarly to the Ising model, we show that for fixed $T>0$ the $3$-state Potts model for the ferromagnetic case exhibits a phase transition at one critical value of $h$ or not at all, depending on $T$. However, an interesting new phenomenon occurs for the $3$-state Potts model because the critical value of $h$ can be non-zero for some range of temperatures. The $3$-state Potts model for the antiferromagnetic case exhibits a phase transition at up to two critical values of $h$. </p><p><br></p><p dir="ltr">The recursive constructions of the $(n+1)^{st}$ level Cayley tree from two copies of the $n^{th}$ level Cayley tree allows one to write a relatively simple rational function relating the Lee-Yang zeros at one level to the next. This allows us to use techniques from dynamical systems.</p>