We map stock market interactions to spin models to recover their hierarchical structure using a simulated annealing based Super-Paramagnetic Clustering (SPC) algorithm. This is directly compared to a modified implementation of a maximum likelihood approach to fast-Super-Paramagnetic Clustering (f-SPC). The methods are first applied standard toy test-case problems, and then to a dataset of 447 stocks traded on the New York Stock Exchange (NYSE) over 1249 days. The signal to noise ratio of stock market correlation matrices is briefly considered. Our result recover approximately clusters representative of standard economic sectors and mixed clusters whose dynamics shine light on the adaptive nature of financial markets and raise concerns relating to the effectiveness of industry based static financial market classification in the world of real-time data-analytics. A key result is that we show that the standard maximum likelihood methods are confirmed to converge to solutions within a Super-Paramagnetic (SP) phase. We use insights arising from this to discuss the implications of using a Maximum Entropy Principle (MEP) as opposed to the Maximum Likelihood Principle (MLP) as an optimization device for this class of problems.

<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>