Statistical Mechanics of Monomer-Dimer Models on Complete and Erdös-Rényi Graphs

Nella tesi sono trattate due famiglie di modelli meccanico statistici su vari grafi: i modelli di spin ferromagnetici (o di Ising) e i modelli di monomero-dimero. Il primo capitolo è dedicato principalmente allo studio del lavoro di Dembo e Montanari, in cui viene risolto il modello di Ising su grafi aleatori. Nel secondo capitolo vengono studiati i modelli di monomero-dimero, a partire dal lavoro di Heilemann e Lieb,con l'intento di dare contributi nuovi alla teoria. I principali temi trattati sono disuguaglianze di correlazione, soluzioni esatte su alcuni grafi ad albero e sul grafo completo, la concentrazione dell'energia libera intorno al proprio valor medio sul grafo aleatorio diluito di Erdös-Rényi.

Modeling Ultrafast Spectroscopy of the NADH Dimer: From UV-VIS to X-rays

Ultrafast pump-probe spectroscopy is a conceptually simple and versatile tool for resolving photoinduced dynamics in molecular systems. Due to the fast development of new experimental setups, such as synchrotron light sources and X-ray free electron lasers (XFEL), new spectral windows are becoming accessible. On the one hand, these sources have enabled scientist to access faster and faster time scales and to reach unprecedent insights into dynamical properties of matter. On the other hand, the complementarity of well-developed and novel techniques allows to study the same physical process from different points of views, integrating the advantages and overcoming the limitations of each approach. In this context, it is highly desirable to reach a clear understanding of which type of spectroscopy is more suited to capture a certain facade of a given photo-induced process, that is, to establish a correlation between the process to be unraveled and the technique to be used. In this thesis, I will show how computational spectroscopy can be a tool to establish such a correlation. I will study a specific process, which is the ultrafast energy transfer in the nicotinamide adenine dinucleotide dimer (NADH). This process will be observed in different spectral windows (from UV-VIS to X-rays), accessing the ability of different spectroscopic techniques to unravel the system evolution by means of state-of-the-art theoretical models and methodologies. The comparison of different spectroscopic simulations will demonstrate their complementarity, eventually allowing to identify the type of spectroscopy that is best suited to resolve the ultrafast energy transfer.

Statistical mechanics of polymer models: rigorous results and applications

Monomer-dimer models are amongst the models in statistical mechanics which found application in many areas of science, ranging from biology to social sciences. This model describes a many-body system in which monoatomic and diatomic particles subject to hard-core interactions get deposited on a graph. In our work we provide an extension of this model to higher-order particles. The aim of our work is threefold: first we study the thermodynamic properties of the newly introduced model. We solve analytically some regular cases and find that, differently from the original, our extension admits phase transitions. Then we tackle the inverse problem, both from an analytical and numerical perspective. Finally we propose an application to aggregation phenomena in virtual messaging services.