Crossover effects on the phase transitions phenomena translated by arborecences and spectral properties

This study investigates how visibility graphs constructed from Monte Carlo Markov Chain time series of spin models capture the critical behavior of the system. More precisely, we show that this approach identifies continuous phase transitions as well as important nuances, such as crossover effects occurring in the transition from a critical line to a first-order line through a tricritical point, as observed, for example, in the Blume–Emery–Griffiths model or, in a simpler setting, in the Blume–Capel model. By applying Kirchhoff’s theorem, we show that the number of spanning trees of the resulting graphs serves as a sensitive indicator of these phase transitions. Furthermore, a qualitative analysis of the adjacency matrices based on random matrix theory provides additional evidence for these phenomena. The methodology developed here can potentially be extended to the analysis of criticality in empirical time series from complex systems, such as climate, financial, and epidemiological data, where the Hamiltonian governing the dynamics is not necessarily known.