Henrique Almeida Fernandes

Visibility graphs are spatial interpretations of time series. When derived from the time evolution of physical systems, the graphs associated with such series may exhibit properties that can reflect aspects such as ergodicity, criticality, or other dynamical behaviors. It is important to describe how the criticality of a system is manifested in the structure of the corresponding graphs or, in a particular way, in the spectra of certain matrices constructed from them. In this paper, we show how the critical behavior of an Ising spin system manifests in the spectra of the adjacency and Laplacian matrices constructed from an ensemble of time evolutions simulated via Monte Carlo (MC) Markov chains, even for small systems and short MC steps. In particular, we show that the number of spanning trees-or its logarithm-which represents a kind of structural entropy or topological complexity here obtained from Kirchhoff's theorem, can, in an alternative way, describe the criticality of the spin system. These findings parallel those obtained from the spectra of correlation matrices, which similarly encode signatures of critical and chaotic behavior.

In this work, we investigated how the use of complex networks as catalytic surfaces can affect the phase diagram of the Yaldram–Khan model, as well as how the order of the phase transitions present in the seminal work behaves when randomness is added to the model. The study was conducted by taking into consideration two well‐known random networks, the Erdős‐Rényi network (ERN), with its long‐range randomness, and the random geometric graph (RGG), with its spatially constrained randomness. We perform extensive steady‐state Monte Carlo simulations for , the NO dissociation rate, and show the behavior of the reactive window as a function of the average degree of the networks. Our results also show that, different from the ERN, which preserves the nature of the phase transitions of the original model for all considered average degrees, the RGG seems to have two second‐order phase transitions for small values of average degree.

The contact process is a nonequilibrium Hamiltonian model that, even in one dimension, lacks an exact solution and has been extensively studied via Monte Carlo simulations, both in steady-state and time-dependent scenarios. Although the effects of particle mobility and diffusion on criticality have been preliminarily explored, they remain poorly understood in many aspects. In this work, we examine how the critical rate of the model varies with the probability of particle mobility. By analyzing different stochastic evolutions of the system, we employ two modern approaches: (1) Random Matrix Theory (RMT): By building on the success of RMT, particularly Wishart-like matrices, in studying statistical physics of systems with up-down symmetry via magnetization dynamics [R. da Silva, IJMPC 2022], we demonstrate its applicability to models with an absorbing state; (2) Optimized Temporal Power Laws: By using short-time dynamics, we optimize power laws derived from ensemble-averaged evolutions of the system. Both methods consistently reveal that the critical rate decays with mobility according to a simple Belehradek function. Additionally, a straightforward mean-field analysis supports the decay of the critical parameter with mobility, although it predicts a simpler linear dependence. We also demonstrate that the more sophisticated pair approximation mean-field model developed by ben-Avraham and Köhler aligns closely with the Belehradek function, precisely matching our lattice simulation results.

In this work, we revisited the Ziff-Gullari-Barshad (ZGB) model in order to investigate its critical behavior when carbon monoxide (CO) molecules are allowed to desorb from the catalytic surface. As shown by several authors, when this kind of desorption takes place, the first-order phase transition of the standard model disappears, and an Ising-like critical point is found for a very small value of the desorption rate. However, our time-dependent Monte Carlo simulations reveal that, instead of a single critical point, there exists a critical line that appears to encompass multiple universality classes, passing through the Ising and three-state Potts points, as well as providing reasonable evidence for the four-state Potts point, resulting in an unprecedented critical line exhibiting weak universality.

In this paper, we revisit the ZGB model and explore the effects of the presence of inert sites on the catalytic surface. The continuous and discontinuous phase transitions of the model are studied via time-dependent Monte Carlo simulations. In our study, we are concerned with building a refinement procedure, based on a simple concept known as coefficient of determination r , in order to find the possible phase transition points given by the two parameters of the model: the adsorption rates of carbon monoxide, y , and the density of inert sites, . First, we obtain 10 ⁶ values of r by sweeping the whole set of possible values of the parameters with an increment 10 ⁻³ , i.e. and with . Then, with the possible phase transition points in hand, we turn our attention to some fixed values of and perform a more detailed refinement considering larger lattices and increasing the increment by one order of magnitude to estimate the critical points with higher precision. Finally, we estimate the static critical exponents , , and , as well as the dynamic critical exponents z and .