We consider the Ising model on the hexagonal lattice evolving according to Metropolis dynamics. We study its metastable behavior in the limit of vanishing temperature when the system is immersed in a small external magnetic field. We determine the asymptotic properties of the transition time from the metastable to the stable state up to a multiplicative factor and study the mixing time and the spectral gap of the Markov process. We give a geometrical description of the critical configurations and show how not only their size but their shape varies depending on the thermodynamical parameters. Finally we provide some results concerning polyiamonds of maximal area and minimal perimeter.

We study the numerical simulation of the shaken dynamics, a parallel Markovian dynamics for spin systems with local interaction and transition probabilities depending on the two parameters q and J that “tune” the geometry of the underlying lattice. The analysis of the mixing time of the Markov chain and the evaluation of the spin-spin correlations as functions of q and J, make it possible to determine in the (q, J) plane a phase transition curve separating the disordered phase from the ordered one. The relation between the equilibrium measure of the shaken dynamics and the Gibbs measure for the Ising model is also investigated. Finally two different coding approaches are considered for the implementation of the dynamics: a multicore CPU approach, coded in Julia, and a GPU approach coded with CUDA.

We define a class of Markovian parallel dynamics for spin systems on arbitrary graphs with nearest neighbor interaction described by a Hamiltonian function $$H(\sigma )$$H ( σ ) . These dynamics turn out to be reversible and their stationary measure is explicitly determined. Convergence to equilibrium and relation of the stationary measure to the usual Gibbs measure are discussed when the dynamics is defined on $$\mathbb {Z}^2$$Z 2 . Further it is shown how these dynamics can be used to define natively parallel algorithms to face problems in the context of combinatorial optimization.

Quadratic Unconstrained Binary Optimization (QUBO or UBQP) is concerned with maximizing/minimizing the quadratic form $$H(J, \eta ) = W \sum _{i,j} J_{i,j} \eta _{i} \eta _{j}$$ H ( J , η ) = W ∑ i , j J i , j η i η j with J a matrix of coefficients, $$\eta \in \{0, 1\}^N$$ η ∈ { 0 , 1 } N and W a normalizing constant. In the statistical mechanics literature, QUBO is a lattice gas counterpart to the (generalized) Sherrington–Kirkpatrick spin glass model. Finding the optima of H is an NP-hard problem. Several problems in combinatorial optimization and data analysis can be mapped to QUBO in a straightforward manner. In the combinatorial optimization literature, random instances of QUBO are often used to test the effectiveness of heuristic algorithms. Here we consider QUBO with random independent coefficients and show that if the $$J_{i,j}$$ J i , j ’s have zero mean and finite variance then, after proper normalization, the minimum and maximum per particle of H do not depend on the details of the distribution of the couplings and are concentrated around their expected values. Further, with the help of numerical simulations, we study the minimum and maximum of the objective function and provide some insight into the structure of the minimizer and the maximizer of H. We argue that also this structure is rather robust. Our findings hold also in the diluted case where each of the $$J_{i,j}$$ J i , j ’s is allowed to be zero with probability going to 1 as $$N \rightarrow \infty $$ N → ∞ in a suitable way.

The analysis and processing of satellite remote sensing data have established new paradigms to observe the Earth and to study climate changes. Data collected by sensors and systems operating across the entire frequency spectrum carry on enormous information content. They enable systematic retrieval and characterization of bio-geophysical variables. The retrieval of these variables, however, is often hindered by limitations such as missing data, low spatial and temporal resolutions, and insufficient co-located ancillary data. These challenges arise due to factors such as sensor limitations, dead pixels, cloud cover, and discontinuous data acquisition. To overcome these issues, numerous techniques have been proposed, ranging from classical interpolation methods and spectral transforms to modern artificial intelligence-based approaches. This paper presents a comprehensive and critical review of satellite remote-sensing data augmentation methods, focusing on how interpolation theory and artificial intelligence techniques can be effectively applied to reconstruct missing data and enhance the quality of retrieved bio-geophysical variables. This roadmap offers a comprehensive framework that categorizes and evaluates existing methods, highlights their strengths and limitations, and identifies promising areas for future exploration.

Human actions have accelerated changes in global temperature, precipitation patterns, and other critical Earth systems. Key markers of these changes can be linked to the dynamic of Essential Climate Variables (ECVs) and related measures, such as Soil Moisture (SM), Above Ground Biomass (AGB), and Freeze-Thaw (FT) Dynamics. ECVs are crucial for understanding global climate changes, including hydrological and carbon cycles. Moreover, monitoring ECVs helps to validate climate models and inform policy decisions. Monitoring activities can be carried out at a global scale by using technologies like microwave remote sensing. However, other than proper technological developments, the study of ECVs requires suitable theoretical retrieval tools, which leads to the solutions of inverse problems. In this survey, we analyze and summarize the main retrieval techniques available in the literature for SM, AGB, and FT, performed on data collected with microwave remote sensing sensors. Furthermore, we present the project RETINA (REmote sensing daTa INversion with multivariate functional modeling for essential climAte variables characterization), recently funded by the European Union under the Italian National Recovery and Resilience Plan of NextGenerationEU, under the Italian Ministry of University and Research. The main goal of RETINA, is to create innovative techniques for analyzing data generated by the interaction of electromagnetic waves with the Earth’s surface, applying theoretical insights to address real-world challenges.