V Conference of the Italian Society of Statistical Physics - SIFS

Complex component systems are collections of discrete units such as species, words, genes, whose observed realizations are naturally summarized by component counts. Many empirical laws have been observed in those systems, such as Taylor's law, Zipf's law, and Heaps' law. Domain-specific mechanisms are often employed to explain their emergence but, despite their ubiquity, a unifying framework remains elusive. Here we propose such framework showing that, when observables are in the form of component counts, many scaling laws emerge as generic consequences of statistical convergence in large, heterogeneous systems. Taylor's law, for instance, reflects a crossover between sampling noise and genuine system heterogeneity, while Zipf's and Heaps' laws arise from the convergence of order statistics and distinct component counts under heavy-tailed but otherwise generic priors. Moreover, the resulting scaling laws, which we predict theoretically and verify across dozens of complex component systems from biology to linguistics, are largely distribution-agnostic and often transient. Our work thus suggests that these ubiquitous patterns are better interpreted as baseline statistical limit theorems instead of fundamental principles that require tailored generative explanations.

In the context of complex networks, designing a null model that preserves certain properties of the original network is a fundamental problem. While for binary bipartite networks, the problem has been solved by the Curveball algorithm, which preserves the degree of each node exactly, for weighted bipartite networks, finding a null model that strictly preserves some features of the original network is still an open challenge. We developed a microcanonical algorithm that preserves exactly the degree and the strength of each node in a weighted bipartite network. The algorithm is based on an edge-swap procedure that combines three different moves: weight shuffling, simple edge swap, and bridge edge swap. The algorithm has been built to guarantee that the underlying transition matrix of the Monte Carlo Markov chain is symmetric, which is a crucial property to obtain an unbiased sample of random graphs. We also empirically validated our algorithm, showing that it produces an unbiased sampling of graphs. Finally, we also provide a heuristic method to estimate the mixing time of the MCMC.

Feedback between collective behavior and the environment is central to many complex systems, from climate change to financial markets, yet it is typically absent in standard evolutionary games. Here, we introduce a model in which strategies and incentives coevolve: the game played at each time is a convex combination of two social dilemmas, weighted by the current level of cooperation. This endogenous feedback generates apparently irrational regimes—chimera games—where stable cooperation emerges even when classical theory would predict instability. We further show that sufficiently large delays in the feedback induce sustained oscillations around both the equilibrium level of cooperation and the effective game, while variations in feedback strength reshape fixed points and long-term outcomes. Together, these results demonstrate how endogenous feedback fundamentally alters evolutionary predictions, opening new pathways for the emergence and persistence of cooperation.

Cities keep growing and host more than half of the global population. Despite their importance, there are no robust empirical laws describing city spatial dynamics. In [1], employing recent remote sensing data, we studied the growth of 19 cities over 30 years using population as control variable. When focusing on the largest component, we find that variations of population and built area are essentially proportional. However, when looking in a given direction, there are strong deviations to the square-root law expected in the isotropic case. By employing the framework of surface growth, we reveal that cities interface follow are anomalously intrinsically rough. Their local roughness is characterized by a scaling exponent equal to about 0.54. Other exponents are widely spread. We also observed seemingly universal, stretched-exponential fluctuations. These observations raise many physical questions; we discuss some of them concerning surface growth through strongly polydisperse deposition in [2]; a recent modified Eden based model reproducing the empirically observed results was proposed in [3]. [1] Marquis, U., Artime, O., Gallotti, R., & Barthelemy, M. (2025). Universal Roughness and the Dynamics of Urban Expansion. Phys. Rev. Lett., 135(18), 187403. [2] Marquis, U., Gallotti, R., & Barthelemy, M. (2026). Beyond dynamic scaling: rare events break universality. https://arxiv.org/abs/2604.01820 [3] Hendrick, M., Trique, M., & Manoli, G. (2026). Disorder Crossover in Urban-Front Growth. https://arxiv.org/abs/2604.21437

Complex systems tend to exhibit out-of-equilibrium dynamics over a broad range of timescales. A key theory challenge is to understand the features of this out-of-equilibrium behavior from the properties of the attractors of the system’s dynamical equations. Mean-field theories of spin glass dynamics offer elegant examples, linking phenomena such as aging to the properties of special families of stationary points of the underlying free-energy landscape, that attract the dynamics at large times. However, these insights apply mainly to systems to which one can associate such landscapes. It is an open question how to extend these ideas to high-dimensional non-conservative systems (such as biological neural networks, large ecosystems) whose dynamics is not landscape optimization. In this talk, I will present a simple model of a high-dimensional system with non-reciprocal interactions whose out-of-equilibrium, chaotic dynamics can be characterized analytically at long times. I will discuss its dynamical phase diagram and I will compare it to the statistical distribution of the many, unstable fixed points of the dynamical equations. The talk is based on joint work with Samantha Fournier, Alessandro Pacco and Pierfrancesco Urbani.

Understanding how complex networks respond to perturbations is essential to reveal their functional organization in all possible dynamical regimes. Existing information–thermodynamic approaches rely largely on diffusive descriptions, which implicitly constrain the diversity of responses that a system can exhibit. However, real-world networks often operate far from equilibrium and display rich, transient behaviors under perturbations. We show that diffusion-based frameworks impose a fundamental limitation: the entropy associated with network responses cannot increase in time, ruling out the emergence of enhanced functional diversity. To overcome this restriction, we develop a general information-thermodynamic theory of network response along arbitrary dynamical trajectories, based on a density-matrix formalism with a time-dependent evolution operator. We derive a normalization-preserving flow equation and demonstrate that it admits a variational foundation in terms of an information-theoretic least dissipation principle, establishing a direct correspondence with non-equilibrium thermodynamics. Applying the framework to synthetic and empirical networks coupled to nonlinear dynamics, we find regimes of positive entropy production that are inaccessible to diffusive approximations. Our results provide a unified theoretical basis to quantify how structure and dynamics jointly shape network responses, opening the way to trajectory-based extensions of information thermodynamics for complex systems.

We study rare events in time-correlated stochastic processes, focusing on the emergence of single big jump effects. The main part of this contribution concerns time-integrated observables in the Ornstein-Uhlenbeck (OU) process. Using an excursion-based mapping to a continuous-time random walk, we analyze the statistics of A=∫vn(t) dt where v(t) is the velocity following OU dynamics and n>0 positive integer number. We show that the large-deviation properties of A are governed by the statistics of the area accumulated during a single excursion above the origin. The associated rate function exhibits anomalous scaling and reveals a dynamical phase transition between a regime dominated by typical Gaussian fluctuations and a rare-event regime controlled by a single large excursion. In the latter case, the tail of the distribution is fully captured by a single big jump mechanism, providing a clear physical interpretation of the breakdown of standard large-deviation scaling in correlated dynamics. We conclude by briefly outlining recent extensions of the single big jump framework beyond the OU process, where perturbative corrections to the big jump principle allow one to describe moderate deviations in systems with stretched-exponential statistics, bridging typical fluctuations and extreme events.

In the study of stochastic processes, identifying the parity under time-reversal is essential to verify detailed balance, and to compute the entropy production rate (which is, otherwise, ambiguously defined). While in many cases the correct time-reversal symmetry is suggested by physical intuition, for generic processes the identification is not trivial: as a result, systems at thermal equilibrium may be mistakenly interpreted as non-equilibrium ones. We focus on the reversible deterministic dynamics of a slow variable coupled to many degrees of freedom acting as a thermal bath. We show that the time-reversal symmetry of the slow variable is preserved when passing to an effective stochastic description, independently of the nature of the bath. In turn, for generic 2-dimensional continuous Markov processes, we provide a criterion to identify the time-reversal parity rules under which the dynamics is at equilibrium (if any). The case of the Lotka-Volterra model is discussed as an example.

We study a finite one-dimensional lattice where nearest neighbour sites can exchange mass with an asymmetric local rule whose preferred di- rection is a random quenched variable à la Sinai. For any strength of the asymmetry, the total mass localizes around the minimum of a suitable poten- tial. For any realization of the disorder we obtain the stationary mass profile. For closed boundary conditions, the average over disorder profile is the arcsin law.

Neural networks are the fundamental tools of modern machine learning. They process information by loading it in the activations of the so-called input neurons: the information flows along the architecture of nested layers until the output is reached, where the decision takes place. This latter information processing can be interpreted as a discrete mapping that is ultimately shaped by the underlying network's topology. For this reason, studying neural networks from the fundamental standpoint is of interest to statistical physicists and for the complex systems' community at large. One obvious extension of the usual neural models amounts to consider hypergraphs as the backbone for the interactions. Working in this generalized setting, we will show that neural hypergraphs, if properly parameterized, have performances which are at least comparable to those reported for standard neural networks, but crucially without the need of deploying a neural activation function (usually indispensable for modern neural models). Furthermore, we will prove that, in some learning contexts, hypergraph models perform better than their network counterparts, achieving better accuracies despite having access to the same training samples.

Perturbations in disordered systems have dramatic effects both in the structure of low energy states that get completely reshuffled by small perturbations. This leads to chaotic dependence of the Gibbs state and aging reinitialization upon changes in control parameters. This fragility of the Gibbs states is strongly related to the emergence of critical Long Range Order of Spin Glass type. When two systems with independent disorders are coupled, their overlaps become similar. A crucial question is how this effect depends on the strength of the coupling between the two systems. Non-perturbative phenomena are present when 1 ~ ∆H ≪ N , being ∆H the coupling Hamiltonian and N the system size. In this talk I will consider the effect of very tiny perturbations in disordered systems and study (1) the formation of strong correlations in weakly coupled systems, (2) the fluctuations leading to size corrections in the Sherrington-Kirkpatrick model, (3) The stability of spin-glass RSB states in Dyson-lattice hierarchical spin glasses.

The scaling limit where both the size of the training set $P$ and the width $N$ of a deep neural network grow at the same rate, the so-called proportional-width regime, has been intensely studied for shallow, single-hidden-layer networks. However, extending these non-perturbative results from shallow architectures to deep non-linear networks has proven very challenging. Here we present a surprisingly effective \emph{approximate} approach to predict the generalization performance of Bayesian multi-layer perceptrons (MLPs) and of fixed depth $L$ on arbitrary high-dimensional data. We propose an equivalent Wishart Ansatz to capture the dominant stochastic fluctuations of the hierarchical empirical kernels of MLPs. This allows us to perform a large deviation analysis for the partition function of MLPs in the proportional limit, expressed in terms of a renormalized NNGP kernel. In this description, even strong representation learning in the proportional limit is encoded in at most $L$ scalar order parameters, determined self-consistently. Extending the approach to convolutional architectures (CNNs), we identify a hierarchical local kernel renormalization mechanism, which encodes more complex data-dependent transformations of the large-width kernel in CNNs due to finite-width effects. We test our effective theory against sampling experiments from the Bayesian posterior of finite deep neural networks with depths up to $L =10$ and $P~10^3$ on classic benchmark datasets, finding overall very good agreement together with two distinct types of systematic deviations.

Multitask learning, in which multiple auxiliary tasks are used to train a network to improve the generalization error of a specific task, has been used in many machine learning fields to improve network performance. We study this problem in the large-input limit of the multi-output teacher–student setting, in which the dynamics of online stochastic gradient descent are well described by a system of differential equations. We find the analytical solution for the corresponding linear model, expanding on the adaptive filters literature, and study the behavior of models trained with different activation functions, showing that dynamical effects dominate the solution.

During this session the Assembly of the Members of the Italian Society of Statistical Physics - SIFS will take place and only SIFS members can attend.

In this talk, I will review some of Amos Maritan’s many contributions to polymer physics with a special emphasis on the most recent results.

Inspired by the formation of bigels, we developed a bond percolation model that yields multiple percolating clusters in three dimensions not only at the critical point, but also above it. Our simulations suggest that, in the thermodynamic limit, there is no upper limit to the number of percolating clusters. We show that in finite systems the maximum number of percolating clusters that can be obtained grows logarithmically with the lattice size. For equal initial densities in the thermodynamic limit, all clusters percolate at the same threshold and exhibit critical exponents consistent with the critical exponents of standard percolation. The threshold depends linearly on the initial density of species and the maximum and minimum initial densities decay exponentially with the maximal number of spanning clusters. We also study a percolation model in which we occupy bonds randomly and each time a spanning cluster appears we remove it. The maximum number n_max of spanning clusters one can harvest in this way grows with the system size like a power-law with exponent d-d_f. Also, the variance of n_max and the size distribution of the remaining finite clusters grow like power-laws.

We study the rheology of dense granular materials subjected to vertical vibration using numerical simulations of a stress-imposed vane rheometer. The effective viscosity increases with confining pressure, decreases with vibration amplitude, and exhibits a non-monotonic dependence on frequency: weakening is observed at intermediate frequencies but is lost at high frequencies. We show that the rheological response is governed by the balance between grain-scale agitation energy and the stabilizing effect of confinement. This framework reconciles previously observed trends in viscosity and friction weakening and emphasizes the central role of energy injection and dissipation in determining granular flow properties under vibration.

We develop a density-functional theory for solutions of associating polymers where attractions among charged monomers (stickers) are represented by local binary degrees of freedom, which are randomly placed along the chains. Extending the original Garel and Orland's field-theoretic scheme for single-chain systems to an ensemble of interacting chains, we give the exact formulations of the model in both cases of annealed and quenched distributions of charges which we solve at the mean field level. The solution roduces qualitatively different free energy functionals. In the annealed case, the theory naturally yields a nontrivial scalar order parameter for the fraction of bonded sticker monomers and a self-consistent mass-action law at the saddle point. By contrast, In the quenched case no independent bonding order parameter emerges and the main effect is a renormalization of the effective two-body interaction parameter. The formulation is microscopic at the Hamiltonian level and, within the same field-theoretic framework, provides a systematic starting point for fluctuation corrections beyond the mean field approximation.

Particle-based simulations of dense suspensions and jammed soft solids provide unique insight into self-organization during shear-thickening and upon flow cessation. I'll discuss how spatio-temporal analysis of microscopic structures, stresses and rearrangements reveals the origin of stress transmission, stress fluctuations, and residual stresses.

Neuronal cultures exhibit complex activity characterized by bursts and avalanches, displaying a bistable coexistence of scale invariance and synchronization that remains robust across different percentages of inhibitory neurons. In this work, we analyze experimental data from human-derived neuronal cultures with controlled excitation/inhibition ratios, providing a systematic characterization of avalanche statistics and their temporal organization. To describe this phenomenology, we propose a novel Kuramoto model for two interacting populations, introducing an inhibitory feedback term that captures the effective coupling between excitatory and inhibitory neurons while preserving the key features of single-population dynamics. The model reproduces the experimental findings, including the coexistence of scale-free avalanches and synchronization, their statistical properties and temporal correlations, and the emergence of an amplification-attenuation regime.

Microbiomes are intricate, dynamic communities of microorganisms that influence nearly every aspect of life. In this talk, I will outline key challenges in understanding their collective behavior and summarize recent results suggesting universal regularities in species-abundance patterns. I will discuss how interactions and non-trivial correlations shape community-level outcomes, and how microbial interaction networks often exhibit balanced structures that appear to be disrupted in dysbiotic states associated with disease. Many fundamental questions remain open, making microbial network complexity a rich and rapidly developing field for statistical physics and complex-systems research.

The increasing availability of large-scale textual corpora has opened new possibilities for data-driven, quantitative approaches to historical analysis using Natural Language Processing (NLP). However, diachronic corpora with historical relevance from the pre-digital era remain scarce and often incomplete. We present a quantitative approach to historical analysis based on the reconstruction and exploration of a diachronic corpus of around 600,000 articles from the Italian newspaper ”La Repubblica”, covering all the articles published from the 1st of January 1985 to the 31st of December 2000 - a period of major political, social, and geopolitical change in Italy and globally. Using NLP techniques, we analyze the text at both lexical and semantic levels across multiple temporal resolutions; we then apply tools from complex systems and statistical physics to trace shifts in media discourse over time. This unsupervised approach allows us to detect key transition periods, such as the transition from the First Republic to the Second Republic in Italy, or major international conflicts like the Gulf War or the Kosovo War, without relying on prior labeling. The results show how combining computational linguistics with ideas from complex systems can offer new quantitative insight into historical changes, opening up new paths for studying the dynamics of media and society through large-scale textual data.

The remarkable predictive power of the asymptotic state theory applied to nonequilibrium quantum dynamics relies on the ability to the cast the problem in the formalism of asymptotic states of quantum field theory, a framework which allowed the general analysis of systems with short-range interactions. At the same time, numerical studies of long-range interacting spin chains revealed dynamical features — including undamped oscillations at large time — quite analogous to those observed in short-range systems, suggesting that the domain of application of the asymptotic state theory could be further extended. Despite the additional analytical difficulties posed by long-range interactions, we show for the first time that such an extension can indeed be achieved in the case of quantum quenches. Considering d-dimensional homogeneous systems, we treat the long-range interaction as switched on by the quench and perform the derivation within the basis of asymptotic states of the short-range interacting pre-quench theory. This approach allows us to analytically determine the post-quench state and the one-point functions of local observables and to characterize their large-time behavior.

We discuss the $\nu=1$ Quantum Symmetric Simple Exclusion Process, a one-dimensional system of free fermions with time-dependent, spatially uniform stochastic hopping amplitudes. We investigate the out-of-equilibrium entanglement spreading following both global and inhomogeneous quantum quenches. For global quenches, we develop a diffusive quasiparticle picture that captures the full-time probability distribution of the entanglement entropy. For inhomogeneous quenches, we extend the framework of quantum generalized hydrodynamics to account for the underlying stochastic dynamics of the local quasiparticle occupation function.

Graph states play a central role in quantum information science, providing a unifying framework for measurement–based quantum computation, quantum error correction, and the study of many–body entanglement. When a graph structure is used to prescribe a set of pairwise interactions among qubits, the resulting state captures how the connectivity pattern influences the generation and distribution of quantum correlations. Understanding how entanglement responds to structural properties of the underlying graph is therefore of broad interest, especially when the graph is drawn from an ensemble. In this talk, I will rely on the entanglement distance, an entanglement quantifier whose geometric formulation has been summarized in [1] and recently applied to quantum systems defined on graph structures [2,3], to study graph–structured quantum states with non-local couplings induced by Katz-weighted interactions. In this setting, Katz weights encode the contribution of walks at all lengths, so that indirect connectivity, cycles, triangles and other recurrent patterns can contribute to the effective couplings. This provides a natural framework to investigate how entanglement is affected by network geometry beyond the nearest-neighbour structure encoded by the original graph. I will then apply this approach to random graph ensembles and study how the entanglement profile changes across different classes of random networks. [1] De Simone, L.; Capra, L.; Vesperini, A.; Rossi, L.; Di Cairano, L.; Franzosi, R. Geometric Aspects of Entanglement. Entropy 2026, 28, 299. https://doi.org/10.3390/e28030299 [2] De Simone L and Franzosi R 2025 Journal of Physics A: Mathematical and Theoretical 58 415302 URL: https://doi.org/10.1088/1751-8121/ae0bcb [3] De Simone L and Franzosi R Advanced Quantum Technologies n/a e00514 (Preprint https://advanced.onlinelibrary.wiley.com/doi/pdf/10.1002/qute.202500514)

Quantifying the impact of hereditary transmission within lineage trees remains a fundamental challenge universal to a wide array of biological domains. In this talk, I will introduce and discuss the new concept of inheritance entropy, a quantity designed to gauge the hereditary structure of inactive cells across a lineage. We measured this entropy in 32 human stem cell clonal colonies, obtained from high-definition single-cell lineage tracing experiments, and showed that in the greatest majority of clones the entropy is decisively smaller than that of the corresponding non-hereditary ensemble, hence proving that variations in the proliferative power of stem cell lineages are determined by hereditary epigenetic factors that regulate cell-cycle exit. The method can also be employed to locate the specific node of the tree where a mutation in the probability of inactivity occurs, together with a determination of the lag between the mutation ultimately leading to inactivity and its actual expression. This framework can be used to assess in a robust, simple and model-agnostic way the hereditary origin of differential growth in any type of lineage trees.

Understanding how epigenetic patterns are established and maintained in living cells, and how these couple to 3D chromatin organisation, is a central open problem in biophysics. Here we introduce diluted chromatin-like magnetic polymer models in which only a fraction of monomers can carry epigenetic marks, while the remaining ones are neutral. This minimal polymer model captures the finite availability of epigenetic modifiers, and moves the system beyond an Ising description toward a different universality class. By combining mean-field theory and simulations, we show that dilution enables epigenetic phase separation, providing a robust physical pathway to first order transitions, which create epigenetic memory through metastability and phase coexistence. Epigenetic memory decays as a stretched exponential, revealing the coexistence of distinct timescales associated with mark spreading and chromatin dynamics. Our results highlight dilution as a key control parameter governing both the phase behavior and the memory capacity of epigenetic polymers.

We report the emergence of a novel macroscopic phase separation in a two-dimensional system of thermal inertial dumbbells driven into persistent rotation by a constant applied torque, interacting solely through repulsive forces. Through extensive molecular dynamics simulations, we map the phase diagram of the model and demonstrate that, above a threshold rotational activity, the system spontaneously demixes into a dilute liquid coexisting with a dense hexatic phase, with well-defined binodal and spinodal boundaries. By analyzing the Irving–Kirkwood pressure, we show that the underlying mechanism is fundamentally controlled by translational inertia: locally injected rotational kinetic energy is converted into translational motion at collisions and stored in the system in a density-dependent manner, producing a "hot dilute, cool dense" state. The resulting non-monotonic kinetic temperature gives rise to a Van der Waals loop in the pressure, rendering the homogeneous phase mechanically unstable. Quantitative agreement between the kinetic and Irving–Kirkwood pressures confirms that inertia is the dominant ingredient, and we show that the phase separation is progressively suppressed and disappears in the overdamped limit. Moreover, the coarsening kinetics follow the diffusive growth law R(t) ∼ t^(1/3) with dynamical scaling of the structure factor. Chirality additionally manifests through persistent edge currents along interfaces and, for isolated compact clusters, spontaneous rigid-body rotation about the center of mass with an angular velocity scaling linearly with the Péclet number — a behavior we rationalize through a simple balance of active and viscous torques.

Understanding how intelligent agents navigate competitive environments is a fundamental challenge in complex systems. We investigate the macroscopic dynamics of "strategic tension" across two structurally distinct model systems: chess and Go. We introduce a generalizable network framework to quantify this tension dynamically. In chess, tension emerges from the spatial network of interacting threats; in Go, from a network of group vulnerabilities and ownership instability under topological perturbations. In both domains, strategic tension is defined via the spectral properties of interaction matrices, rooted in topological entropy to quantify the exponential diversity of strategic trajectories. Analyzing games by human experts and state-of-the-art AI (e.g., Stockfish, AlphaGo), we reveal distinct profiles of complexity management. Engines consistently sustain significantly higher tension for longer durations than elite humans. This behavior reflects AI's capacity to defer immediate resolution and maintain high-dimensional latent options, scaling dynamically with computational depth. Conversely, human players actively simplify high-tension positions—an adaptive response to biological cognitive limits rather than a mere stylistic choice. Extending this framework from the hierarchical interactions of chess to preliminary data on the spatial dynamics of Go suggests that this divergence between artificial and biological problem-solving may be a universal phenomenon. Beyond cognitive insights, this framework enables the design of pedagogical AI tailored to human limits. Extrapolating to broader complex systems, such as economic markets or geopolitics, our findings suggest AI integration could fundamentally alter systemic stability, sustaining highly complex standoffs while drastically increasing the stakes of miscalculation.

In the first part of the talk, I will provide a pedagogical overview of recent literature focusing on quantum circuits and quantum cellular automata (QCA) as useful theoretical models in many-body physics, both in and out of equilibrium. I will highlight in particular the features that allow them to mimic the behaviour of realistic many-body systems, and explain their relevance for implementations in the quantum devices available today. In the second part of the talk, I will focus on some of my recent work on the subject, including exact results pertaining to the dynamics of entanglement in certain random circuits, and an experimental proposal to implement QCA in globally driven Rydberg-atom systems.

The quantum first-detection problem concerns the statistics of the time at which a system, subject to repeated measurements, is observed in a prescribed target state for the first time. Unlike in its classical counterpart, the measurement back action intrinsic to quantum mechanics can profoundly alter the system dynamics. Here, we show that the first-detection-time statistics for the Stark Hamiltonian is particularly sensitive to the the presence of many fermions. In particular, we show that while, in the single-particle scenario, a measurement protocol that detects the particle in its initial state strongly affects the time evolution at every measurement, in the many-particle case only measurements performed at integer multiples of the Bloch-oscillation period affect the system. We further show that measurements progressively spread the fermionic density and can therefore be used for state preparation.

In the last decades, the development of sophisticated cooling techniques has enabled the trapping of atomic gases in optical lattices, thus providing a suitable platform for the study of collective effects in many-body quantum systems. To a very good approximation, the behavior of bosonic atoms on a lattice can be described by an extended Bose–Hubbard (EBH) Hamiltonian, providing a paradigm of the superfluid-insulator transition, which has been extensively studied in the past by theory and simulations. Less attention has been paid to the behavior of the model in truncated lattices, with or without periodic boundary conditions. In this contribution we focus on the hard-core limit of the extended Bose-Hubbard model on small square and triangular grids --- i.e., sections of the square and triangular lattices containing up to 13 sites. By mapping out the zero-temperature phase diagram through exact diagonalization, we find ground-state characteristics that are markedly different from those emerging in the thermodynamic limit. The dichotomy between superfluid-like and insulating-like behavior is then investigated also in two-dimensional systems of a few interacting bosons in the continuum, subject to confining and optical-lattice potentials. Using path-integral Monte Carlo simulations, we compute kinetic and potential energies, as well as superfluid fraction and exchange-cycle statistics, finding hints of Bose-Hubbard behavior even in few-particle systems.

In recent years, Higher-Order Networks have emerged as a framework that extends the formalism of Complex Networks, enabling the description of interactions among multiple agents beyond simple pairwise relations. Examples of such higher-order structures are naturally encoded in simplicial complexes, which have been employed in a variety of contexts related to the modeling of complex systems. Here, we introduce a novel operator to describe a natural random walk process on a simplicial complex that models transitions across multidimensional structures. Walkers are allowed to wander across simplices of various dimensions, bridging nodes to edges, edges to triangles, and so on, within a nested organization that hierarchically extends to higher-order structures of arbitrarily large, yet finite, dimension. The asymptotic distribution of the walkers provides a natural ranking, enabling the assessment of the relative importance of simplices of different orders and types. We further apply this framework to compute the explorability of a simplicial complex, restricted to nodes, in the presence of a long-range stochastic teleportation term. The effect of noise appears highly beneficial for optimal search strategies when the walk occurs on higher-order structures, compared to what is observed on the corresponding binary graph. This illustrates an interesting example of the interplay between topological structures and the dynamical processes defined on them.

The study of spins and particles on graphs has applications across many areas, from time dynamics on networks to combinatorial optimization. In this talk, I will discuss the large n limit of the O(n) model on general graphs. The absence of translational invariance leads to an infinite set of saddle point constraints in the thermodynamic limit. I will show that the free energy at low and high temperatures T is controlled by two central graph-theoretic objects: the Laplacian matrix at low T and the Adjacency matrix at high T. Their interplay will be illustrated across several classes of graphs. On trees, on can obtain an exact solution in which the Lagrange multipliers depend only on the local coordination numbers. I will highlight the physical consequences using the example of a Y-junction. For graphs allowing a phase transition I will show that the singular part of the free energy is governed by the Laplacian spectrum, whereas the full free energy coincides with it only in the zero-temperature limit. I will conclude by presenting analogous results for the quantum version of the model.