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Minority languages are vital to preserving cultural heritage, yet they face growing risks of extinction due to limited digital resources and the dominance of artificial intelligence models trained on high-resource languages. This white paper proposes a framework to generate linguistic tools for low-resource languages, focusing on data creation to support the development of language models that can aid in preservation efforts. Sardinian, an endangered language, serves as the case study to demonstrate the framework's effectiveness. By addressing the data scarcity that hinders intelligent applications for such languages, we contribute to promoting linguistic diversity and support ongoing efforts in language standardization and revitalization through modern technologies.

We consider a class of non-integrable 2D Ising models obtained by perturbing the nearest-neighbor model via a weak, finite range potential which preserves translation and spin-flip symmetry, and we study its critical theory in the half-plane. We prove that the leading order long-distance behavior of the correlation functions for spins on the boundary is the same as for the nearest-neighbor model, up to an analytic multiplicative renormalization constant. In particular, the scaling limit is the Pfaffian of an explicit matrix. The proof is based on an exact representation of the generating function of correlations in terms of a Grassmann integral and on a multiscale analysis thereof, generalizing previous results to include observables located on the boundary., Comment: 59 pages, 9 figures

We consider the Renormalization Group (RG) fixed-point theory associated with a fermionic $\psi^4_d$ model in $d=1,2,3$ with fractional kinetic term, whose scaling dimension is fixed so that the quartic interaction is weakly relevant in the RG sense. The model is defined in terms of a Grassmann functional integral with interaction $V^*$, solving a fixed-point RG equation in the presence of external fields, and a fixed ultraviolet cutoff. We define and construct the field and density scale-invariant response functions, and prove that the critical exponent of the former is the naive one, while that of the latter is anomalous and analytic. We construct the corresponding (almost-)scaling operators, whose two point correlations are scale-invariant up to a remainder term, which decays like a stretched exponential at distances larger than the inverse of the ultraviolet cutoff. Our proof is based on constructive RG methods and, specifically, on a convergent tree expansion for the generating function of correlations, which generalizes the approach developed by three of the authors in a previous publication [A. Giuliani, V. Mastropietro, S. Rychkov, JHEP 01 (2021) 026]., Comment: 56 pages, 11 figures

We discuss rigorous results about the response functions of gapped and gapless interacting fermionic lattice models, relevant for the study of transport in condensed matter physics., Comment: 27 pages. Article commissioned by the Encyclopedia of Mathematical Physics

In the current digitalization era, capturing and effectively representing knowledge is crucial in most real-world scenarios. In this context, knowledge graphs represent a potent tool for retrieving and organizing a vast amount of information in a properly interconnected and interpretable structure. However, their generation is still challenging and often requires considerable human effort and domain expertise, hampering the scalability and flexibility across different application fields. This paper proposes an innovative knowledge graph generation approach that leverages the potential of the latest generative large language models, such as GPT-3.5, that can address all the main critical issues in knowledge graph building. The approach is conveyed in a pipeline that comprises novel iterative zero-shot and external knowledge-agnostic strategies in the main stages of the generation process. Our unique manifold approach may encompass significant benefits to the scientific community. In particular, the main contribution can be summarized by: (i) an innovative strategy for iteratively prompting large language models to extract relevant components of the final graph; (ii) a zero-shot strategy for each prompt, meaning that there is no need for providing examples for "guiding" the prompt result; (iii) a scalable solution, as the adoption of LLMs avoids the need for any external resources or human expertise. To assess the effectiveness of our proposed model, we performed experiments on a dataset that covered a specific domain. We claim that our proposal is a suitable solution for scalable and versatile knowledge graph construction and may be applied to different and novel contexts.

We consider classical $O(N)$ vector models in dimension three and higher and investigate the nature of the low-temperature expansions for their multipoint spin correlations. We prove that such expansions define asymptotic series, and derive explicit estimates on the error terms associated with their finite order truncations. The result applies, in particular, to the spontaneous magnetization of the 3D Heisenberg model. The proof combines a priori bounds on the moments of the local spin observables, following from reflection positivity and the infrared bound, with an integration-by-parts method applied systematically to a suitable integral representation of the correlation functions. Our method generalizes an approach, proposed originally by Bricmont and collaborators [6] in the context of the rotator model, to the case of non-abelian symmetry and non-gradient observables., Comment: Comments welcome

We consider a gas of bosons interacting through a hard-sphere potential with radius $\frak{a}$ in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term $4\pi \rho \frak{a}$ and shows that corrections are smaller than $C \rho \frak{a} (\rho \frak{a}^3)^{1/2}$, for a sufficiently large constant $C > 0$. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy is, in fact, of the order $\rho \frak{a} (\rho \frak{a}^3)^{1/2}$, in agreement with the Lee-Huang-Yang prediction., Comment: 15 pages, new version with improved rate

All-cause mortality is a very coarse grain, albeit very reliable, index to check the health implications of lifestyle determinants, systemic threats and socio-demographic factors. In this work we adopt a statistical-mechanics approach to the analysis of temporal fluctuations of all-cause mortality, focusing on the correlation structure of this index across different regions of Italy. The correlation network among the 20 Italian regions was reconstructed using temperature oscillations and travellers' flux (as a function of distance and region's attractiveness, based on GDP), allowing for a separation between infective and non-infective death causes. The proposed approach allows monitoring of emerging systemic threats in terms of anomalies of correlation network structure., Comment: For (Python + Pandas + Jax) code, see https://github.com/GuidoGigante/All-cause-mortality-fluctuation-across-Italian-regions/

We study a model of fully-packed dimer configurations (or perfect matchings) on a bipartite periodic graph that is two-dimensional but not planar. The graph is obtained from $\mathbb Z^2$ via the addition of an extensive number of extra edges that break planarity (but not bipartiteness). We prove that, if the weight $\lambda$ of the non-planar edges is small enough, a suitably defined height function scales on large distances to the Gaussian Free Field with a $\lambda$-dependent amplitude, that coincides with the anomalous exponent of dimer-dimer correlations. Because of non-planarity, Kasteleyn's theory does not apply: the model is not integrable. Rather, we map the model to a system of interacting lattice fermions in the Luttinger universality class, which we then analyze via fermionic Renormalization Group methods., Comment: 44 pages, 9 figures. Final version accepted for publication on Probability and Mathematical Physics

We review the problem of determining the ground states of 2D Ising models with nearest neighbor ferromagnetic and dipolar interactions, and prove a new result supporting the conjecture that, if the nearest neighbor coupling $J$ is sufficiently large, the ground states are periodic and `striped'. More precisely, we prove a restricted version of the conjecture, by constructing the minimizers within the variational class of states whose domain walls are arbitrary collections of horizontal and/or vertical straight lines., Comment: 23 pages, minor corrections with respect to previous version

Protein Contact Network (PCN) is a powerful tool for analysing the structure and function of proteins. In particular, PCN has been used for disclosing the molecular features of allosteric regulation through PCN clustering. Such analysis is relevant in many applications, such as the recent study of SARS-CoV-2 Spike Protein. Despite its relevance, methods for the analysis of PCN are spread into a set of different libraries and tools. Therefore, the introduction of a tool that incorporates all the function may help researchers. We present PCN-Miner a software tool implemented in the Python programming language able to import protein in the Protein Data Bank format and generate the corresponding protein contact network. Then it offers a set of algorithms for the analysis of PCS that cover a large set of applications: from clustering to embedding and subsequent analysis. Software is available at \url{https://github.com/hguzzi/ProteinContactNetworks}

After having introduced the notion of universality in statistical mechanics and its importance for our comprehension of the macroscopic behavior of interacting systems, I review recent progress in the understanding of the scaling limit of lattice critical models, including a quantitative characterization of the limiting distribution and the robustness of the limit under perturbations of the microscopic Hamiltonian. Specifically, I focus on two classes of non-exactly-solvable two-dimensional systems: non-planar Ising models and interacting dimers. In both settings, I describe the conjectures on the expected structure of the scaling limit, review the progress towards their proof, and state some of the recent results on the universality of the limit, which I contributed to. Finally, I outline the ideas and methods involved in the proofs, describe some of the perspectives opened by these results, and propose several open problems., Comment: 32 pages, submitted to the proceedings of ICM 2022; v2: typos corrected

In this paper, meant as a companion to arXiv:2006.04458, we consider a class of non-integrable $2D$ Ising models in cylindrical domains, and we discuss two key aspects of the multiscale construction of their scaling limit. In particular, we provide a detailed derivation of the Grassmann representation of the model, including a self-contained presentation of the exact solution of the nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic estimates of the fermionic Green's function in the cylinder, required for the multiscale analysis of the model. We also review the multiscale construction of the effective potentials in the infinite volume limit, in a form suitable for the generalization to finite cylinders. Compared to previous works, we introduce a few important simplifications in the localization procedure and in the iterative bounds on the kernels of the effective potentials, which are crucial for the adaptation of the construction to domains with boundaries., Comment: 69 pages. Minor revisions. Version accepted by Annales Henri Poincar\'e. arXiv admin note: substantial text overlap with arXiv:2006.04458

In the last years, scientific and industrial research has experienced a growing interest in acquiring large annotated data sets to train artificial intelligence algorithms for tackling problems in different domains. In this context, we have observed that even the market for football data has substantially grown. The analysis of football matches relies on the annotation of both individual players' and team actions, as well as the athletic performance of players. Consequently, annotating football events at a fine-grained level is a very expensive and error-prone task. Most existing semi-automatic tools for football match annotation rely on cameras and computer vision. However, those tools fall short in capturing team dynamics, and in extracting data of players who are not visible in the camera frame. To address these issues, in this manuscript we present FootApp, an AI-based system for football match annotation. First, our system relies on an advanced and mixed user interface that exploits both vocal and touch interaction. Second, the motor performance of players is captured and processed by applying machine learning algorithms to data collected from inertial sensors worn by players. Artificial intelligence techniques are then used to check the consistency of generated labels, including those regarding the physical activity of players, to automatically recognize annotation errors. Notably, we implemented a full prototype of the proposed system, performing experiments to show its effectiveness in a real-world adoption scenario.

Conformal field theory (CFT) is an extremely powerful tool for explicitly computing critical exponents and correlation functions of statistical mechanics systems at a second order phase transition, or of condensed matter systems at a quantum critical point. Conformal invariance is expected to be a feature of the fixed point theory obtained from a microscopic model at criticality, under appropriate averaging and rescaling operations: the action of the Wilsonian Renormalization Group (RG). Unfortunately, an explicit connection between critical microscopic models and their conformally invariant scaling limit is still lacking in general. Nevertheless, the last decades witnessed significant progress on this topic, both from the mathematical and physics sides, where several new tools have been introduced and their ranges of applications have constantly and significantly increased: I refer here, e.g., to discrete holomorphicity, SLE, the use of lattice Ward Identities in constructive RG, the conformal bootstrap program and its recent applications to 3D CFT. In an effort to make further progress on these problems, the one-day workshop "Emergent CFTs in statistical mechanics" was organized: the goal was to bring together probabilists, mathematical physicists and theoretical physicists, working on various aspects of critical statistical mechanics systems with complementary tools, both at the discrete and the continuum level, in the hope of creating new connections between the different approaches. This paper is based on an introductory talk given at the workshop: after a summary of the main topics discussed in the meeting, I illustrate the approach to the problem based on constructive RG methods, by reviewing recent results on the existence and the explicit characterization of the scaling limit of critical 2D Ising models with finite range interactions in cylindrical geometry., Comment: 21 pages, based on the talk given at the workshop Emergent CFTs in statistical mechanics (part of the Inhomogeneous Random Systems series), held at Institut Curie, Paris, on January 29, 2020. Compared to v1, two references have been added, others have been updated and typos have been corrected. Version accepted for publication on the Markov Processes Relat. Fields IRS 2020 special issue

Much of our understanding of critical phenomena is based on the notion of Renormalization Group (RG), but the actual determination of its fixed points is usually based on approximations and truncations, and predictions of physical quantities are often of limited accuracy. The RG fixed points can be however given a fully rigorous and non-perturbative characterization, and this is what is presented here in a model of symplectic fermions with a nonlocal ("long-range") kinetic term depending on a parameter $\varepsilon$ and a quartic interaction. We identify the Banach space of interactions, which the fixed point belongs to, and we determine it via a convergent approximation scheme. The Banach space is not limited to relevant interactions, but it contains all possible irrelevant terms with short-ranged kernels, decaying like a stretched exponential at large distances. As the model shares a number of features in common with $\phi^4$ or Ising models, the result can be used as a benchmark to test the validity of truncations and approximations in RG studies. The analysis is based on results coming from Constructive RG to which we provide a tutorial and self-contained introduction. In addition, we prove that the fixed point is analytic in $\varepsilon$, a somewhat surprising fact relying on the fermionic nature of the problem., Comment: 55 pages + appendices, many figures; v2: refs added; v3: refs added, intro greatly expanded, App. J.1 added, version accepted by JHEP; v4: minor misprints corrected

We consider a class of non-integrable 2D Ising models, whose Hamiltonian, in addition to the nearest neighbor couplings, includes weak multi-spin interactions, even under spin flip. We study the model in cylindrical domains of arbitrary aspect ratio and prove that, in the scaling limit, the multipoint energy correlations converge to the same limiting correlations as those of the nearest-neighbor Ising model in the cylinder with renormalized couplings, up to an overall multiplicative constant, independent of the shape and size of the domain. The proof is based on a representation of the generating function of correlations in terms of a non-Gaussian Grassmann integral, and a constructive Renormalization Group (RG) analysis thereof. A key technical novelty compared with previous works is a systematic analysis of the effect of the boundary corrections to the RG flow, in particular a proof that the scaling dimension of boundary operators is better by one dimension than their bulk counterparts. A cancellation mechanism based on an approximate image rule for the fermionic Green's function is of crucial importance for controlling the RG flow of the marginal boundary terms., Comment: 89 pages. Substantial revisions throughout the manuscript

Coronaviruses are a class of virus responsible of the recent outbreak of Human Severe Acute Respiratory Syndrome. The molecular machinery behind the viral entry and thus infectivity is based on the formation of the complex of virus spike protein with the angiotensin-converting enzyme 2 (ACE2). The detection of putative allosteric sites on the viral spike protein can trace the path to develop allosteric drugs to weaken the strength of the spike-ACE2 interface and, thus, reduce the viral infectivity. In this work we present results of the application of the Protein Contact Network (PCN) paradigm to the complex SARS-CoV spike - ACE2 relative to both 2003 SARS and the recent 2019 - CoV. Results point to a specific region, present in both structures, that is predicted to act as allosteric site modulating the binding of the spike protein with ACE2., Comment: 13 pages, 2 figures, 2 tables

In this paper we revisit and extend some mathematical aspects of Onsager's theory of liquid crystals that have been investigated in recent years by different communities (statistical mechanics, analysis and probability). We introduce a model of anisotropic molecules with three-dimensional orientations interacting via a Kac-type interaction. We prove that, in the limit in which the range of the interaction is sent to infinity after the thermodynamic limit, the free energy tends to the infimum of an effective energy functional \`a la Onsager. We then prove that, if the spherical harmonic transform of the angular interaction has a negative minimum, this effective free energy functional displays a first order phase transition as the total density of the system increases.

The emergence of long-range order at low temperatures in atomistic systems with continuous symmetry is a fundamental, yet poorly understood phenomenon in Physics. To address this challenge we study a discrete microscopic model for an elastic crystal with dislocations in three dimensions, previously introduced by Ariza and Ortiz. The model is rich enough to support some realistic features of three-dimensional dislocation theory, most notably grains and the Read-Shockley law for grain boundaries, which we rigorously derive in a simple, explicit, geometry. We analyze the model at positive temperatures, in terms of a Gibbs distribution with energy function given by the Ariza-Ortiz Hamiltonian plus a contribution from the dislocation cores. Our main result is that the model exhibits long range positional order at low temperatures. The proof is based on the tools of discrete exterior calculus, together with cluster expansion techniques., Comment: 54 pages, 8 figures. Final version accepted for publication on JEMS. Compared to version v2: minor corrections and new latex format

In the last few years, the methods of constructive Fermionic Renormalization Group have been successfully applied to the study of the scaling limit of several two-dimensional statistical mechanics models at the critical point, including: weakly non-integrable 2D Ising models, Ashkin-Teller, 8-Vertex, and close-packed interacting dimer models. In this note, we will focus on the illustrative example of the interacting dimer model and review some of the universality results derived in this context. In particular, we will discuss the massless Gaussian free field (GFF) behavior of the height fluctuations. It turns out that GFF behavior is connected with a remarkable identity (`Haldane' or 'Kadanoff relation') between an amplitude and an anomalous critical exponent, characterizing the large distance behavior of the dimer-dimer correlations., Comment: to appear on J. Math. Phys., special volume dedicated to the ICMP 2018. 23 pages, 3 figures

We study a class of close-packed dimer models on the square lattice, in the presence of small but extensive perturbations that make them non-determinantal. Examples include the 6-vertex model close to the free-fermion point, and the dimer model with plaquette interaction previously analyzed in \cite{A,AL,GMT17a,GMT17b}. By tuning the edge weights, we can impose a non-zero average tilt for the height function, so that the considered models are in general not symmetric under discrete rotations and reflections. In the determinantal case, height fluctuations in the massless (or `liquid') phase scale to a Gaussian log-correlated field and their amplitude is a universal constant, independent of the tilt. When the perturbation strength $\lambda$ is sufficiently small we prove, by fermionic constructive Renormalization Group methods, that log-correlations survive, with amplitude $A$ that, generically, depends non-trivially and non-universally on $\lambda$ and on the tilt. On the other hand, $A$ satisfies a universal scaling relation (`Haldane' or `Kadanoff' relation), saying that it equals the anomalous exponent of the dimer-dimer correlation., Comment: 78 pages, 5 figures. Version accepted for publication on Comm. Math. Phys. Compared to previous version, a few more details have been added in Sect.6.5 on the finite size corrections to the correlation functions, see in particular eqs.(6.132), (6.133) and following discussion

We consider a system of anisotropic plates in the three-dimensional continuum, interacting via purely hard core interactions. We assume that the particles have a finite number of allowed orientations. In a suitable range of densities, we prove the existence of a uni-axial nematic phase, characterized by long range orientational order (the minor axes are aligned parallel to each other, while the major axes are not) and no translational order. The proof is based on a coarse graining procedure, which allows us to map the plate model into a contour model, and in a rigorous control of the resulting contour theory, via Pirogov-Sinai methods., Comment: 29 pages, 4 figures

The Haldane model is a paradigmatic 2d lattice model exhibiting the integer quantum Hall effect. We consider an interacting version of the model, and prove that for short-range interactions, smaller than the bandwidth, the Hall conductivity is quantized, for all the values of the parameters outside two critical curves, across which the model undergoes a `topological' phase transition: the Hall coefficient remains integer and constant as long as we continuously deform the parameters without crossing the curves; when this happens, the Hall coefficient jumps abruptly to a different integer. Previous works were limited to the perturbative regime, in which the interaction is much smaller than the bare gap, so they were restricted to regions far from the critical lines. The non-renormalization of the Hall conductivity arises as a consequence of lattice conservation laws and of the regularity properties of the current-current correlations. Our method provides a full construction of the critical curves, which are modified (`dressed') by the electron-electron interaction. The shift of the transition curves manifests itself via apparent infrared divergences in the naive perturbative series, which we resolve via renormalization group methods., Comment: 33 pages, 1 figure