arXiv subjects

Pierluigi Contucci

Publications and source records attributed to Pierluigi Contucci.

At least 19 recordsLinked to original sources

arXiv2026

The solution of the Hopfield model with Gaussian patterns

We prove that the free energy of the Hopfield model with Gaussian patterns has a thermodynamic limit at every load $α$ and temperature $T$, and we give it as an explicit variational principle. The Hubbard--Stratonovich transformation maps the model onto $N$ Ising spins coupled to $M=αN$ Gaussian hidden units, whose radius, the empirical variance of the hidden layer, is a thermodynamic variable with an entropic cost. At fixed radius the model is an Ising--spherical spin glass with Gaussian couplings, which we solve. The principle (i) reproduces the Amit-Gutfreund-Sompolinsky theory as its replica-symmetric restriction, (ii) identifies $T_g=1+\sqrtα$ as the rigorous boundary of the annealed region, (iii) selects a unique spin-glass solution of the AGS equations, and (iv) gives the radius an equation of state. Moreover, (v) below $T_g$ we prove that replica symmetry is broken throughout: the instability of the AGS solution reduces to the zero-field de Almeida-Thouless condition of the Sherrington-Kirkpatrick model.

math-ph↗

arXiv2026

Testing Transformer Learnability on the Arithmetic Sequence of Rooted Trees

We study whether a transformer network can learn the deterministic sequence of trees generated by the iterated prime factorization of the natural numbers. Each integer is mapped into a rooted planar tree and the resulting sequence $\mathbb{N}\mathscr{T}$ defines an arithmetic text with measurable statistical structure. A transformer network (the GPT-2 architecture) is trained from scratch on the first $10^{11}$ elements and evaluated on next-token prediction and masked-token prediction tasks, with a Hidden Markov Model as baseline and a scaling analysis over context window, dataset size, vocabulary size and model size. The model reaches a word accuracy of about $0.4$, well above the baseline, and its performance remains stable on test blocks located at $10^{13}$-$10^{15}$, far beyond the training interval. Moreover, the likelihood assigned by the trained model separates the arithmetic text from two controls: synthetic token sequences that preserve the marginal token frequencies but not their order, with a separation that widens with the context, and sequences containing more than three consecutive square-free integers, a configuration that arithmetic forbids. These results indicate that the transformer captures regularities of the arithmetic text that go beyond its frequency profile.

cs.AI↗

arXiv2026

From entropic constraints to reinforced processes: a probabilistic origin of multiscale measures

We study multiscale measures, defined by a hierarchy of partial Gibbs equilibria, from variational and probabilistic perspectives. We first show that they are the unique solutions of a generalized maximum-entropy principle in which conditional entropies at different levels carry different weights. We then construct a hierarchical reinforced multinomial process proving a large-deviation principle for its empirical measure. The stochastic process leaves the typical empirical distribution unchanged but modifies the exponential cost of its fluctuations. The resulting rate function is a weighted sum of conditional Kullback-Leibler divergences, establishing a direct link with the entropic principle above. The relation between the reinforcement and Poisson-Dirichlet processes is discussed.

math-ph↗

arXiv2026

Variational Bounds for Perceptron Learning from Structured Data

We introduce a variational approach to a finite-temperature continuous-spin perceptron trained on a Gaussian mixture. The model allows for a broad class of concave utilities and log-concave separable prior measures on the spins. By combining the interpolation method with log-concavity and concentration estimates, we derive lower and upper minimax variational bounds for the limiting quenched pressure. Remarkably, the two bounds differ only in the order of optimization of two variational parameters, while all remaining extrema are controlled by the concave--convex structure of the variational potential. Whenever the two optimizations commute, the two bounds match and identify the solution of the model. The same potential yields the fixed-point equations as stationarity conditions and provides a unified route to the computation of the ground-state energy, training loss, and generalization error.

cs.LG↗

arXiv2026

On the phase diagram of the multiscale mean-field spin-glass

In this paper we study the phase diagram of a Sherrington-Kirkpatrick (SK) model where the couplings are forced to thermalize at different time scales. Besides being a challenging generalization of the SK model, such settings may arise naturally in physics whenever part of the many degrees of freedom of a system relaxes to equilibrium considerably faster than the others. For this model we compute the asymptotic value of the second moment of the overlap distribution. Furthermore, we provide a rigorous sufficient condition for an annealed solution to hold, identifying a high temperature, or weak coupling, region. In addition, we also prove that for sufficiently strong couplings the solution must present a number of replica symmetry breaking levels at least equal to the number of time scales already present in the multiscale model. Finally, we give a sufficient condition for the existence of gaps in the support of the functional order parameters.

math-ph↗

arXiv2025

Tree asymptotic densities in number theory

We study the asymptotic distribution of integers sharing the same rooted-tree structure that encodes their complete prime factorization tower. For each tree we derive an explicit density formula depending only on a pair $(m,k)$, the density signature of the tree, up to a suitable multiplicative scalar factor and introduce the corresponding tree zeta function, which generalizes the prime zeta function. Classical results such as the prime number theorem and later work by Landau appear as special cases.

math.NT↗

arXiv2025

Statistical Properties of the Rooted-Tree Encoding of $\mathbb{N}$

We prime-encode the natural numbers via recursive factorisation, iterated to the exponents, generating a corpus of planar rooted trees equivalently represented as Dyck words. This forms a deterministic text endowed with internal rules. Statistical analysis of the corpus reveals that the dictionary and the entropy grow sublinearly, compression shows non-monotonic trend, and the rank-frequency curves assume a stable parabolic form deviating from Zipf's law. Correlation analysis using mean-squared displacement reveals a transition from normal diffusion to superdiffusion in the associated walk. These findings characterise the tree-encoded sequence as a statistically structured text with long-range correlations grounded in its generative arithmetic law, providing an empirical basis for subsequent theoretical and learnability

math-ph↗

arXiv2025

A New Lens on the Sustainability of the AI Revolution

We introduce the Economic Productivity of Energy (EPE), GDP generated per unit of energy consumed, as a quantitative lens to assess the sustainability of the Artificial Intelligence (AI) revolution. Historical evidence shows that the first industrial revolution, pre-scientific in the sense that technological adoption preceded scientific understanding, initially disrupted this ratio: EPE collapsed as profits outpaced efficiency, with poorly integrated technologies, and recovered only with the rise of scientific knowledge and societal adaptation. Later industrial revolutions, such as electrification and microelectronics, grounded in established scientific theory, did not exhibit comparable declines. Today's AI revolution, highly profitable yet energy-intensive, remains pre-scientific and may follow a similar trajectory in EPE. We combine this conceptual discussion with cross-country EPE data spanning the last three decades. We find that the advanced economies exhibit a consistent linear growth in EPE: those countries are the ones that contribute most to global GDP production and energy consumption, and are expected to be the most affected by the AI transition. Therefore, we advocate for regular monitoring of EPE: transparent reporting of AI-related energy use and productivity-linked incentives can expose hidden energy costs and prevent efficiency-blind economic expansion. Embedding EPE within sustainability frameworks would help align technological innovation with energy productivity, a critical condition for sustainable growth.

physics.soc-ph↗

arXiv2024

Limit theorems for the cubic mean-field Ising model

We study a mean-field spin model with three- and two-body interactions. The equilibrium measure for large volumes is shown to have three pure states, the phases of the model. They include the two with opposite magnetization and an unpolarized one with zero magnetization, merging at the critical point. We prove that the central limit theorem holds for a suitably rescaled magnetization, while its violation with the typical quartic behavior appears at the critical point.

math-ph↗

arXiv2024

Ferromagnetic Ising Model on multiregular random graphs

A family of multispecies Ising models on generalized regular random graphs is investigated in the thermodynamic limit. The architecture is specified by class-dependent couplings and magnetic fields. We prove that the magnetizations, neighbours correlations and free energy converge to suitable functions evaluated at the solution of a belief propagation fixed point equation. In absence of magnetic fields, a phase transition is identified and the corresponding critical parameters are determined by the spectral radius of a low-dimensional matrix.

math-ph↗

arXiv2024

Bounds on tree distribution in number theory

The occurrence and the distribution of patterns of trees associated to natural numbers are investigated. Bounds from above and below are proven for certain natural quantities.

math.NT↗

arXiv2023

Central limit theorem for the overlaps on the Nishimori line

The overlap distribution of the Sherrington-Kirkpatrick model on the Nishimori line has been proved to be self averaging for large volumes. Here we study the joint distribution of the rescaled overlaps around their common mean and prove that it converges to a Gaussian vector.

math-ph↗

arXiv2023

Voter-like dynamics with conflicting preferences on modular networks

Two of the main factors shaping an individual's opinion are social coordination and personal preferences, or personal biases. To understand the role of those and that of the topology of the network of interactions, we study an extension of the voter model proposed by Masuda and Redner (2011), where the agents are divided into two populations with opposite preferences. We consider a modular graph with two communities that reflect the bias assignment, modeling the phenomenon of epistemic bubbles. We analyze the models by approximate analytical methods and by simulations. Depending on the network and the biases' strengths, the system can either reach a consensus or a polarized state, in which the two populations stabilize to different average opinions. The modular structure has generally the effect of increasing both the degree of polarization and its range in the space of parameters. When the difference in the bias strengths between the populations is large, the success of the very committed group in imposing its preferred opinion onto the other one depends largely on the level of segregation of the latter population, while the dependency on the topological structure of the former is negligible. We compare the simple mean-field approach with the pair approximation and test the goodness of the mean-field predictions on a real network.

physics.soc-ph↗

arXiv2023

A ${\mathbb N}$atural Avenue

We consider an infinite sequence of rooted trees naturally emerging in a number-theoretical context. We advance some ideas on its structure by discussing some elementary properties. Some of those properties are shown to be related to classical results or conjectures in number theory.

math.NT↗

arXiv2022

The inverse problem beyond two-body interaction: the cubic mean-field Ising model

In this paper we solve the inverse problem for the cubic mean-field Ising model. Starting from configuration data generated according to the distribution of the model we reconstruct the free parameters of the system. We test the robustness of this inversion procedure both in the region of uniqueness of the solutions and in the region where multiple thermodynamics phases are present.

cond-mat.stat-mech↗

arXiv2022

Human-AI ecosystem with abrupt changes as a function of the composition

The progressive advent of artificial intelligence machines may represent both an opportunity or a threat. In order to have an idea of what is coming we propose a model that simulate a Human-AI ecosystem. In particular we consider systems where agents present biases, peer-to-peer interactions and also three body interactions that are crucial and describe two humans interacting with an artificial agent and two artificial intelligence agents interacting with a human. We focus our analysis by exploring how the relative fraction of artificial intelligence agents affect that ecosystem. We find evidence that for suitable values of the interaction parameters, arbitrarily small changes in such percentage may trigger dramatic changes for the system that can be either in one of the two polarised states or in an undecided state.

cs.HC↗

arXiv2022

An inference problem in a mismatched setting: a spin-glass model with Mattis interaction

The Wigner spiked model in a mismatched setting is studied with the finite temperature Statistical Mechanics approach through its representation as a Sherrington-Kirkpatrick model with added Mattis interaction. The exact solution of the model with Ising spins is rigorously proved to be given by a variational principle on two order parameters, the Parisi overlap distribution and the Mattis magnetization. The latter is identified by an ordinary variational principle and turns out to concentrate in the thermodynamic limit. The solution leads to the computation of the Mean Square Error of the mismatched reconstruction. The Gaussian signal distribution case is investigated and the corresponding phase diagram is identified.

cond-mat.dis-nn↗

arXiv2021

The solution of the deep Boltzmann machine on the Nishimori line

The deep Boltzmann machine on the Nishimori line with a finite number of layers is exactly solved by a theorem that expresses its pressure through a finite dimensional variational problem of min-max type. In the absence of magnetic fields the order parameter is shown to exhibit a phase transition.

math-ph↗