arXiv subjects

Emanuele Mingione

Publications and source records attributed to Emanuele Mingione.

At least 19 recordsLinked to original sources

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

Limit theorems for the non-convex multispecies Curie-Weiss model

We study the thermodynamic properties of the generalized non-convex multispecies Curie-Weiss model, where interactions among different types of particles (forming the species) are encoded in a generic matrix. For spins with a generic prior distribution, we compute the pressure in the thermodynamic limit using simple interpolation techniques. For Ising spins, we further analyze the fluctuations of the magnetization in the thermodynamic limit under the Boltzmann-Gibbs measure. It is shown that a central limit theorem holds for a rescaled and centered vector of species magnetizations, which converges to either a centered or non-centered multivariate normal distribution, depending on the rate of convergence of the relative sizes of the species.

math-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↗

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

On the convergence to the non-equilibrium steady state of a Langevin dynamics with widely separated time scales and different temperatures

We study the solution of the two-temperatures Fokker-Planck equation and rigorously analyse its convergence towards an explicit non-equilibrium stationary measure for long time and two widely separated time scales. The exponential rates of convergence are estimated assuming the validity of logarithmic Sobolev inequalities for the conditional and marginal distributions of the limit measure. We show that these estimates are sharp in the exactly solvable case of a quadratic potential. We discuss a few examples where the logarithmic Sobolev inequalities are satisfied through simple, though not optimal, criteria. In particular we consider a spin-glass model with slowly varying external magnetic fields whose non-equilibrium measure corresponds to Guerra's hierarchical construction appearing in Talagrand's proof of the Parisi formula.

math-ph↗

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↗

arXiv2021

Stationarization and Multithermalization in spin glasses

We develop further the study of a system in contact with a multibath having different temperatures at widely separated timescales. We consider those systems that do not thermalize in finite times when in contact with an ordinary bath but may do so in contact with a multibath. Thermodynamic integration is possible, thus allowing one to recover the stationary distribution on the basis of measurements performed in a `multi-reversible' transformation. We show that following such a protocol the system is at each step described by a generalization of the Boltzmann-Gibbs distribution, that has been studied in the past. Guerra's bound interpolation scheme for spin-glasses is closely related to this: by translating it into a dynamical setting, we show how it may actually be implemented in practice. The phase diagram plane of temperature vs "number of replicas", long studied in spin-glasses, in our approach becomes simply that of the two temperatures the system is in contact with. We suggest that this representation may be used to directly compare phenomenological and mean-field inspired models.Finally, we show how an approximate out of equilibrium probability distribution may be inferred experimentally on the basis of measurements along an almost reversible transformation.

cond-mat.dis-nn↗

arXiv2020

The multi-species mean-field spin-glass on the Nishimori line

In this paper we study a multi-species disordered model on the Nishimori line. The typical properties of this line, a set of identities and inequalities among correlation functions, allow us to prove the replica symmetry i.e. the concentration of the order parameter. When the interaction structure is elliptic we rigorously compute the exact solution of the model in terms of a finite dimensional variational principle and we study its properties.

math-ph↗

arXiv2020

Deep Boltzmann machines: rigorous results at arbitrary depth

A class of deep Boltzmann machines is considered in the simplified framework of a quenched system with Gaussian noise and independent entries. The quenched pressure of a $K$-layers spin glass model is studied allowing interactions only among consecutive layers. A lower bound for the pressure is found in terms of a convex combination of $K$ Sherrington-Kirkpatrick models and used to study the annealed and replica symmetric regimes of the system. A map with a one dimensional monomer-dimer system is identified and used to rigorously control the annealed region at arbitrary depth $K$ with the methods introduced by Heilmann and Lieb. The compression of this high noise region displays a remarkable phenomenon of localisation of the processing layers. Furthermore a replica symmetric lower bound for the limiting quenched pressure of the model is obtained in a suitable region of the parameters and the replica symmetric pressure is proved to have a unique stationary point.

math-ph↗

arXiv2020

Annealing and replica-symmetry in Deep Boltzmann Machines

In this paper we study the properties of the quenched pressure of a multi-layer spin-glass model (a deep Boltzmann Machine in artificial intelligence jargon) whose pairwise interactions are allowed between spins lying in adjacent layers and not inside the same layer nor among layers at distance larger than one. We prove a theorem that bounds the quenched pressure of such a K-layer machine in terms of K Sherrington-Kirkpatrick spin glasses and use it to investigate its annealed region. The replica-symmetric approximation of the quenched pressure is identified and its relation to the annealed one is considered. The paper also presents some observation on the model's architectural structure related to machine learning. Since escaping the annealed region is mandatory for a meaningful training, by squeezing such region we obtain thermodynamical constraints on the form factors. Remarkably, its optimal escape is achieved by requiring the last layer to scale sub-linearly in the network size.

cond-mat.stat-mech↗

arXiv2019

Equilibrium and Dynamics of a Multi-Bath Sherrington-Kirkpatrick Model

In this paper we study the equilibrium statistical mechanical as well as the dynamical properties of a Sherrington and Kirkpatrick model in a multi-bath setting introduced in [4]. We show that the free energy per particle in the thermodynamical limit obeys a variational principle of Parisi type. The relation between the resulting order parameters is discussed.

cond-mat.dis-nn↗

arXiv2018

A Multi-Scale Spin-Glass Mean-Field Model

In this paper a multi-scale version of the Sherrington and Kirkpatrick model is introduced and studied. The pressure per particle in the thermodynamical limit is proved to obey a variational principle of Parisi type. The result is achieved by means of lower and upper bounds. The lower bound is obtained with a Ruelle cascade using the interpolation technique, while the upper bound exploits factorisation properties of the equilibrium measure and the synchronisation technique

math-ph↗

arXiv2017

Two populations mean-field monomer-dimer model

A two populations mean-field monomer-dimer model including both hard-core and attractive interactions between dimers is considered. The pressure density in the thermodynamic limit is proved to satisfy a three-dimensional variational principle. A detailed analysis is made in the limit in which one population is much smaller than the other and a ferromagnetic mean-field phase transition is found.

math-ph↗

arXiv2016

Mean-Field Monomer-Dimer models. A review

A collection of rigorous results for a class of mean-field monomer-dimer models is presented. It includes a Gaussian representation for the partition function that is shown to considerably simplify the proofs. The solutions of the quenched diluted case and the random monomer case are explained. The presence of the attractive component of the Van der Waals potential is considered and the coexistence phase coexistence transition analysed. In particular the breakdown of the central limit theorem is illustrated at the critical point where a non Gaussian, quartic exponential distribution is found for the number of monomers centered and rescaled with the volume to the power 3/4.

math-ph↗