arXiv subjects

Giulio Biroli

Publications and source records attributed to Giulio Biroli.

At least 19 recordsLinked to original sources

arXiv2026

Overcoming critical slowing down in frustrated spin systems by learned multiscale sampling

Cluster algorithms, such as the Swendsen--Wang and Wolff methods, are among the most successful MCMC methods for mitigating critical slowing down in statistical systems. These constructive cluster algorithms, however, fail in the presence of even extremely weak frustration. Here, we sidestep this fundamental limitation by learning rather than constructing the relevant clusters. Specifically, we use the wavelet conditional renormalization group (WCRG) sampling method to learn the probability distribution of collective fluctuations of a frustrated two-dimensional soft-spin model. Configurations are then generated recursively from coarse to fine scales by sampling conditional wavelet distributions. The WCRG method reproduces the main statistical properties of the system across different phases, including the local-field distribution and the structure factor. At an Ising-like critical point, the conditional dynamics remains decorrelated within $\mathcal{O}(1)$ sweeps at each scale, yielding an overall sampling complexity of $\mathcal{O}(\log_2 L)$, thus making WCRG much more efficient than standard local MCMC methods. These results show that learned multiscale sampling can overcome critical slowing down in frustrated systems for which conventional cluster algorithms fail. By assessing the sampling accuracy of different observables, we also clarify the main tradeoff of the WCRG method: the accuracy of the fast sampling scheme depends on the expressiveness of the energy-based model used to estimate the wavelet conditional distributions.

cond-mat.stat-mech↗

arXiv2026

Nucleation beyond Equilibrium: Fronts Control Invasion in Bistable Ecosystems

Bistability, the existence of two alternative stable states with distinct basins of attraction, is common across ecology and many other biological, chemical, and physical systems. In spatially extended systems, the invasion of one state by the other proceeds through nucleation, the fluctuation-driven growth of a droplet beyond a critical size. Classical Nucleation Theory (CNT) quantifies this process using the energy landscape, but this description relies on detailed balance, the condition of microscopic reversibility that holds at equilibrium. Ecological dynamics generally violate detailed balance and are described not by a single scalar field but by several coupled, non-conserved abundances--a vector order parameter--for which no general nucleation theory exists. Here we derive such a theory for reaction-diffusion systems with a non-conserved vector order parameter, valid both close to the binodal, where invasion proceeds through propagating fronts, and close to the spinodal, where the metastable state loses stability. Extensive numerical computations of the quasipotential confirm the theory in both regimes. We show that the mathematical structure of CNT survives out of equilibrium, once energetic quantities are replaced by dynamical properties of fronts: the front speed plays the role of the bulk free-energy difference between phases, and the diffusivity that of the surface tension. Applied to the two-species Lotka-Volterra model, an archetypal system of bistable ecological antagonism, our theory shows that strong interspecific competition generates a pronounced depletion region within fronts, where the total abundance falls well below carrying capacity. This vectorial structure, invisible to a scalar description based on species frequency alone, makes invasion exponentially harder as competition strengthens at fixed competitive advantage--a prediction testable in microbial systems.

cond-mat.stat-mech↗

arXiv2026

WaiT for the Signal: Simple Frequency-Aware Flow-Matching

As image generation models scale to ever higher resolutions, global coherence, local detail, and texture fidelity become critical axes for generation quality. However, standard flow matching treats all spatial frequencies uniformly, ignoring the natural frequency hierarchy where high-frequency bands become indistinguishable from pure noise far earlier than coarse structures. We introduce WaiT, a Wavelet-aware image Transformer that decomposes generation into coarse and fine bands via lossless wavelets. True to its name, the high-frequency bands wait for the signal: staying pure noise until coarse structure has emerged, then joining the flow for joint refinement. Since standard FID discards fine-grained detail through aggressive downsampling, we introduce a more stringent three-axis evaluation protocol to assess quality at native resolution. On ImageNet 512x512, WaiT achieves a pixel-space FID of 1.43 and is Pareto-optimal across all three axes, reducing sampling compute by up to 50%. With our largest 2B model, we set a new state-of-the-art FID of 1.3 for pixel-space models on ImageNet 512 resolution. Our formulation outperforms even the strongest latent-space models on texture fidelity, and scales seamlessly to high-resolution OpenImages and to video generation, achieving a state-of-the-art FVD of 0.84 on Kinetics-600 with no algorithmic modifications.

cs.CV↗

arXiv2026

Inertial Asynchronous Computation

Computation is the controlled evolution of a state. Asynchronous evolutions, where all parts of the state change in their own time without stopping each other, put this control in jeopardy. It is in fact a mystery how natural processes perform asynchronous computations using many units with no global orchestration. Here we demonstrate how collective computational abilities can emerge in asynchronous many-body systems. The key insight is to split the physical "hardware" underlying the computation into two asymmetrically coupled parts, analogous to position and momentum in a harmonic oscillator. The resulting inertia nudges the evolution of the state so that the asynchronous computation proceeds in the right order. By treating our inertial asynchronous computer as a nonequilibrium material, we map out its phase diagram numerically and analytically using a framework we dub loop dynamical mean-field theory. We experimentally demonstrate our approach using analog spiking neuromorphic chips designed to mimic actual neurons in the brain. In addition, we construct software that can run on asynchronous hardware: we denoise movies whose clean versions were never seen during training, an instantiation of the generalization transition underlying modern machine learning. Our results point to a general strategy for reliable, decentralized computation in energy-constrained settings from dynamics self-assembly to cell differentiation.

cond-mat.stat-mech↗

arXiv2026

The critical slowing down in diffusion models

Computational sampling has been central to the sciences since the mid-20th century. While machine-learning-based approaches have recently enabled major advances, their behavior remains poorly understood, with limited theoretical control over when and why they succeed. Here we provide such insight for diffusion models-a class of generative schemes highly effective in practice-by analyzing their application to the $O(n)$ model of statistical field theory in the Gaussian limit $n \to \infty$. In this analytically tractable setting, we show that training a score model with a one-layer network architecture matching the exact solution exhibits a form of critical slowing down in parameter learning. This slowing down also impacts the generation process, indicating that the well-known difficulties of sampling near criticality persist even for learned generative models. To overcome this bottleneck, we demonstrate the power of combining architectural depth with physical locality. We find that using a two-layer architecture drastically reduces the critical slowing down, with the training time scaling logarithmically rather than quadratically with system size. By introducing a local score approximation we show that this acceleration in training time can be achieved without increasing the number of neural network parameters. Taken together, these results demonstrate that diffusion models can overcome the critical slowing down through appropriate architectural design, and establish a controlled framework for understanding and improving learned sampling methods in statistical physics and beyond.

cond-mat.dis-nn↗

arXiv2026

Discontinuous BBP transitions

The Baik-Ben Arous-Peche (BBP) transition sets fundamental limits for detecting low-rank structure in noisy high-dimensional data and underlies a wide range of spectral methods in many fields from physics to statistics and data sciences. In standard settings, this transition is continuous, implying that signal recovery emerges gradually above a sharp threshold. We show that BBP transitions can instead be discontinuous in very general settings and provide a full theory of this phenomenon. When the eigenvalue density vanishes faster than linearly at the spectral edge, the overlap between the leading eigenvector and the signal jumps discontinuously at the critical point. We study this mechanism in deformed Gaussian and reweighted Wishart ensembles. We analyze in detail the finite-size effects, which play a central and qualitatively new role in the discontinuous BBP transition. Unlike the continuous BBP transition, we establish the existence of an extended pre-critical region where informative eigenvectors emerge well before the asymptotic threshold. The main consequence-and difference from the continuous BBP transition-is that signal recovery can occur at significantly lower signal-to-noise ratio and it is accompanied by strong sample-to-sample variability. Our results show the relevance and the novelty of the discontinuous BBP transition, and highlight the practical implications for signal detection.

cond-mat.dis-nn↗

arXiv2026

On the anomalous elasticity in the mechanical response of amorphous solids

The response of amorphous solids to a mechanical perturbation consists in an elastic and a plastic deformation. The latter is mediated by localized irreversible rearrangements associated with Eshelby-like quadrupolar singularities in the displacement field. It has recently been argued that a density of such singularities leads to an anomalous elastic behavior taking the form of screening effects, which goes beyond classical elastic predictions. Here, we reexamine this scenario using general theoretical arguments and a description in terms of an elasto-plastic model, which we compare with atomistic simulations of the canonical Eshelby inclusion geometry. We discuss the conditions under which a finite, i.e., nonvanishing, density of quadrupolar events is created by an imposed perturbation. We argue that, except when the perturbation is macroscopic, there are many situations in which the density of quadrupolar defects is zero in the thermodynamic limit. In these cases, we find that plastically active quadrupoles emerge in a region whose size generically scales as the spatial extent $\ell$ of the mechanical perturbation. This mechanism leads to anomalous elasticity on a scale $\ell$ close to the perturbation and to conventional elasticity beyond. The simulations of the elasto-plastic model reproduce the emergence of plastic quadrupoles in a region set by $\ell$ and the associated renormalization of the effective shear modulus, but they do not exhibit the dipole-screening signatures reported in atomistic and experimental studies. Our analysis delineates the scale-dependent breakdown of long-wavelength elasticity in amorphous materials and suggests directions for incorporating anomalous screening into mesoscopic modeling frameworks.

cond-mat.soft↗

arXiv2026

Topological Exploration of High-Dimensional Empirical Risk Landscapes: general approach, and applications to phase retrieval

We consider the landscape of empirical risk minimization for high-dimensional Gaussian single-index models (generalized linear models). The objective is to recover an unknown signal $\boldsymbol{\theta}^\star \in \mathbb{R}^d$ (where $d \gg 1$) from a loss function $\hat{R}(\boldsymbol{\theta})$ that depends on pairs of labels $(\mathbf{x}_i \cdot \boldsymbol{\theta}, \mathbf{x}_i \cdot \boldsymbol{\theta}^\star)_{i=1}^n$, with $\mathbf{x}_i \sim \mathcal{N}(0, I_d)$, in the proportional asymptotic regime $n \asymp d$. Using the Kac-Rice formula, we analyze different complexities of the landscape -- defined as the expected number of critical points -- corresponding to various types of critical points, including local minima. We first show that some variational formulas previously established in the literature for these complexities can be drastically simplified, reducing to explicit variational problems over a finite number of scalar parameters that we can efficiently solve numerically. Our framework also provides detailed predictions for properties of the critical points, including the spectral properties of the Hessian and the joint distribution of labels. We apply our analysis to the real phase retrieval problem for which we derive complete topological phase diagrams of the loss landscape, characterizing notably BBP-type transitions where the Hessian at local minima (as predicted by the Kac-Rice formula) becomes unstable in the direction of the signal. We test the predictive power of our analysis to characterize gradient flow dynamics, finding excellent agreement with finite-size simulations of local optimization algorithms, and capturing fine-grained details such as the empirical distribution of labels. Overall, our results open new avenues for the asymptotic study of loss landscapes and topological trivialization phenomena in high-dimensional statistical models.

stat.ML↗

arXiv2026

Theory of Speciation Transitions in Diffusion Models with General Class Structure

Diffusion Models generate data by reversing a stochastic diffusion process, progressively transforming noise into structured samples drawn from a target distribution. Recent theoretical work has shown that this backward dynamics can undergo sharp qualitative transitions, known as speciation transitions, during which trajectories become dynamically committed to data classes. Existing theoretical analyses, however, are limited to settings where classes are identifiable through first moments, such as mixtures of Gaussians with well-separated means. In this work, we develop a general theory of speciation in diffusion models that applies to arbitrary target distributions admitting well-defined classes. We formalize the notion of class structure through Bayes classification and characterize speciation times in terms of free-entropy difference between classes. This criterion recovers known results in previously studied Gaussian-mixture models, while extending to situations in which classes are not distinguishable by first moments and may instead differ through higher-order or collective features. Our framework also accommodates multiple classes and predicts the existence of successive speciation times associated with increasingly fine-grained class commitment. We illustrate the theory on two analytically tractable examples: mixtures of one-dimensional Ising models at different temperatures and mixtures of zero-mean Gaussians with distinct covariance structures. In the Ising case, we obtain explicit expressions for speciation times by mapping the problem onto a random-field Ising model and solving it via the replica method. Our results provide a unified and broadly applicable description of speciation transitions in diffusion-based generative models.

cs.LG↗

arXiv2026

High-Dimensional Analysis of Gradient Flow for Extensive-Width Quadratic Neural Networks

We study the high-dimensional training dynamics of a shallow neural network with quadratic activation in a teacher-student setup. We focus on the extensive-width regime, where the teacher and student network widths scale proportionally with the input dimension, and the sample size grows quadratically. This scaling aims to describe overparameterized neural networks in which feature learning still plays a central role. In the high-dimensional limit, we derive a dynamical characterization of the gradient flow, in the spirit of dynamical mean-field theory (DMFT). Under l2-regularization, we analyze these equations at long times and characterize the performance and spectral properties of the resulting estimator. This result provides a quantitative understanding of the effect of overparameterization on learning and generalization, and reveals a double descent phenomenon in the presence of label noise, where generalization improves beyond interpolation. In the small regularization limit, we obtain an exact expression for the perfect recovery threshold as a function of the network widths, providing a precise characterization of how overparameterization influences recovery.

math.OC↗

arXiv2025

Boltzmann generators for amorphous particle systems

Sampling configurations in thermodynamic equilibrium is a long-standing challenge in statistical physics. Boltzmann generators address this problem by employing generative models to propose independent configurations, which are then reweighted via importance sampling using exact likelihood evaluations. Recent Boltzmann Generators based on continuous normalizing flows and flow matching have achieved significant success for particle systems and biomolecules. However, these approaches have not been extended to amorphous materials (glasses), for which equilibrium sampling is notoriously slow. Because of their disordered structure, the invariances and geometrical constraints of amorphous materials differ from those of crystals and biomolecules, preventing the direct use of existing generative models. Here, we develop Boltzmann Generators tailored to amorphous materials by building the required equivariances directly into Riemannian stochastic interpolants. Our framework incorporates periodic boundary conditions and particle symmetries using equivariant graph neural networks. Numerical experiments demonstrate that enforcing physical symmetries significantly improves the accuracy of Boltzmann Generators, but also reveal an intrinsic limitation of the continuous-flow formulation: accumulated numerical errors during likelihood integration break time-reversibility, compromising exact thermodynamic reweighting. These results reveal a fundamental challenge for continuous-flow generative models in statistical mechanics and call for alternative approaches that preserve exact thermodynamic consistency.

stat.ML↗

arXiv2025

Large deviations in the many-body localization transition: The case of the random-field XXZ chain

The effect of rare system-wide resonances in the many-body localization (MBL) transition has recently attracted significant attention. They are expected to play a prominent role in the stability of the MBL phase, prompting the development of new theoretical frameworks to properly account for their statistical weight. We employ a method based on an analogy with mean-field disordered glassy systems to characterize the statistics of transmission amplitudes between distant many-body configurations in Hilbert space, and apply it to the random-field XXZ spin chain. By introducing a Lagrange multiplier, which formally plays the role of an effective temperature controlling the influence of extreme outliers in the heavy-tailed distribution of propagators, we identify three distinct regimes: (i) an ergodic phase with uniform spreading in Hilbert space, (ii) an intermediate regime where delocalization is driven by rare, disorder-dependent long-range resonances, and (iii) a robust MBL phase where such resonances cannot destabilize localization. We derive a finite-size phase diagram in the disorder--interaction plane both in the spin and in the Anderson basis that quantitatively agrees with recent numerical results based on real-space spin-spin correlation functions. We further demonstrate that even infinitesimal interactions can destroy the Anderson insulator at finite disorder, with the critical disorder remaining finite down to small interaction strengths. By visualizing resonant transmission pathways on the Hilbert space graph, we provide a complementary perspective to real-space and spectral probes, revealing how the destabilization of the MBL phase at finite sizes stems from the emergence of resonant paths that become progressively rarer and shorter-ranged deep in the localized phase.

cond-mat.dis-nn↗

arXiv2025

When is nonreciprocity relevant?

Nonreciprocal interactions are widely observed in nonequilibrium systems, from biological or sociological dynamics to open quantum systems. Despite the ubiquity of nonreciprocity, its impact on phase transitions is not fully understood. In this work, we derive criteria to perturbatively assess whether nonreciprocity changes the universality class of two-species systems undergoing a phase transition. These criteria, stated in terms of the unperturbed critical exponents in the spirit of the Harris criterion for disordered systems, assess whether static critical exponents change at first order under a given perturbation. For example, in the case of a nonreciprocal version of model A with two species, a homogeneous nonreciprocal perturbation is relevant whenever the two parts are initially identical and uncoupled, and irrelevant otherwise. Our results agree with existing renormalization group calculations and with numerical simulations.

cond-mat.stat-mech↗

arXiv2025

Eigenvalue spectral tails and localization properties of asymmetric networks

In contrast to the neatly bounded spectra of densely populated large random matrices, sparse random matrices often exhibit unbounded eigenvalue tails on the real and imaginary axis, called Lifshitz tails. In the case of asymmetric matrices, concise mathematical results have proved elusive. In this work, we present an analytical approach to characterising these tails. We exploit the fact that eigenvalues in the tail region have corresponding eigenvectors that are exponentially localised on highly-connected hubs of the network associated to the random matrix. We approximate these eigenvectors using a series expansion in the inverse connectivity of the hub, where successive terms in the series take into account further sets of next-nearest neighbours. By considering the ensemble of such hubs, we are able to characterise the eigenvalue density and the extent of localisation in the tails of the spectrum in a general fashion. As such, we classify a number of different asymptotic behaviours in the Lifshitz tails, as well as the leading eigenvalue and the inverse participation ratio. We demonstrate how an interplay between matrix asymmetry, network structure, and the edge-weight distribution leads to the variety of observed behaviours.

cond-mat.dis-nn↗

arXiv2025

Entropy Production from Density Field Theories for interacting particles systems

Entropy production quantifies the breaking of time-reversal symmetry in non-equilibrium systems. Here, we develop a direct method to obtain closed, tractable expressions for entropy production in a broad class of dynamical density functional theories, from Dean's exact stochastic equation for microscopic densities to coarse-grained fluctuating-hydrodynamics models with density-dependent mobility. The method employs an Onsager-Machlup path-integral formulation. Our results reproduce particle-level calculations and matches recent Doi-Peliti treatments, confirming that the irregular noise structure of Dean's equation poses no obstacle when handled consistently. We further extend the framework to active mixtures with non-reciprocal interactions and to run-and-tumble or active-Brownian suspensions, generalizations that require a careful treatment of the spurious-drift. Our method furnishes a practical route to quantify irreversibility in density functional field theories and paves the way for systematic studies of entropy production in multi-field active fluids that couple density, momentum and orientation.

cond-mat.stat-mech↗

arXiv2025

Numerical investigation of the equilibrium Kauzmann transition in a two-dimensional atomistic glass

Dense liquids gradually transform into non-equilibrium amorphous solids as they pass through the experimental glass transition. Experimentally, ergodicity is lost because measurements are conducted within a finite time window. More than seventy years ago, Kauzmann posed a fundamental question: If experiments could run indefinitely, would there exist a critical temperature at which an ergodicity-breaking phase transition occurs? Random first-order transitions represent the modern theoretical framework for this idea, rigorously established in the mean-field limit of high-dimensional atomistic systems and several idealized physical models. However, achieving theoretical understanding in finite dimensions is challenging, while experimental and numerical limitations on accessible timescales hinder direct observation of the putative Kauzmann transition. Here, we overcome this longstanding barrier by developing a computational strategy that combines three advanced Monte Carlo methods to access the equilibrium thermodynamic properties of a two-dimensional atomistic glass-former down to zero temperature across a range of system sizes. This enables us to directly measure thermodynamic and structural observables that provide unambiguous evidence that the system undergoes a Kauzmann transition at a temperature that vanishes in the thermodynamic limit. This transition is towards an ideal glass state characterized by a complex energy landscape with a hierarchical organization of low-lying states. Our results are the first demonstration that computer simulations can fully probe the statistical mechanics of the bulk transition to a non-ergodic glass state. We anticipate that our study will serve as a foundation for future simulation work on larger systems, three-dimensional materials, and more complex glass-forming models to fully elucidate the nature of the glass state of matter.

cond-mat.soft↗

arXiv2025

Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Training

Diffusion models have achieved remarkable success across a wide range of generative tasks. A key challenge is understanding the mechanisms that prevent their memorization of training data and allow generalization. In this work, we investigate the role of the training dynamics in the transition from generalization to memorization. Through extensive experiments and theoretical analysis, we identify two distinct timescales: an early time $\tau_\mathrm{gen}$ at which models begin to generate high-quality samples, and a later time $\tau_\mathrm{mem}$ beyond which memorization emerges. Crucially, we find that $\tau_\mathrm{mem}$ increases linearly with the training set size $n$, while $\tau_\mathrm{gen}$ remains constant. This creates a growing window of training times with $n$ where models generalize effectively, despite showing strong memorization if training continues beyond it. It is only when $n$ becomes larger than a model-dependent threshold that overfitting disappears at infinite training times. These findings reveal a form of implicit dynamical regularization in the training dynamics, which allow to avoid memorization even in highly overparameterized settings. Our results are supported by numerical experiments with standard U-Net architectures on realistic and synthetic datasets, and by a theoretical analysis using a tractable random features model studied in the high-dimensional limit.

cs.LG↗

arXiv2025

A Differentiable Rank-Based Objective For Better Feature Learning

In this paper, we leverage existing statistical methods to better understand feature learning from data. We tackle this by modifying the model-free variable selection method, Feature Ordering by Conditional Independence (FOCI), which is introduced in \cite{azadkia2021simple}. While FOCI is based on a non-parametric coefficient of conditional dependence, we introduce its parametric, differentiable approximation. With this approximate coefficient of correlation, we present a new algorithm called difFOCI, which is applicable to a wider range of machine learning problems thanks to its differentiable nature and learnable parameters. We present difFOCI in three contexts: (1) as a variable selection method with baseline comparisons to FOCI, (2) as a trainable model parametrized with a neural network, and (3) as a generic, widely applicable neural network regularizer, one that improves feature learning with better management of spurious correlations. We evaluate difFOCI on increasingly complex problems ranging from basic variable selection in toy examples to saliency map comparisons in convolutional networks. We then show how difFOCI can be incorporated in the context of fairness to facilitate classifications without relying on sensitive data.

stat.ML↗