Journal of Statistical Mechanics: Theory and Experiment

Local interactions in many-body quantum systems are generally non-commuting and consequently the Hamiltonian of a local region cannot be measured simultaneously with the global Hamiltonian. The connection between the probability distributions of measurement outcomes of the local and global Hamiltonians will depend on the angles between the diagonalizing bases of these two Hamiltonians. In this paper we characterize the relation between these two distributions. On one hand, we upperbound the probability of measuring an energy τ in a local region, if the global system is in a superposition of eigenstates with energies $\epsilon <\tau$. On the other hand, we bound the probability of measuring a global energy in a bipartite system that is in a tensor product of eigenstates of its two subsystems. Very roughly, we show that due to the local nature of the governing interactions, these distributions are identical to what one encounters in the commuting cases, up to exponentially small corrections. Finally, we use these bounds to study the spectrum of a locally truncated Hamiltonian, in which the energies of a contiguous region have been truncated above some threshold energy. We show that the lower part of the spectrum of this Hamiltonian is exponentially close to that of the original Hamiltonian. A restricted version of this result in 1D was a central building block in a recent improvement of the 1D area-law.

Inequality and its consequences are the subject of intense recent debate. Using a simplified model of the economy, we address the relation between inequality and liquidity, the latter understood as the frequency of economic exchanges. Assuming a Pareto distribution of wealth for the agents, that is consistent with empirical findings, we find an inverse relation between wealth inequality and overall liquidity. We show that an increase in the inequality of wealth results in an even sharper concentration of the liquid financial resources. This leads to a congestion of the flow of goods and the arrest of the economy when the Pareto exponent reaches one.

We propose a simple method to extract the community structure of large networks. Our method is a heuristic method that is based on modularity optimization. It is shown to outperform all other known community detection methods in terms of computation time. Moreover, the quality of the communities detected is very good, as measured by the so-called modularity. This is shown first by identifying language communities in a Belgian mobile phone network of 2 million customers and by analysing a web graph of 118 million nodes and more than one billion links. The accuracy of our algorithm is also verified on ad hoc modular networks.

Approximate message passing (AMP) has been shown to be an excellent statistical approach to signal inference and compressed sensing problems. The AMP framework provides modularity in the choice of signal prior; here we propose a hierarchical form of the Gauss–Bernoulli prior which utilizes a restricted Boltzmann machine (RBM) trained on the signal support to push reconstruction performance beyond that of simple i.i.d. priors for signals whose support can be well represented by a trained binary RBM. We present and analyze two methods of RBM factorization and demonstrate how these affect signal reconstruction performance within our proposed algorithm. Finally, using the MNIST handwritten digit dataset, we show experimentally that using an RBM allows AMP to approach oracle-support performance.

Current automated systems have crucial limitations that need to be addressed before artificial intelligence can reach human-like levels and bring new technological revolutions. Among others, our societies still lack level-5 self-driving cars, domestic robots, and virtual assistants that learn reliable world models, reason, and plan complex action sequences. In these notes, we summarize the main ideas behind the architecture of autonomous intelligence of the future proposed by Yann LeCun. In particular, we introduce energy-based and latent variable models and combine their advantages in the building block of LeCun's proposal, that is, in the hierarchical joint-embedding predictive architecture.

We show that a variety of modern deep learning tasks exhibit a 'double-descent' phenomenon where, as we increase model size, performance first gets worse and then gets better. Moreover, we show that double descent occurs not just as a function of model size, but also as a function of the number of training epochs. We unify the above phenomena by defining a new complexity measure we call the effective model complexity and conjecture a generalized double descent with respect to this measure. Furthermore, our notion of model complexity allows us to identify certain regimes where increasing (even quadrupling) the number of train samples actually hurts test performance.

We investigate a disordered multi-dimensional linear system in which the interaction parameters are colored noises, varying stochastically in time with defined temporal correlations. We refer to this type of disorder as 'annealed', in contrast to quenched disorder in which couplings are fixed over time. Using generating functional methods, we extend dynamical mean-field theory to accommodate annealed disorder and employ it to find the exact solution of the linear model in the limit of a large number of degrees of freedom. Our analysis yields analytical results for the non-stationary autocorrelation, the stationary variance, the power spectral density, and the phase diagram of the model. Some unexpected features emerge upon changing the correlation time of the interactions. The stationary variance of the system and the critical variance of the disorder are generally found to be non-monotonic functions of the correlation time of the interactions. We also find that a re-entrant phase transition can take place when this correlation time is varied.

Recent years have been marked with the fast-pace diversification and increasing ubiquity of machine learning (ML) applications. Yet, a firm theoretical understanding of the surprising efficiency of neural networks (NNs) to learn from high-dimensional data still proves largely elusive. In this endeavour, analyses inspired by statistical physics have proven instrumental, enabling the tight asymptotic characterization of the learning of NNs in high dimensions, for a broad class of solvable models. This manuscript reviews the tools and ideas underlying recent progress in this line of work. We introduce a generic model—the sequence multi-index model, which encompasses numerous previously studied models as special instances. This unified framework covers a broad class of ML architectures with a finite number of hidden units—including multi-layer perceptrons, autoencoders, attention mechanisms, and tasks –(un)supervised learning, denoising, contrastive learning, in the limit of large data dimension, and comparably large number of samples. We explicate in full detail the analysis of the learning of sequence multi-index models, using statistical physics techniques such as the replica method and approximate message-passing algorithms. This manuscript thus provides a unified presentation of analyses reported in several previous works, and a detailed overview of central techniques in the field of statistical physics of ML. This review should be a useful primer for ML theoreticians curious of statistical physics approaches; it should also be of value to statistical physicists interested in the transfer of such ideas to the study of NNs.

The Louvain method was proposed 15 years ago as a heuristic method for the fast detection of communities in large networks. During this period, it has emerged as one of the most popular methods for community detection: the task of partitioning vertices of a network into dense groups, usually called communities or clusters. Here, after a short introduction to the method, we give an overview of the different generalizations, modifications and improvements that have been proposed in the literature, and also survey the quality functions, beyond modularity, for which it has been implemented. Finally, we conclude with a discussion on the limitations of the method and perspectives for future research.

We present a detailed study of the finite-size one-dimensional quantum XY chain in a transverse field in the presence of boundary fields coupled with the order-parameter spin operator. We consider fields located at the chain boundaries that have the same strength and that are oppositely aligned. We derive exact expressions for the gap Δ as a function of the model parameters for large values of the chain length L. These results allow us to characterize the nature of the ordered phases of the model. We find a magnetic (M) phase ($\Delta \sim \textrm{e}^{-aL}$), a magnetic-incommensurate (MI) phase ($\Delta \sim \textrm{e}^{-aL} f_{\textrm{MI}}(L)$), a kink (K) phase ($\Delta \sim L^{-2}$) and a kink-incommensurate (KI) phase ($\Delta \sim L^{-2} f_{\textrm{KI}}(L)$); $f_{\textrm{MI}}(L)$ and $f_{\textrm{KI}}(L)$ are bounded oscillating functions of L. We also analyze the behavior along the phase boundaries. In particular, we characterize the universal crossover behavior across the K-KI phase boundary. On this boundary, the dynamic critical exponent is z = 4, i.e. $\Delta \sim L^{-4}$ for large values of L.

We present an ansatz of generalising the construction of recursion relations for the correlation functions of the $\mathfrak{sl}_2$-invariant fundamental exchange model in the thermodynamic limit by Jimbo, Miwa, Smirnov, Takeyama and one of our present authors in 2004 for higher rank. Due to the structure of the correlators as functions of their inhomogeneity parameters, a recursion formula for the reduced density matrix was proven. In the case of $\mathfrak{sl}_3$, we use the explicit results of Klümper and Ribeiro, and Nirov, Hutsalyuk and one of our present authors for the reduced density matrix of up to operator length three to verify whether it is possible to relate the residues of the density matrix of length n to the density matrix of length smaller than n as in $\mathfrak{sl}_2$. This is unclear, since the reduced quantum Knizhnik–Zamolodchikov equation splits into two parts for higher rank. In fact, we show two relations, one of which is a straightforward generalisation to the $\mathfrak{sl}_2$ case and one which is completely new. This allows us to construct an analogue of the operator X_k, which we call the snail operator. In the $\mathfrak{sl}_2$ case, this operator has many useful properties, including in particular the fact that only one irreducible representation of the Yangian $Y(\mathfrak{sl}_2)$, the Kirillov–Reshetikhin module W_k, contributes to the residue at $\lambda_i-\lambda_j = -(k+1)$. Here, we give an overview of the mathematical background, T-systems, and show a new application of the extended T-systems introduced by Mukhin and Young in 2012 regarding the snail operator.

We characterize quantum dynamics in triangular billiards in terms of five properties: (1) the level spacing ratio (LSR), (2) spectral complexity (SC), (3) Lanczos coefficient variance, (4) energy eigenstate localisation in the Krylov basis, and (5) dynamical growth of spread complexity. The billiards we study are classified as integrable, pseudointegrable or non-integrable, depending on their internal angles which determine properties of classical trajectories and associated quantum spectral statistics. A consistent picture emerges when transitioning from integrable to non-integrable triangles: (1) average LSRs increase; (2) SC growth slows down; (3) Lanczos coefficient variances decrease; (4) energy eigenstates delocalize in the Krylov basis; and (5) spread complexity increases, displaying a peak prior to a plateau instead of recurrences. Pseudo-integrable triangles deviate by a small amount in these characteristics from non-integrable ones, which in turn approximate models from the Gaussian orthogonal ensemble (GOE). Isosceles pseudointegrable and non-integrable triangles have independent sectors that are symmetric and antisymmetric under a reflection symmetry. These sectors separately reproduce characteristics of the GOE, even though the combined system approximates characteristics expected from integrable theories with Poisson distributed spectra.

We theoretically investigate the many-body dynamics of a tight-binding chain with dephasing noise on an infinite interval. We obtain the exact solution of the average particle-density profile for the domain wall and the alternating initial conditions via the Bethe ansatz, analytically deriving the asymptotic expressions for the long-time dynamics. For the domain wall initial condition, we obtain the scaling form of the average density, elucidating that diffusive transport always emerges in the long-time dynamics if the strength of the dephasing, no matter how small, is positive. For the alternating initial condition, our exact solution leads to the fact that the average density displays oscillatory decay or overdamped decay depending on the strength of the dissipation. Furthermore, we demonstrate that the asymptotic forms approach those of the symmetric simple exclusion process, identifying corrections from it.

We consider a one-dimensional system comprising of N run-and-tumble particles confined in a harmonic trap interacting via a repulsive inverse-square power-law interaction. We numerically compute the global density profile in the steady state which shows interesting crossovers between three different regimes: as the activity increases, we observe a change from a density with sharp peaks characteristic of a crystal region to a smooth bell-shaped density profile, passing through the intermediate stage of a smooth Wigner semi-circle characteristic of a liquid phase. We also investigate analytically the crossover between the crystal and the liquid regions by computing the covariance of the positions of these particles in the steady state in the weak noise limit. It is achieved by using the method introduced in Touzo et al (2024 Phys. Rev. E 109 014136) to study the active Dyson Brownian motion. Our analytical results are corroborated by thorough numerical simulations.

This paper addresses the theoretical foundations of pedestrian models for crowd dynamics. While the topic gains momentum, current models differ widely in their mathematical structure, even if we only consider continuous agent-based models. To clarify their underpinning, we first lay the mathematical foundations of the common hierarchical decomposition into strategic, tactical, and operational levels and underline the practical interest in preserving the continuity between the latter two levels by working with a floor field, rather than way-points. Turning to local navigation, we clarify how three archetypical approaches, namely, purely reactive models, anticipatory models based on the idea of time to collision, and game theory, differ in the way they extrapolate trajectories in the near future. We also insist on the oft-overlooked distinction between processes pertaining to decision-making and physical contact forces. Notably, the implications of these differences are illustrated with a comparison of the numerical predictions of these models in the simple scenario of head-on collision avoidance between agents, by varying the walking speed, the reaction times, and the degree of courtesy of the agents.

The Louvain method was proposed 15 years ago as a heuristic method for the fast detection of communities in large networks. During this period, it has emerged as one of the most popular methods for community detection: the task of partitioning vertices of a network into dense groups, usually called communities or clusters. Here, after a short introduction to the method, we give an overview of the different generalizations, modifications and improvements that have been proposed in the literature, and also survey the quality functions, beyond modularity, for which it has been implemented. Finally, we conclude with a discussion on the limitations of the method and perspectives for future research.

Network meta-analysis (NMA) is a technique used in medical statistics to combine evidence from multiple medical trials. NMA defines an inference and information processing problem on a network of treatment options and trials connecting the treatments. We believe that statistical physics can offer useful ideas and tools for this area, including from the theory of complex networks, stochastic modelling and simulation techniques. The lack of a unique source that would allow physicists to learn about NMA effectively is a barrier to this. In this article we aim to present the 'NMA problem' and existing approaches to it coherently and in a language accessible to statistical physicists. We also summarise existing points of contact between statistical physics and NMA, and describe our ideas of how physics might make a difference for NMA in the future. The overall goal of the article is to attract physicists to this interesting, timely and worthwhile field of research.

The phenomenon of spontaneous synchronization, particularly within the framework of the Kuramoto model, has been a subject of intense research over the years. The model comprises oscillators with distributed natural frequencies interacting through a mean-field coupling, and serves as a paradigm to study synchronization. In this review, we put forward a general framework in which we discuss in a unified way known results with more recent developments obtained for a generalized Kuramoto model that includes inertial effects and noise. We describe the model from a different perspective, highlighting the long-range nature of the interaction between the oscillators, and emphasizing the equilibrium and out-of-equilibrium aspects of its dynamics from a statistical physics point of view. In this review, we first introduce the model and discuss both for the noiseless and noisy dynamics and for unimodal frequency distributions the synchronization transition that occurs in the stationary state. We then introduce the generalized model, and analyze its dynamics using tools from statistical mechanics. In particular, we discuss its synchronization phase diagram for unimodal frequency distributions. Next, we describe deviations from the mean-field setting of the Kuramoto model. To this end, we consider the generalized Kuramoto dynamics on a one-dimensional periodic lattice on the sites of which the oscillators reside and interact with one another with a coupling that decays as an inverse power-law of their separation along the lattice. For two specific cases, namely, in the absence of noise and inertia, and in the case when the natural frequencies are the same for all the oscillators, we discuss how the long-time transition to synchrony is governed by the dynamics of the mean-field mode (zero Fourier mode) of the spatial distribution of the oscillator phases.

We present an ansatz of generalising the construction of recursion relations for the correlation functions of the $\mathfrak{sl}_2$-invariant fundamental exchange model in the thermodynamic limit by Jimbo, Miwa, Smirnov, Takeyama and one of our present authors in 2004 for higher rank. Due to the structure of the correlators as functions of their inhomogeneity parameters, a recursion formula for the reduced density matrix was proven. In the case of $\mathfrak{sl}_3$, we use the explicit results of Klümper and Ribeiro, and Nirov, Hutsalyuk and one of our present authors for the reduced density matrix of up to operator length three to verify whether it is possible to relate the residues of the density matrix of length n to the density matrix of length smaller than n as in $\mathfrak{sl}_2$. This is unclear, since the reduced quantum Knizhnik–Zamolodchikov equation splits into two parts for higher rank. In fact, we show two relations, one of which is a straightforward generalisation to the $\mathfrak{sl}_2$ case and one which is completely new. This allows us to construct an analogue of the operator X_k, which we call the snail operator. In the $\mathfrak{sl}_2$ case, this operator has many useful properties, including in particular the fact that only one irreducible representation of the Yangian $Y(\mathfrak{sl}_2)$, the Kirillov–Reshetikhin module W_k, contributes to the residue at $\lambda_i-\lambda_j = -(k+1)$. Here, we give an overview of the mathematical background, T-systems, and show a new application of the extended T-systems introduced by Mukhin and Young in 2012 regarding the snail operator.

Predicting the future behavior of complex systems exhibiting critical-like dynamics is often considered to be an intrinsically hard task. Here, we study the predictability of the depinning dynamics of elastic interfaces in random media driven by a slowly increasing external force, a paradigmatic complex system exhibiting critical avalanche dynamics linked to a continuous non-equilibrium depinning phase transition. To this end, we train a variety of machine learning models to infer the mapping from features of the initial relaxed line shape and the random pinning landscape to predict the sample-dependent staircase-like force–displacement curve that emerges from the depinning process. Even if for a given realization of the quenched random medium the dynamics are in principle deterministic, we find that there is an exponential decay of the predictability with the displacement of the line as it nears the depinning transition from below. Our analysis on how the related displacement scale depends on the system size and the dimensionality of the input descriptor reveals that the onset of the depinning phase transition gives rise to fundamental limits to predictability.

We employ the quasiparticle picture of entanglement evolution to obtain an effective description for the out-of-equilibrium entanglement Hamiltonian at the hydrodynamical scale following quantum quenches in free fermionic systems in two or more spatial dimensions. Specifically, we begin by applying dimensional reduction techniques in cases where the geometry permits, building directly on established results from one-dimensional systems. Subsequently, we generalize the analysis to encompass a wider range of geometries. We obtain analytical expressions for the entanglement Hamiltonian valid at the ballistic scale, which reproduce the known quasiparticle picture predictions for the Renyi entropies and full counting statistics. We also numerically validate the results with excellent precision by considering quantum quenches from several initial configurations.

Particle–particle correlation functions in ionic systems control many of their macroscopic properties. In this work, we use stochastic density functional theory to compute these correlations, and then we analyze their long-range behavior. In particular, we study the system's response to a rapid change (quench) in the external electric field. We show that the correlation functions relax diffusively toward the non-equilibrium stationary state and that in a stationary state, they present a universal conical shape. This shape distinguishes this system from systems with short-range interactions, where the correlations have a parabolic shape. We relate this temporal evolution of the correlations to the algebraic relaxation of the total charge current reported previously.

Long-range correlations manifested as power spectral density scaling $1/f^\beta$ for frequency f and a range of exponents β are investigated for a superposition of uncorrelated pulses with distributed durations τ. Closed-form expressions for the frequency power spectral density are derived for a one-sided exponential pulse function and several variants of bounded and unbounded power-law distributions of pulse durations ${P_\tau(\tau)\sim1/\tau^\alpha}$ with abrupt and smooth cutoffs. The asymptotic scaling relation $\beta = 3-\alpha$ is demonstrated for $1 \lt \alpha \lt 3$ in the limit of an infinitely broad distribution $P_\tau(\tau)$. Logarithmic corrections to the frequency scaling are exposed at the boundaries of the long-range dependence regime, β = 0 and β = 2. Analytically demonstrated finite-size effects associated with distribution truncations are shown to reduce the frequency ranges of scale invariance by several decades. The regimes of validity of the $\beta = 3-\alpha$ relation are clarified.

Recent years have been marked with the fast-pace diversification and increasing ubiquity of machine learning (ML) applications. Yet, a firm theoretical understanding of the surprising efficiency of neural networks (NNs) to learn from high-dimensional data still proves largely elusive. In this endeavour, analyses inspired by statistical physics have proven instrumental, enabling the tight asymptotic characterization of the learning of NNs in high dimensions, for a broad class of solvable models. This manuscript reviews the tools and ideas underlying recent progress in this line of work. We introduce a generic model—the sequence multi-index model, which encompasses numerous previously studied models as special instances. This unified framework covers a broad class of ML architectures with a finite number of hidden units—including multi-layer perceptrons, autoencoders, attention mechanisms, and tasks –(un)supervised learning, denoising, contrastive learning, in the limit of large data dimension, and comparably large number of samples. We explicate in full detail the analysis of the learning of sequence multi-index models, using statistical physics techniques such as the replica method and approximate message-passing algorithms. This manuscript thus provides a unified presentation of analyses reported in several previous works, and a detailed overview of central techniques in the field of statistical physics of ML. This review should be a useful primer for ML theoreticians curious of statistical physics approaches; it should also be of value to statistical physicists interested in the transfer of such ideas to the study of NNs.

The uncanny ability of over-parameterised neural networks to generalise well has been explained using various 'simplicity biases'. These theories postulate that neural networks avoid overfitting by first fitting simple, linear classifiers before learning more complex, non-linear functions. Meanwhile, data structure is also recognised as a key ingredient for good generalisation, yet its role in simplicity bias is not yet understood. Here, we show that neural networks trained using stochastic gradient descent initially classify their inputs using lower-order input statistics, such as mean and covariance, and exploit higher-order statistics only later during training. We first demonstrate this distributional simplicity bias (DSB) in a solvable model of a single neuron trained on synthetic data. We then demonstrate DSB empirically in a range of deep convolutional networks and visual transformers trained on CIFAR10, and show that it even holds in networks pre-trained on ImageNet. We discuss the relation of DSB to other simplicity biases and consider its implications for the principle of Gaussian universality in learning.

We present a detailed study of the finite-size one-dimensional quantum XY chain in a transverse field in the presence of boundary fields coupled with the order-parameter spin operator. We consider fields located at the chain boundaries that have the same strength and that are oppositely aligned. We derive exact expressions for the gap Δ as a function of the model parameters for large values of the chain length L. These results allow us to characterize the nature of the ordered phases of the model. We find a magnetic (M) phase ($\Delta \sim \textrm{e}^{-aL}$), a magnetic-incommensurate (MI) phase ($\Delta \sim \textrm{e}^{-aL} f_{\textrm{MI}}(L)$), a kink (K) phase ($\Delta \sim L^{-2}$) and a kink-incommensurate (KI) phase ($\Delta \sim L^{-2} f_{\textrm{KI}}(L)$); $f_{\textrm{MI}}(L)$ and $f_{\textrm{KI}}(L)$ are bounded oscillating functions of L. We also analyze the behavior along the phase boundaries. In particular, we characterize the universal crossover behavior across the K-KI phase boundary. On this boundary, the dynamic critical exponent is z = 4, i.e. $\Delta \sim L^{-4}$ for large values of L.

We investigate a disordered multi-dimensional linear system in which the interaction parameters are colored noises, varying stochastically in time with defined temporal correlations. We refer to this type of disorder as 'annealed', in contrast to quenched disorder in which couplings are fixed over time. Using generating functional methods, we extend dynamical mean-field theory to accommodate annealed disorder and employ it to find the exact solution of the linear model in the limit of a large number of degrees of freedom. Our analysis yields analytical results for the non-stationary autocorrelation, the stationary variance, the power spectral density, and the phase diagram of the model. Some unexpected features emerge upon changing the correlation time of the interactions. The stationary variance of the system and the critical variance of the disorder are generally found to be non-monotonic functions of the correlation time of the interactions. We also find that a re-entrant phase transition can take place when this correlation time is varied.

We consider the problem of learning a target function corresponding to a deep, extensive-width, non-linear neural network with random Gaussian weights. We consider the asymptotic limit where the number of samples, the input dimension and the network width are proportionally large. We propose a closed-form expression for the Bayes-optimal test error, for regression and classification tasks. We further compute closed-form expressions for the test errors of ridge regression, kernel and random features regression. We find, in particular, that optimally regularized ridge regression, as well as kernel regression, achieve Bayes-optimal performances, while the logistic loss yields a near-optimal test error for classification. We further show numerically that when the number of samples grows faster than the dimension, ridge and kernel methods become suboptimal, while neural networks achieve test error close to zero from quadratically many samples.