Posters

Greivin Alfaro - LPTHE

SWAP algorithm for lattice spin models

We adapted the SWAP molecular dynamics algorithm for use in lattice Ising spin models. We dressed the spins with a randomly distributed length and we alternated long-range spin exchanges with conventional single spin flip Monte Carlo updates, both accepted with a stochastic rule which respects detailed balance. We show that this algorithm, when applied to the bi-dimensional Edwards-Anderson model, speeds up significantly the relaxation at low temperatures and manages to find ground states with high efficiency and little computational cost. The exploration of spin models should help in understanding why SWAP accelerates the evolution of particle systems and shed light on relations between dynamics and free-energy landscapes.

Santiago Aranguri - Courant Institute

Phase transitions of diffusion models in large dimensions

Diffusion models are the state-of-the-art algorithm to generate images. In this talk, I will present recent work on exact characterizations of phase transitions of diffusion models when the data to be generated is a spin configuration from the Curie-Weiss model. With these characterizations, we provide two ways of interpolating between noise and data which lead to more accurate data generation. This gives rise to a limiting model (in the large dimension limit) that is not obtainable from other ways of interpolating, and gives a theoretical foundation for techniques that practitioners use in ML. Based on joint work with Eric Vanden-Eijnden.

Stefano Bae - Università di Roma Sapienza

A Very Effective and Simple Diffusion Reconstruction for the Diluted Ising Model

Diffusion-based generative models (DM) are deep learning models that use diffusion process to learn the probability distribution of high-dimensional data. In recent years, they have become extremely successful in generating multimedia contents. However, is still unknown if such models can be used to generate high-quality datasets of physical models. In this work, we use a Landau-Ginzburg like model to infer the distribution of the bond-diluted Ising model. We show that the generated samples reproduce correctly the statistical and critical properties of the physical model.

Joseph Baron - LPENS

A path integral approach to sparse non-Hermitian random matrices

The theory of large random matrices has proved an invaluable tool for the study of systems with disordered interactions in many quite disparate research areas. Widely applicable results, such as the celebrated elliptic law for dense random matrices, allow one to draw broad conclusions about what salient statistical properties of a large system determine, for instance, dynamical stability. However, such simple and universal results have been more difficult to come by in the case of sparse random matrices, especially when the interactions are asymmetric. In this poster, I present a new approach, which maps the problem of finding the spectral properties of a non-Hermitian random matrix onto evaluating the response functions of a particular linear dynamical system. The response functions are then evaluated using the MSRJD path integral formalism, enabling one to construct Feynman diagrams and perform a perturbative analysis. This approach provides simple modified versions of the classic elliptic and semi-circle laws that take into account the sparse corrections. One is thus able to draw quite general qualitative conclusions about the effect of sparsity on the eigenvalue spectrum.

Claudio Chilin - Universidad Complutense de Madrid

Daydreaming Hopfield Networks and their surprising effectiveness on correlated data

To improve the storage capacity of the Hopfield model, we develop a version of the dreaming algorithm that perpetually reinforces the patterns to be stored (as in the Hebb rule), and erases the spurious memories (as in dreaming algorithms). For this reason, we called it Daydreaming. Daydreaming is not destructive and it converges asymptotically to stationary retrieval maps. When trained on random uncorrelated examples, the model shows optimal performance in terms of the size of the basins of attraction of stored examples and the quality of reconstruction. We also train the Daydreaming algorithm on correlated data obtained via the random-features model and argue that it spontaneously exploits the correlations thus increasing even further the storage capacity and the size of the basins of attraction. Moreover, the Daydreaming algorithm is also able to stabilize the features hidden in the data. Finally, we test Daydreaming on the MNIST dataset and show that it still works surprisingly well, producing attractors that are close to unseen examples and class prototypes. The presentation will be jointed together with Ludovica Serricchio.

Francesco D'Amico - Università di Roma Sapienza

A Self-attention-based dynamical system stores examples in marginal states

Transformers are one of the most successful architectures of modern deep neural networks. At their core there is the so-called attention mechanism, which recently interested the physics community as it can be written as the derivative of some energy function in certain cases. While it is possible to write the cross-attention layer as a modern Hopfield network, the same is not possible for the self-attention, which is used in the GPT architectures and other autoregressive models. In this work we show that it is possible to obtain the self-attention layer as the derivative of local energy terms, which resemble a pseudo-likelihood. First we show numerically that the self-attention dynamics has attractors even if there is no global energy function. Second, we leverage the analogy with pseudo-likelihood to design a recurrent model that can be trained without backpropagation: the dynamics shows transient states that are strongly correlated with both train and test examples. In conclusion, we present a novel framework that interprets self-attention as an attractor network, potentially paving the way for new theoretical approaches to understanding transformers.

Luca Maria Del Bono - Università di Roma Sapienza

Nearest-Neighbours Neural Network architecture for efficient sampling of statistical physics models

Efficiently sampling the Gibbs-Boltzmann distribution for spin glass systems is a hard task that is important both for the theoretical understanding of these models and for solving practical optimization problems. Recently, many attempts have been made to tackle the problem by mixing classical Monte Carlo schemes with newly devised Neural Networks. In this work, we introduce the Nearest-Neighbours Neural Network, a physically interpretable architecture that has a number of parameters scaling linearly with the size of the system and can be applied to a large variety of topologies. We show that the architecture is able to learn the Gibbs-Boltzmann distribution for the two-dimensional Edwards-Anderson model and reproduce properties such as the energy, the correlation function, and the overlap probability distribution. Finally, we show that the performance increases in relation to the number of layers used, in a way that clearly connects to the correlation length of the system.

Ivan Di Terlizzi - Max Planck Institute for the Physics of Complex Systems

Variance Sum Rule for Entropy Production

Non-equilibrium steady states, from the planetary scale to biological processes, are characterized by entropy production via energy dissipation to the environment, which is often challenging to measure. We introduce a variance sum rule (VSR) for displacement and impulse variances that permits us to measure the entropy production rate σ in nonequilibrium steady states. We first illustrate it for directly measurable forces, such as an active Brownian particle in an optical trap. By further introducing a model-dependent fitting procedure, we develop a method based on the VSR to derive σ from one-dimensional stochastic traces without measuring forces. In particular, we apply this inference procedure to flickering experiments in human red blood cells. We find that σ is spatially heterogeneous with a finite correlation length, and its average value agrees with calorimetry measurements.

Samantha Fournier - IPhT, CEA, Université de Paris-Saclay

Learning with high dimensional chaotic systems

Chaotic dynamics can naturally arise in high-dimensional heterogeneous systems of interacting variables. The simplest examples are random recurrent neural networks. I will discuss how to study simplified models of this kind through dynamical mean field theory (DMFT) and show that the corresponding chaotic dynamics can be tuned and shaped by synaptic connections to perform a set of interesting tasks. I will show how DMFT can be used to explore and describe the space of synaptic connections leading to good performances of the corresponding trained dynamical systems. This is based on:

Fournier, Urbani, Statistical physics of learning in high-dimensional chaotic systems, JSTAT 2023

Fournier, Urbani, to appear, 2024."

Leonardo Galliano - Università di Trieste

Policy-guided Monte Carlo for general state space: application to glass-forming mixtures

Traditional Monte Carlo methods provide a powerful tool to sample the equilibrium Boltzmann distribution of classical interacting systems, but they often struggle with glass-forming liquids and their slow relaxation.

To tackle this issue, we extend and refine a smart Monte Carlo algorithm inspired by reinforcement learning.

The algorithm dynamically adjusts the proposal distribution of the MH kernel in order to optimize the sampling efficiency.

We investigate how the method performs when applied to paradigmatic mixture models of glass-forming liquids.

We explore different smart Monte Carlo moves drawn from simple distributions, whose parameters are optimized in a continuing task through on-policy learning.

The method is at least 2 orders of magnitude faster than the swap Monte Carlo for discrete mixture models, and offers a promising solution to study more complex glassy systems.

Giulia Garcia-Lorenzana - ENS, Université Paris-Cité

Interactions and migration rescuing ecological diversity

How diversity is maintained in natural ecosystems is a long-standing question in Theoretical Ecology. By studying a system that combines ecological dynamics, heterogeneous interactions and spatial structure, we uncover a new mechanism for the survival of diversity-rich ecosystems in the presence of demographic fluctuations. For a single species, one finds a continuous phase transition between an extinction and a survival state, that falls into the universality class of Directed Percolation. Here we show that the case of many species with heterogeneous interactions is different and richer. By merging theory and simulations, we demonstrate that with sufficiently strong demographic noise, the system exhibits behavior akin to the single-species case, undergoing a continuous transition. Conversely, at low demographic noise, we observe unique features indicative of the ecosystem's complexity. The combined effects of the heterogeneity in the interaction network and migration enable the community to thrive, even in situations where demographic noise would lead to the extinction of isolated species. The emergence of mutualism induces the development of global bistability, accompanied by sudden tipping points. We present a way to predict the catastrophic shift from high diversity to extinction by probing responses to perturbations as an early warning signal.

Federico Ghimenti - Laboratoire MSC, Université Paris-Cité

What do clever algorithms for glasses do? The uses of time reparametrization invariance

Glass formers are characterized by increasingly sluggish dynamics as external parameters, such as density or temperature, are varied beyond a certain point, resulting in response times that increase by several orders of magnitude. To study these dynamics numerically, sophisticated methods are employed to accelerate the processes. We claim, based on analytical arguments and simulations, that algorithms with vastly different effectiveness work by reparametrizing the time over which the relaxation of the system occurs. Thus, the choice of algorithm has little to no impact on the relationships between observables at any given time. Our findings reconcile two views of glassy dynamics thus far considered to be mutually exclusive: one attributes the slowness to locally and collectively hindered mobility, while the other attributes it to the complexity of the global phase-space landscape. We argue that local constraints significantly reparameterize the flow of time, while the global landscape determines the relationships between operators at any given moment. The local constraints thus 'dress' the global landscape's properties. Our results imply that very fast algorithms, based on clever yet artificially designed dynamics, can be used to infer the time reparameterization-invariant properties of natural physical systems.

Chandraniva Guha Ray - MPI-PKS

Continuum mechanics of epithelial (un)buckling

Mechanics play a crucial role in driving the deformation of tissues during development. An example of this is apical constriction, during which the apical cell sides contract, splaying the cells and hence bending the cell sheet. Apical constriction is thus associated with the geometric singularity in which cell cross-sections become triangular and hence the cell sides are maximally bent. We explore the mechanical consequences of this geometric singularity in what is perhaps the simplest problem of tissue buckling: a monolayer of cells is described by a surface tensions of the apical, basal, and lateral sides of the cell cross-sections. Under external compression, the monolayer buckles, and, as the compression is increased further, fans of triangular cells expand from the crests and troughs of the buckled shape. Taking a continuum limit, we reveal and explain an intriguing secondary bifurcation beyond the onset of triangular cells: The buckling amplitude of a thin monolayer increases with increasing compression until the cells touch sterically, but, surprisingly, the buckling amplitude of a thick monolayer decreases with increasing compression.

Meeson Ha - Chosun University

Crossover Phenomena in Quasi long-range Order and Local Faceting

We introduce a model of Ising spins coupled to (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) growth, exhibiting local faceting and quasi-long-range Ising order. Ising spins, mimicking surface reconstructions, move with the interface and indirectly interact via its height. Local disorder hinders growth, while Ising Bloch walls introduce an energy bias. This interplay leads to mesoscopic facets with persistent (but renormalized) KPZ growth and quasi-long-range Ising order within facets. Through a Cole-Hopf transformation, we connect our model to the question of sign-order development in a directed path through a random medium. Our model demonstrates the existence of only quasi-long-range sign-order, providing a clearer picture than previous studies.

Roi Holtzmann - Weizmann Institute of Science

The Mpemba effect for phase transitions in Landau theory

The Mpemba effect describes the situation in which a hot system cools faster than an identical copy that is initiated at a colder temperature. In many of the experimental observations of the effect, e.g., in water and clathrate hydrates, it is defined by the occurrence of the phase transition. However, none of the theoretical investigations so far considered the timing of the phase transition, and most of the abstract models used to explore the Mpemba effect do not have a phase transition. In this work, we suggest a definition of phase transition time in a non-equilibrium state. In a Landau phenomenological model, we show, that a Mpemba effect with respect to phase transitions exists, namely that the hotter system undergoes the transition before the colder one when quenched to a cold temperature.

Jaeseok Hur - KAIST

Impact of Talent Configuration Properties on Wealth Dynamics

An individual’s success is determined by talent and luck of every moment. Moreover, it is also determined by social interactions. To analyze these three factors, we propose a new agent-based model that talent, luck, and social interactions with other people influence its capital by combining the ‘Talent versus Luck model’ [1] with the ‘Bouchaud-Mezard model’ [2].

Unlike the Bouchaud-Mezard model, we consider that agents have heterogeneous talent. As a result, the capital dynamics of system does not depend only on the network structure, but also on the all possible talent allocations on the network [3]. So we define it as the “talent configuration (TC) effect”. As quantifying the nature of talent configuration in two metrics: “talent assortativity” (TA) and “talent-degree correlation” (TD), we analyze the TC effect based on the impact of TA and TD on three major economic indices: growth rates ($n_{\rm rate}$), inequality (Gini coefficient, $n_{\rm Gini}$), and the correlation between capital level ‘L’ and talent ‘T’ (LT correlation, $n_{\rm LT}$).

We find that, in general, there is a positive correlation between TA and TD for growth rate, Gini coefficient, and LT correlation, where interesting outcomes are followings: First, high TA gives high growth rate, but it also gives high inequality. This is due to the trade-off relation between two quantities, so that we must reinforce social mix for equality but must weaken it for economic growth. Second, the LT correlation does not only depend on total amount of opportunity but also on interactions in the network with high TA or TD. This implies that meritocratic society is not only the result of individual’s independent accomplishment but also interactions among them. Lastly, the influence of nature of talent configuration on each index varies against the network structure. While the ‘lattice-like’ (Watt-Strogatz, WS) network that almost all agents have same degree, is more sensitive to TA, the ‘scale-free’ (Barabasi-Albert, BA) network that the distribution of degrees follows almost power-law, is more sensitive to TD. Our findings provide us the convincing explanation for the formation of centralized economy by selection and focus to some elites, as well as the formation of concentrated economy by clustering of high socioeconomic homophily groups.

[1] Pluchino, A., Biondo, A. E., \& Rapisarda, A. (2018). Talent versus luck: The role of randomness in success and failure. Advances in Complex systems, 21(03n04), 1850014.

[2] Bouchaud, J. P., \& Mézard, M. (2000). Wealth condensation in a simple model of economy. Physica A: Statistical Mechanics and its Applications, 282(3-4), 536-545.

[3] Stojkoski, V., Utkovski, Z., Basnarkov, L., \& Kocarev, L. (2019). Cooperation dynamics in networked geometric Brownian motion. Physical Review E, 99(6), 062312.

[4] J. Hur, M. Ha, \& H. Jeong, {\em Interplay of network structure and talent configuration on wealth dynamics}, \href{https://arxiv.org/abs/2404.04175}{arXiv:2404.04175}"

Persia Jana Kamali - Università Commerciale Luigi Bocconi

Dynamical mean field theory of confluent tissues and beyond

Confluent tissues are biological complex systems of interacting and self-propelling cells. Experiments show that such systems undergo phase transitions that play an important role in tissue formation and in the spread of metastatic cancer. Furthermore, the transition shares many properties similar to the jamming transition observed in particulate matter. Inspired by these observations, a model has been proposed where the Voronoi description of confluent tissues is mapped to a random Continuous Constraint Satisfaction Problem (CCSP) with equality constraints: by solving the Hamiltonian with the replica method, the model predicts the same rigidity transition of Vertex/Voronoi models for confluent tissues. In this paper, we re-propose the same model to study the dynamical properties of confluent tissues, under Gradient Descent (GD) and in the mean field limit. With the help of the Dynamical Mean Field Theory (DMFT) description for statistical mechanics, we derive the dynamical equations and we propose an efficient algorithm for their integration. By comparing the results with numerical simulations, we confirm the correctness of the theory and the existence of the rigidity transition observed both experimentally and theoretically. In addition and in the context of optimization science, we show that GD is blind to Replica Symmetry Breaking (RSB), when it occurs at zero temperature.

Jaron Kent-Dobias - INFN Sezione di Roma I

Algorithm-independent bounds on the performance of optimization using marginal complexity

Optimization seeks extremal points in a function. When there are superextensively many optima, optimization algorithms are liable to get stuck. Under these conditions, generic algorithms tend to find marginal optima, which have many nearly flat directions. We introduce a technique to count marginal optima in random landscapes and use the statistics of marginal optima calculated using this technique to produce generic bounds on optimization, based on the simple principle that algorithms will overwhelmingly tend to get stuck only where marginal optima are found. We demonstrate the idea on a simple non-Gaussian problem of maximizing the sum of squared random functions on a compact space. Numeric experiments using both gradient descent and generalized approximate message passing algorithms fall inside the expected bounds.

Tobias Kühn - Institut de la Vision

Computing free energies as diagrammatic expansions around non-Gaussian theories

In condensed-matter and high-energy physics, diagrammatic rules have been used since the days of Feynman to compute free energies and quantities derived from them. In statistical physics, the application of the corresponding techniques were hampered for a long time because many of them rely on the assumption of a Gaussian theory to expand around. We here present steps towards lifting this restriction, deriving diagrammatic rules to compute free energies as expansions around non-Gaussian theories. Furthermore, we show how to apply this method in systems such as interacting spins and simple liquids and suggest more applications in the realm of soft matter and inference.

Qinyi Liao - Wenzhou Institute, University of Chinese Academy of Sciences

Unifying vitrification, crystallisation and glass-forming ability in an exactly solved model

Structural orders are vital in vitrification, crystallisation and glass-forming ability. Despite numerous experimental and numerical investigations, the stringent relation between structural orders and glass transition remains elusive. On the other hand, the mean-field theory based on replica symmetry (RSB) breaking is clear in physics but less helpful in guiding practical applications. Here, we merge the two lines for the first time. We propose a synthetic theory combining Ising symmetry breaking and RSB by studying a spin-glass model borrowed from information science. This model has a locally favoured order competing against ferromagnetic order, i.e., crytsalline order, which gives rise to internal frustrations and provides a minimum paradigm of the competition between structural orders. Upon cooling the supercooled paramagnetic liquids, it either develops the competing order continuously and ends up in a glass phase or crystalises depending on preordering and cooling rate as the bifurcation of crystallisation and vitrification in structural glass formers. This growing order in supercooled liquids captures the structural ordering found in broad realistic systems. It thus unambiguously clarifies the relationship between competing ordering and vitrification. In addition, by mimicking the glass formation, we enhance glass-forming ability by introducing local defects to encourage the frustrations in liquids.

Claudio Chilin - Universidad Complutense de Madrid

Daydreaming Hopfield Networks and their surprising effectiveness on correlated data

David Machado Pérez - University of Havana

Approximated Master Equations for local search algorithms in random K-SAT instances

The theoretical description of local search algorithms used to solve combinatorial optimization tasks is an open problem in the physics of disordered systems and in computer science. Despite the efforts, this issue has persisted for over 20 years. Today, the links between algorithmic dynamics in random graphs and the thresholds obtained from statistical physics are still not clear. We apply two systems of approximated master equations, the Cavity Master Equation (CME) and the Approximate Conditional Dynamics (CDA), to an emblematic case of combinatorial optimization: Random K-Satisfiability. Both techniques are used to reproduce the behavior of the well-known algorithms WalkSAT and Focused Metropolis Search (FMS). We get qualitative agreement in broad areas of their phase diagrams. The description of the dynamics is accurate below the clustering transition. The single instance version of our equations noticeably outperforms the average case version. However, even the same averaged equations possess notable advantages over other known approximations in the literature. With a detailed comparison between techniques, this work paves the way for future studies of algorithmic dynamics.

Antoine Maillard - ETH Zürich

Bayes-optimal learning of an extensive-width neural network from quadratically many samples

I will present how recent progress on the problem of ellipsoid fitting in high-dimensional probability led to sharply characterize Bayes-optimal learning in a neural network with a quadratic activation function, with width growing linearly in the data dimension, and a number of data samples quadratic in said dimension. To the best of our knowledge, this is the first characterization of optimal learning in neural networks in this challenging regime. This is joint work with E. Troiani, S. Martin, F. Krzakala, and L. Zdeborová.

John Mateus - Universidad de los Andes, Bogotà

Wigner Surmise: How general it can be?

We study the time-evolution from an initial state to an equilibrium state for a 2D-Dyson gas of $N$ charged particles interacting through a 2D-logarithmic Coulomb potential surrounded by a thermal bath at a reduced temperature $\beta=q^2/(k_BT)$, with $q_0$ the charge per particle, $T$ the temperature of the bath and $k_B$ the Boltzmann’s constant, for $\beta\in[0.1, 4.0]$. We analyze the standard deviation of two-particle distances using a standard growth model in logarithmic independent variable, and the spacing distribution between nearest neighbors using a generalized Wigner’s distribution model from which we can know the standard deviation to compare with the initial analysis. We show how a logarithmic-time-law scale governs the time-evolution of this process and prove the validity of Wigner’s Surmise for $\beta\geq1.0$ compared with those values used in Gaussian ensembles for times greater than relaxation time $\tau\gg\tau_{\text{Eq}}$, i.e., when the system has reached the thermal equilibrium.

Matteo Negri - Università di Roma Sapienza

Random Feature Hopfield Networks generalize retrieval to previously unseen examples

It has been recently shown that, when an Hopfield Network stores examples generated as superposition of random features, new attractors appear in the model corresponding to such features. In this work we expand that result to superpositions of a finite number of features and we show numerically that the network remains capable of learning the features. Furthermore, we reveal that the network also develops attractors corresponding to previously unseen examples generated with the same set of features. We support this result with a simple signal-to-noise argument and we conjecture a phase diagram.

Alessandro Pacco - LPTMS

Local arrangement of stationary points in high-dimensional random landscapes: correlations and signatures of clustering

In this work we aim at making progress in the understanding of the energy landscape of prototypical high-dimensional random functions, namely spherical spin glasses. Our goal is to infer important insights on the activated dynamics of such models, by computing the three-point complexity. Starting from the assumption that the optimal energy barrier between stationary points of the landscape grows with the distance between them in configuration space, we identify two possible scenarios. We observe that jumps at equal energy are "memoryless", and characterized by the same typical barrier, whereas jumps at higher energy are correlated, and large barriers are followed by smaller ones. This second scenario is observed in the form of clustering, meaning that close to a deep local minimum, higher energy stationary points are more densely packed than usual. This makes a link with previously observed simulations in finite dimensional glasses and driven interfaces, where large activated jumps are followed by smaller activations.

Ebo Peerboms - University of Amsterdam

On the choice of basis for maximum entropy models for binary variables

We comment on and clarify the difference between two representations of binary variables used in the statistical inference of maximum entropy spin models with higher-order interactions of any order. We show that the choice of representation corresponds to a particular way of partitioning the observed state space. This leads to two fundamentally different notions of the meaning of higher-order interactions between binary variables.

Ludovica Serriccio - Università di Roma Sapienza

Daydreaming Hopfield Networks and their surprising effectiveness on correlated data

Dan Shafir - Bar-Ilan University

Disorder-Induced Anomalous Mobility Enhancement in Confined Geometries

The diffusion of particles with passage times significantly slower than regular Brownian motion is observed in various systems, such as amorphous materials, living cells and rheology. This behavior is typically attributed to trapping or waiting times that are scale-free and uncorrelated. Our work demonstrates that correlated waiting times, termed quenched disorder, can redefine our understanding of transport properties.

We show that the mobility of a driven particle anomalously depends on channel width, increasing as it grows narrower. Remarkably, this effect suggests a reduction in friction for flow as the channel constricts, opposing expectations based on regular or even anomalous transport dynamics. We further reveal that modifying geometrical constraints in the presence of quenched disorder alters the statistics of rare events, notably extremely large trapping times, resulting in surprising alterations to motion dynamics.

D. Shafir and S. Burov, arXiv:2403.01665 (2024)

Gangmin Son - KIAS

Phase transitions of mutual percolation on one-dimensional multiplex networks with long-range connections

We numerically study the phase transitions of mutual percolation in one-dimensional multiplex networks with long-range connections. In particular, the connection probability follows $\sim 1/r^{(1+\sigma)}$ where $r$ is the distance between two nodes and $\sigma$ controls the extent of long-range connections. It has been argued that the necessary condition for continuous transitions is $0<\sigma<1$, with $\sigma=0$ indicating the point where clustering, i.e., the abundance of triangles, vanishes. Our results suggest that the lower bound of $\sigma$ for continuous transitions can vary from approximately $0$ to $1/3$, depending on subtle problem settings involving edge overlaps. Based on the fact that $\sigma$ is related to the spectral dimension, we discuss our findings and related works from this perspective.

Davide Straziota - Bocconi University

Sampling through Algorithmic Diffusion in Non-Convex Perceptron Problems

We analyze the problem of sampling from a known but intractable distribution by employing a denoising diffusion process, with the score function provided by an oracle such as Approximate Message Passing. We introduce a formalism based on the replica method that can be applied to a generic ensemble of problems and allows us to characterize the process and its limitations in the infinite dimensional limit.

We then apply our theoretical framework considering as target the uniform measure over solutions in two non-convex random constraint satisfaction problems. For the spherical perceptron with negative stability, we show that approximate uniform sampling is achievable across the entire replica symmetric region of the phase diagram. Conversely, in the binary perceptron, uniform sampling via diffusion invariably fails due to the overlap gap property exhibited by the typical set of solutions. We circumvent this issue by introducing a tilted measure, which favors clusters of solutions. With this new target for the diffusion process, we are able to perform controlled sampling of binary solutions when the density of constraints is sufficiently low.

Tommaso Tonolo - GSSI

Generalized Lotka Volterra model on the Bethe lattice

Recent times have witnessed a burst of activity on the application of equilibrium and non-equilibrium statistical mechanics to study the behaviour of large ecosystems, in particular their stability and the nature of their equilibria. In particular, many results on the coexistence of many species have been obtained using the Generalized Lotka-Volterra model. The latter, under appropriate hypothesis on the shape of the interaction matrix between species and the stochasticity concourring to the dynamics (demographic noise), allows to recast the dynamical stability problem in terms of equilibrium statistical mechanics. We present here for the first time results on the equilibrium statistical mechanics of the Generalized Lotka-Volterra model with an interaction network between species which is sparse (Bethe lattice). Our analysis, at variance with the standard approach which makes use of a dense interaction network, reveals novel and highly non-trivial heterogeneity effects in the populations distributions, as for instance strong deviations from Gaussianity when increasing the heterogeneity of intra-species interactions. These results are in accordance with data from real ecosystems and also with other different models for ecological communities, as in [J. Grilli, Nature communications 11(1), 4743 (2020)]. In this talk I will review how the effective Hamiltonian for species interactions derived in [A. Altieri, F. Roy, C. Cammarota, G. Biroli, Phys. Rev. Lett. 126, 258301 (2021)] can be used to generate local marginals for populations distribution abundance when interactions are sparse, and I will present the main results obtained varying both the temperature (strength of demographic noise) and the heterogeneity of species interactions.

Pietro Valigi - Università di Roma Sapienza

Local sign stability and its implications for spectra of sparse random graphs and stability of ecosystems

In the very recent years, an increasing amount of interest has been devoted to the study of models of ecosystems defined on sparse random graphs. In this scenario both network topology and interactions nature play a relevant role in establishing the properties of the ecological model. In particular, differently from what usually happens in dense random matrices, the spectra of locally tree-like graphs with purely predator-prey interactions remain confined in a bounded region on the real axis instead of growing with the system size. Accordingly, these sparse models does not exhibit a trade-off between size and stability. This feature can provide insights into the modelling of ecosystems, especially with regard to the so-called complexity-stability debate.