arXiv subjects

Clélia de Mulatier

Publications and source records attributed to Clélia de Mulatier.

13 recordsLinked to original sources

arXiv2026

A formal framework for higher-order spin models via hypergraphs, polymatroids, and the Tutte polynomial

We develop a rigorous mathematical framework for statistical mechanics models on hypergraphs, and give conditions for lifting the classical connection between Potts model partition functions and the Tutte polynomial from graphs to hypergraphs. We define hypergraphical models and their associated partition functions, and extend these to special classes of models induced by families of interaction functions. For boolean interaction families, whose interaction functions map to $\{0,1\}$, we show that the partition function is determined by a combinatorial rank function, and we establish sufficient conditions for a hypergraph deletion-contraction recurrence and for when the rank function defines a polymatroid. We illustrate the theory by applying it to three hypergraph interaction families: Parity Ising, Delta Potts, and And Ising. The induced hypergraphical models are not isomorphic to each other, but the first two reduce to the same graphical Ising models. We identify three polymatroids naturally associated with hypergraphs for these models: respectively, the binary matroid of the incidence matrix over $\mathbb{F}_2$, the hypergraphical polymatroid, and the boolean polymatroid. For graphs, the first two reduce to the classical graphical matroid, and their partition functions recover the multivariate Tutte polynomial. The Tutte polynomial thus admits at least two distinct generalizations for hypergraphs, both satisfying a deletion-contraction recurrence: the Tutte polynomial of the binary matroid of the hypergraph incidence matrix, and a multivariate version of the Poincaré polynomial of the hypergraphical polymatroid. The partition functions of And Ising models are likewise multivariate versions of the Poincaré polynomial, here of the boolean polymatroid. These examples illustrate the much greater range of hypergraphical models and underscore the need for the unifying theory.

math.CO↗

arXiv2026

Modeling Discrete Data with High-Order Vector Potts Models

Modeling high-dimensional data is challenging, yet essential to understanding many complex systems. Maximum entropy models such as Ising and Potts models have been used extensively to capture pairwise interactions from correlation patterns in data, allowing to infer graphical representations of complex systems from observations (e.g., from protein sequences or neural population activity). Recently, there has been growing interest in modeling higher-order correlation patterns involving simultaneously three or more variables. While progress has been made in binary data with high-order Ising models, we extend this framework to the more general case of discrete data. We introduce q-state spin models, a complete family of maximum entropy models that generalize the vector Potts model to include long-range and arbitrary high-order interactions. In the pairwise case, our models allow for more diverse interaction types compared to the standard vector Potts model. We discuss their statistical interpretation with examples and relate them to discrete Fourier analysis. Using a loop expansion of the partition function, we show that the statistical properties of spin models are fully captured by the algebraic structure of their interactions. We define gauge transformations under which this structure, and thus the partition function, remains invariant. Models equivalent under gauge transformations can be seen as different representations of the same abstract statistical model, despite generally having interactions of different orders, extending results from the binary case. For practical application to data analysis, we focus on a subset of models known in the binary case as Minimally Complex Models, generalizing them to discrete data. We obtain a closed-form expression for the marginal likelihood of these models, enabling fast model selection. We illustrate their use with simple real-world examples.

stat.ME↗

arXiv2026

Abstract Markov Random Fields

Markov random fields are known to be fully characterized by properties of their information diagrams, or I-diagrams. In particular, for Markov random fields, regions in the I-diagram corresponding to disconnected vertex sets in the graph vanish. Recently, I-diagrams have been generalized to F-diagrams, for a larger class of functions F satisfying the chain rule beyond Shannon entropy, such as Kullback-Leibler divergence and cross-entropy. In this work, we generalize the notion and characterization of Markov random fields to this larger class of functions F and investigate preliminary applications. We define F-independences, F-mutual independences, and F-Markov random fields and characterize them by their F-diagram. In the process, we also define F-dual total correlation and prove that its vanishing is equivalent to F-mutual independence. We then apply our results to information functions F that are applied to probability mass functions. We show that if the probability distributions of a set of random variables are Markov random fields for the same graph, then we formally recover the notion of an F-Markov random field for that graph. We then study the Kullback-Leibler diagrams on specific Markov chains, leading to a visual representation of the second law of thermodynamics and a simple explicit derivation of the decomposition of the evidence lower bound for diffusion models.

cs.IT↗

arXiv2025

Effective dimensional reduction of complex systems based on tensor networks

The exact treatment of Markovian models of complex systems requires knowledge of probability distributions exponentially large in the number of components $n$. Mean-field approximations provide an effective reduction in complexity of the models, requiring only a number of phase space variables polynomial in system size. However, this comes at the cost of losing accuracy close to critical points in the systems dynamics and an inability to capture correlations in the system. In this work, we introduce a tunable approximation scheme for Markovian spreading models on networks based on Matrix Product States (MPS). By controlling the bond dimensions of the MPS, we can investigate the effective dimensionality needed to accurately represent the exact $2^n$ dimensional steady-state distribution. We introduce the entanglement entropy as a measure of the compressibility of the system and find that it peaks just after the phase transition on the disordered side, in line with the intuition that more complex states are at the 'edge of chaos'. We compare the accuracy of the MPS with exact methods on different types of small random networks and with Markov Chain Monte Carlo methods for a simplified version of the railway network of the Netherlands with 55 nodes. The MPS provides a systematic way to tune the accuracy of the approximation by reducing the dimensionality of the systems state vector, leading to an improvement over second-order mean-field approximations for sufficiently large bond dimensions.

cond-mat.stat-mech↗

arXiv2024

Bayesian Inference of Minimally Complex Models with Interactions of Arbitrary Order

Finding the model that best describes a high-dimensional dataset is a daunting task, even more so if one aims to consider all possible high-order patterns of the data, going beyond pairwise models. For binary data, we show that this task becomes feasible when restricting the search to a family of simple models, that we call Minimally Complex Models (MCMs). MCMs are maximum entropy models that have interactions of arbitrarily high order grouped into independent components of minimal complexity. They are simple in information-theoretic terms, which means they can only fit well certain types of data patterns and are therefore easy to falsify. We show that Bayesian model selection restricted to these models is computationally feasible and has many advantages. First, the model evidence, which balances goodness-of-fit against complexity, can be computed efficiently without any parameter fitting, enabling very fast explorations of the space of MCMs. Second, the family of MCMs is invariant under gauge transformations, which can be used to develop a representation-independent approach to statistical modeling. For small systems (up to 15 variables), combining these two results allows us to select the best MCM among all, even though the number of models is already extremely large. For larger systems, we propose simple heuristics to find optimal MCMs in reasonable times. Besides, inference and sampling can be performed without any computational effort. Finally, because MCMs have interactions of any order, they can reveal the presence of important high-order dependencies in the data, providing a new approach to explore high-order dependencies in complex systems. We apply our method to synthetic data and real-world examples, illustrating how MCMs portray the structure of dependencies among variables in a simple manner, extracting falsifiable predictions on symmetries and invariance from the data.

cs.AI↗

arXiv2023

Efficient simulations of epidemic models with tensor networks: application to the one-dimensional SIS model

The contact process is an emblematic model of a non-equilibrium system, containing a phase transition between inactive and active dynamical regimes. In the epidemiological context, the model is known as the susceptible-infected-susceptible (SIS) model, and widely used to describe contagious spreading. In this work, we demonstrate how accurate and efficient representations of the full probability distribution over all configurations of the contact process on a one-dimensional chain can be obtained by means of Matrix Product States (MPS). We modify and adapt MPS methods from many-body quantum systems to study the classical distributions of the driven contact process at late times. We give accurate and efficient results for the distribution of large gaps, and illustrate the advantage of our methods over Monte Carlo simulations. Furthermore, we study the large deviation statistics of the dynamical activity, defined as the total number of configuration changes along a trajectory, and investigate quantum-inspired entropic measures, based on the second Rényi entropy.

cond-mat.stat-mech↗

arXiv2018

The Stochastic complexity of spin models: Are pairwise models really simple?

Models can be simple for different reasons: because they yield a simple and computationally efficient interpretation of a generic dataset (e.g. in terms of pairwise dependences) - as in statistical learning - or because they capture the essential ingredients of a specific phenomenon - as e.g. in physics - leading to non-trivial falsifiable predictions. In information theory and Bayesian inference, the simplicity of a model is precisely quantified in the stochastic complexity, which measures the number of bits needed to encode its parameters. In order to understand how simple models look like, we study the stochastic complexity of spin models with interactions of arbitrary order. We highlight the existence of invariances with respect to bijections within the space of operators, which allow us to partition the space of all models into equivalence classes, in which models share the same complexity. We thus found that the complexity (or simplicity) of a model is not determined by the order of the interactions, but rather by their mutual arrangements. Models where statistical dependencies are localized on non-overlapping groups of few variables (and that afford predictions on independencies that are easy to falsify) are simple. On the contrary, fully connected pairwise models, which are often used in statistical learning, appear to be highly complex, because of their extended set of interactions.

cond-mat.dis-nn↗

arXiv2017

Fractality in nonequilibrium steady states of quasiperiodic systems

We investigate the nonequilibrium response of quasiperiodic systems to boundary driving. In particular we focus on the Aubry-André-Harper model at its metal-insulator transition and the diagonal Fibonacci model. We find that opening the system at the boundaries provides a transparent and experimentally viable technique to probe its underlying fractality, which is reflected in the fractal spatial dependence of simple observables (such as magnetization) in the nonequilibrium steady state. We also find that the dynamics in the nonequilibrium steady state depends on the length of the chain chosen: generic length chains harbour qualitatively slower transport (different scaling exponent) than Fibonacci length chains, which is in turn slower than in the closed system. We conjecture that such fractal nonequilibrium steady states should arise in generic driven critical systems that have fractal properties.

cond-mat.dis-nn↗

arXiv2015

The critical catastrophe revisited

The neutron population in a prototype model of nuclear reactor can be described in terms of a collection of particles confined in a box and undergoing three key random mechanisms: diffusion, reproduction due to fissions, and death due to absorption events. When the reactor is operated at the critical point, and fissions are exactly compensated by absorptions, the whole neutron population might in principle go to extinction because of the wild fluctuations induced by births and deaths. This phenomenon, which has been named critical catastrophe, is nonetheless never observed in practice: feedback mechanisms acting on the total population, such as human intervention, have a stabilizing effect. In this work, we revisit the critical catastrophe by investigating the spatial behaviour of the fluctuations in a confined geometry. When the system is free to evolve, the neutrons may display a wild patchiness (clustering). On the contrary, imposing a population control on the total population acts also against the local fluctuations, and may thus inhibit the spatial clustering. The effectiveness of population control in quenching spatial fluctuations will be shown to depend on the competition between the mixing time of the neutrons (i.e., the average time taken for a particle to explore the finite viable space) and the extinction time.

cond-mat.stat-mech↗

arXiv2014

Clustering of branching Brownian motions in confined geometries

We study the evolution of a collection of individuals subject to Brownian diffusion, reproduction and disappearance. In particular, we focus on the case where the individuals are initially prepared at equilibrium within a confined geometry. Such systems are widespread in physics and biology and apply for instance to the study of neutron populations in nuclear reactors and the dynamics of bacterial colonies, only to name a few. The fluctuations affecting the number of individuals in space and time may lead to a strong patchiness, with particles clustered together. We show that the analysis of this peculiar behaviour can be rather easily carried out by resorting to a backward formalism based on the Green's function, which allows the key physical observables, namely, the particle concentration and the pair correlation function, to be explicitly derived.

cond-mat.stat-mech↗

arXiv2014

Universal properties of branching random walks in confined geometries

Characterizing the occupation statistics of a radiation flow through confined geometries is key to such technological issues as nuclear reactor design and medical diagnosis. This amounts to assessing the distribution of the travelled length $\ell$ and the number of collisions $n$ performed by the underlying stochastic transport process, for which remarkably simple Cauchy-like formulas were established in the case of branching Pearson random walks with exponentially distributed jumps. In this Letter, we show that such formulas strikingly carry over to the much broader class of branching processes with arbitrary jumps, provided that scattering is isotropic and the average jump size is finite.

cond-mat.stat-mech↗

arXiv2014

Cauchy's formulas for random walks in bounded domains

Cauchy's formula was originally established for random straight paths crossing a body $B \subset \mathbb{R}^{n}$ and basically relates the average chord length through $B$ to the ratio between the volume and the surface of the body itself. The original statement was later extended in the context of transport theory so as to cover the stochastic paths of Pearson random walks with exponentially distributed flight lengths traversing a bounded domain. Some heuristic arguments suggest that Cauchy's formula may also hold true for Pearson random walks with arbitrarily distributed flight lengths. For such a broad class of stochastic processes, we rigorously derive a generalized Cauchy's formula for the average length travelled by the walkers in the body, and show that this quantity depends indeed only on the ratio between the volume and the surface, provided that some constraints are imposed on the entrance step of the walker in $B$. Similar results are obtained also for the average number of collisions performed by the walker in $B$, and an extension to absorbing media is discussed.

cond-mat.stat-mech↗

arXiv2013

Asymmetric Lévy flights in the presence of absorbing boundaries

We consider a one dimensional asymmetric random walk whose jumps are identical, independent and drawn from a distribution ϕ(η) displaying asymmetric power law tails (i.e. ϕ(η) \sim c/η^{α+1} for large positive jumps and ϕ(η) \sim c/(γ|η|^{α+1}) for large negative jumps, with 0 < α< 2). In absence of boundaries and after a large number of steps n, the probability density function (PDF) of the walker position, x_n, converges to an asymmetric Lévy stable law of stability index αand skewness parameter β=(γ-1)/(γ+1). In particular the right tail of this PDF decays as c n/x_n^{1+α}. Much less is known when the walker is confined, or partially confined, in a region of the space. In this paper we first study the case of a walker constrained to move on the positive semi-axis and absorbed once it changes sign. In this case, the persistence exponent θ_+, which characterizes the algebraic large time decay of the survival probability, can be computed exactly and we show that the tail of the PDF of the walker position decays as c \, n/[(1-θ_+) \, x_n^{1+α}]. This last result can be generalized in higher dimensions such as a planar Lévy walker confined in a wedge with absorbing walls. Our results are corroborated by precise numerical simulations.

cond-mat.stat-mech↗