arXiv subjects

Valentin Lallemant

Publications and source records attributed to Valentin Lallemant.

3 recordsLinked to original sources

arXiv2026

Structure and statistical organization of the stationary state of the Oslo model

In most driven-dissipative sandpile models, the dynamics of the system reaches a critical stationary state. This state displays organization features such as a power-law avalanche spectrum and hyperuniformity, but these features often emerge without a clear path from the microscopic evolution rules. Only in a few cases is there an available description of the stationary state, in other sandpile models the question is open. In this article, we present our result on the stationary state of the Oslo model, a driven-dissipative sandpile model with intrinsic randomness. In order to do so, we use different representations of the system configurations and of the dynamical process. Moving back and forth between these representations allows to identify invariant quantities for each configurations. Moreover, we obtain the detailed statistical description of the stationary state by considering all paths leading to a given configuration at once, and by summing their contributions under the constraint specified by the invariants. As a result, we find that the configurations of the stationary state are structured into a small number of equivalence classes, and that their statistical weights are related to the counting of colored diagrams respecting a small set of rules.

cond-mat.stat-mech↗

arXiv2025

Time transport correlations in abelian sandpile models

Sandpiles form one of the largest class of models displaying a critical stationary state. Despite a few decades of research, a comprehensive and systematic rigorous characterisation of their spatial and, even more, time dependent properties has remained elusive. Among the obstacles, we can mention their out of equilibrium and non-linear dynamics features which prevent, in general, the access to the stationary properties explicitly. In fact, even the knowledge of the stationary state is quite exceptional in sandpiles. In that respect, it has become standard to develop a model to model strategy and, so to say, general results or tools applicable to these systems are missing. In this paper, we unveil general and simple properties of time transport correlations in certain classes of abelian sandpile models. We proceed gradually, starting from results applicable in a broad context, to more and more specific ones, consequently valid to smaller and smaller classes. For instance, we show, under a few hypothesis, that the number of particles dissipated displays mostly anticorrelation in time. Besides, on a more integrable point of view, the approach followed might culminate with the proof of a link between 2-points time transport correlations and the second moment of the integrated transport over time. To be clear, these two quantities are related through a linear system of equations which is explicitly solved and applies to at least three 1D sandpile models, namely the Directed Stochastic Sandpile, the Oslo and the Activated Random Walk (in a peculiar setup) models.

cond-mat.stat-mech↗

arXiv2025

Space-time correlations in the 1D Directed Stochastic Sandpile model

Sandpile models are known to resist exact results. In this direction, space-time correlations between avalanches have proven to be especially difficult to access. One of the main obstacle to do so comes from taking memory effects in a systematic way along the computation. In this paper, we partially fill this gap and derive recursive relations for the particle filling and avalanche 2-points correlation function in the 1D Directed Stochastic Sandpile. These expressions allow to characterize the sign of the correlations and estimates are provided in the particle filling case. In fact, density correlations are shown to be positively correlated. This behavior is directly related to persistence of the local particle filling. On the other hand, we show that avalanches are anticorrelated in the model. This is interpreted by the fact that avalanches disrupt the system and the damage can only be fully compensated after injecting a sufficiently high number of particles. These results indicate an underlying trade off, between static and dynamic observable, for the system to sit in its stationary state. It appears that this balance is controlled by the conservation of the particle number along the avalanches.

cond-mat.stat-mech↗