Reservoir Machine Learning and the Sandpile Identity: Fast Surrogates and Field-Theoretic Insights

Abstract: The Abelian sandpile model exemplifies self-organized criticality, but computing its identity configuration becomes costly at large scales. I will show how echo state networks, trained with scale-aware features and GPU stabilization, can efficiently predict both finite-size identities and their scaling-limit structure. A field-theoretic perspective reveals why this works: in the stable regime, reservoirs act as random neural operators whose kernels resemble Green’s functions, bridging cellular automata, statistical physics, and machine learning.*

Power law in a specific parametric case followed from one-dimensional tropical sandpile model

Abstract: Power law-like distributions have been observed in real-world phenomena and are a common hallmark of self-organized criticality. In the one-dimensional tropical sandpile model, specific parametric case exhibit power-law behavior for the distribution of relaxation length, where a parameter can influence the deformation of the system’s hypersurface and the associated probability distribution of relaxation length. Theoretical research and numerical experiments confirm this inference, suggesting that the relaxation process, which involves deforming the hypersurface to a set of finite points, can be described by a power law, indicating a universal critical behavior in the system.

Clash of States

Abstract: We propose a different approach to studying the sandpile model, involving the superposition of initial configurations and the analysis of their relaxation. Using two random states, obtained through random walks and spanning trees, we observe the activation dynamics of the vertices and quantify their frequency with an odometer, thus revealing stability patterns and critical activity zones. This methodology allows the competition between configurations to be visualized using heat maps and three-dimensional representations, where the characteristic structures of the self-organization process emerge.

Scaling limits in sandpiles

Abstract: Abelian sandpile model is a prototype of self-organised criticality, one structural characteristic feature of which is its scale transcendence. At a formal mathematical level, this property manifests in the form of scaling limit theorems, which can have very different flavours, from purely algebraic, notably for the extended sandpile group, to inherently graphical, with a particular example being the emergence of the tropical sandpile model. This talk serves as a brief overview of several such results.

On Steiner Trees

Abstract: In this talk, I will explore various aspects of Steiner trees, starting with the classical formulation of the problem. My focus will be on some irregular examples, specifically Steiner trees that may have an infinite number of branching points. I will also discuss the uniqueness of solutions, examining cases where multiple minimal trees can exist. Essentially, I aim to share everything I know about Steiner trees — or at least as much as time allows.

Statistics of Tropical Sandpiles: Degree Growth, Coefficient Laws, and Tree Structure of Curves

Smalls order elements in the Teotitlan group

Abstract: We look into a cyclic subgroup of the sandpile group, generated by the so-called Teotitlan element. We will discuss its generation, its growth and how various small-order elements in this group look. Some hypothesis for the properties of this group will be discussed, as well as potential approaches for further insight into the group.