Symmetry Emergence in Self-Organized Criticality

Abstract: We describe a mechanism of affine symmetry emergence in the maximal density regime of the prototypical model of self-organized criticality when the inverse square of the mesh of the underlying lattice is much larger than the number of random perturbation points distributed according to a prescribed probability measure supported in the interior of the ambient convex domain. Moreover, an appropriate scaling limit of the toppling function (aka odometer), which counts the number of operations per site, is a solution to a non-linear partial differential equation well known in the context of optimal transport and differential geometry, making it possible to accurately estimate the deviation of the density from its maximal value in any macroscopic window. The mechanism for the affine symmetry emergence is due to the novel empirical fact, supported in addition by inductive arguments that have recently being upgraded to a rigorous proof, that the scaling limit of the toppling function is the unique concave solution of the Monge-Ampère equation with Dirichlet boundary condition on the convex domain with the potential given by the probability measure used above as the infinite-perturbation profile.

• The paper establishes a hierarchy of sandpile scaling limits in which discrete D₄ symmetry expands to GL₂(ℤ) in the tropical model and SL₂(ℝ) in the Monge–Ampère continuum limit.
• The authors show that, for generic perturbations with density ρ, the normalized tropical odometer N⁻¹ᐟ²Gₚ₁,…,ₚₙ0 converges to the unique concave solution of MA(f)=ρ with Dirichlet boundary conditions.
• The framework predicts microscopic density deviations, including uniformity for a uniformly perturbed disc when N<h⁻², while identifying boundary-supported profiles and nongeneric perturbation sequences as major open challenges.

The paper "Symmetry Emergence in Self-Organized Criticality" by Ernesto Lupercio and Mikhail Shkolnikov (2608.13500) describes a hierarchy of scaling limits for the Abelian sandpile model in its maximal-density regime, culminating in a continuous limit governed by the real Monge–Ampère equation. The central claim is that, as one passes from the discrete sandpile through the tropical sandpile to an "infinitely perturbed" state driven by a probability density $\rho$, the local symmetry group of the model enlarges from the dihedral group $D_4$ to $\mathrm{GL}_2(\mathbb{Z})$ and finally to the full special linear group $\mathrm{SL}_2(\mathbb{R})$. This yields an equi-affine invariant macroscopic description and a quantitative tool for estimating density deviations at the microscopic level.

Setup: the Abelian sandpile in the maximal-density regime

The authors work with the Bak–Tang–Wiesenfeld construction on $h\mathbb{Z}^2 \cap \Omega$, where $\Omega$ is a compact convex domain. States are integer-valued functions stabilized by topplings; the stabilization is characterized via the least action principle [fey2010growth], which identifies the toppling function (odometer) $H_\phi$ as the unique minimizer of the total action among stabilizing functions. Dhar's sandpile group of recurrent states [dhar1990self], whose order counts spanning forests by the matrix-tree theorem, provides the algebraic backbone; Creutz's image of its neutral element exhibits approximate scale invariance apart from linear defects.

The relevant regime is the maximal-density one: the initial state is $\langle 3\rangle$, perturbed by finitely many grains at sites near a fixed set $P$ in the interior of $\Omega$. Prior work of Kalinin and Shkolnikov established that, as $D_4$0, the locus where the stabilized configuration drops below $D_4$1 converges in the Hausdorff sense to a tropical curve $D_4$2 — a rational-slope graph satisfying the balancing condition, extremal for symplectic area in its incidence class, i.e., a solution of a Steiner-type problem. This connects the sandpile strings of Caracciolo, Paoletti and Sportiello [caracciolo2010conservation] to tropical geometry, as anticipated by Sadhu and Dhar [sadhu2012pattern].

Micro-to-meso: the tropical sandpile

The tropical sandpile model assigns to each state a pair $D_4$3, where $D_4$4 is a tropical series (an infimum of affine functions with integer gradients) vanishing on $D_4$5, and stability requires $D_4$6, the corner locus of $D_4$7. Stabilization $D_4$8 is defined variationally as the minimizer of the tropical action $D_4$9 over tropical series dominating $\mathrm{GL}_2(\mathbb{Z})$0 whose corner loci contain $\mathrm{GL}_2(\mathbb{Z})$1. The single-grain operator $\mathrm{GL}_2(\mathbb{Z})$2 is the scaling limit of the microscopic operation of adding, stabilizing, and removing a grain, and decomposes into powers of wave operators; string edges behave solitonically under waves, moving toward the source with speed inversely proportional to their density deviation.

Two consequences follow directly from this variational definition. First, the covariance identity

$\mathrm{GL}_2(\mathbb{Z})$3

expresses the mesoscopic symmetry enlargement from $\mathrm{GL}_2(\mathbb{Z})$4 to $\mathrm{GL}_2(\mathbb{Z})$5. Second, the rescaled density deviation concentrates on $\mathrm{GL}_2(\mathbb{Z})$6 as the push-forward of the symplectic area of the holomorphic lift of the curve. The paper notes a trade-off here: the third formulation of the scaling limit recovers the density but sacrifices tropical covariance.

Meso-to-macro: Monge–Ampère dynamics and $\mathrm{GL}_2(\mathbb{Z})$7 emergence

The new contribution considers the Markov chain of the original Bak–Tang–Wiesenfeld type, but on the tropical sandpile, with perturbation points drawn from a fixed density $\mathrm{GL}_2(\mathbb{Z})$8 supported in the interior of $\mathrm{GL}_2(\mathbb{Z})$9. Unlike the finite recurrent class of the discrete model, no recurrent states exist here; the total action diverges as $\mathrm{SL}_2(\mathbb{R})$0 in two dimensions (observed numerically in prior work), consistent with the general $\mathrm{SL}_2(\mathbb{R})$1 asymptotic argued dimensionally.

The mechanism is clearest in one dimension, where the model is exactly solvable: the negative second derivative of the stabilized series equals a sum of Dirac masses at the perturbation points plus one lattice-determined point, so $\mathrm{SL}_2(\mathbb{R})$2 converges to the solution of $\mathrm{SL}_2(\mathbb{R})$3 with Dirichlet boundary conditions. In two dimensions, the correct replacement is the Monge–Ampère operator $\mathrm{SL}_2(\mathbb{R})$4, where multiplicities are volumes spanned by minimizing gradients. For generic data and smooth boundary, the corner locus $\mathrm{SL}_2(\mathbb{R})$5 is a non-singular trivalent tropical curve with exactly $\mathrm{SL}_2(\mathbb{R})$6 cycles; counting vertices gives $\mathrm{SL}_2(\mathbb{R})$7, with terminal branches vanishing in the limit. Since vertex density tracks perturbation density — supported by simulations and by the theoretical derivation in [KLSS] — one has $\mathrm{SL}_2(\mathbb{R})$8. Because $\mathrm{SL}_2(\mathbb{R})$9 scales quadratically under dilation in $h\mathbb{Z}^2 \cap \Omega$0, the correct normalization is $h\mathbb{Z}^2 \cap \Omega$1, which converges to the unique concave solution $h\mathbb{Z}^2 \cap \Omega$2 of $h\mathbb{Z}^2 \cap \Omega$3 with Dirichlet boundary conditions.

The equi-affine invariance of $h\mathbb{Z}^2 \cap \Omega$4 then yields

$h\mathbb{Z}^2 \cap \Omega$5

which is precisely the claimed symmetry emergence. A useful robustness remark is that the choice of initial tropical series is immaterial: any starting series converges to the same critical state $h\mathbb{Z}^2 \cap \Omega$6. The rigorous proof of this second scaling limit is carried out in the companion theorem [KLSS]; the present note presents the mechanism and supporting evidence rather than a self-contained proof.

Macro-to-micro: quantitative consequences

Solving the macroscopic problem yields concrete microscopic predictions. For the unit disc with uniform $h\mathbb{Z}^2 \cap \Omega$7, the explicit solution $h\mathbb{Z}^2 \cap \Omega$8 implies that the density deviation of the perturbed state on a square window $h\mathbb{Z}^2 \cap \Omega$9 is approximately $\Omega$0 (the same formula holding for all ellipses by affine symmetry). Hence the perturbed disc remains uniformly dense provided $\Omega$1, which also delineates the validity threshold of the regime since density must remain positive. The authors candidly note that this example lies outside the reach of the proven theory, because the support of $\Omega$2 is not disjoint from the boundary; nevertheless numerical simulations agree with the derivation.

In the opposite extreme of a perturbation cloud concentrated near a point $\Omega$3 — modeled macroscopically by $\Omega$4 but implemented microscopically by generic perturbations near $\Omega$5 — the limiting odometer plot is a cone over $\Omega$6 with apex projecting to $\Omega$7. For strictly convex $\Omega$8 the density deviation is non-trivial everywhere, while a straight boundary segment $\Omega$9 produces a sector of vanishing deviation spanned by $H_\phi$0 and $H_\phi$1. The paper also formulates a precise conjecture: for mesh $H_\phi$2 with $H_\phi$3 and empirical measures converging to $H_\phi$4 supported away from straight boundary segments, the rescaled deviation $H_\phi$5 has a weak-* limit $H_\phi$6 characterized by the property that solving $H_\phi$7 returns the unique concave Monge–Ampère solution $H_\phi$8. Anomalous perturbation sequences (e.g., points approaching $H_\phi$9 along a fixed direction) fall outside the conjecture's hypotheses and indeed produce different limits, indicating that the genericity assumption on the perturbation cloud is essential.

Limitations and open questions

Several caveats are stated explicitly. The disc example with uniform profile violates the support condition required by the theory, so its agreement with simulation lacks a proof. The conjecture relating the Poisson equation to the Monge–Ampère solution remains unproven. The vertex-density heuristic $\langle 3\rangle$0 rests on simulations (to be documented elsewhere) and on the derivation in [KLSS]. Non-polygonal domains produce infinitely many vertices globally, though finitely many per compact interior region, complicating global statements. Finally, the idea of averaging over tropical structures to obtain further symmetries, suggested by Conan Leung's remarks, is acknowledged as not fully developed, with only preliminary results in [kalinin2026limits].

Conclusion

The paper organizes the sandpile hierarchy into three scales — Abelian, Tropical, Affine — and interprets the sequence of scaling limits as consecutive symmetry breakings running downward, or equivalently symmetry emergences running upward, terminating in equi-affine invariance governed by the Monge–Ampère equation. Beyond the structural statement, the framework delivers a computable estimate of density deviations in macroscopic windows, including the sharp threshold $\langle 3\rangle$1 for uniformity under uniform perturbation. The main open problems are the proof of the weak-* convergence conjecture and the extension of the theory to profiles whose supports meet the boundary.