Dissipative Abelian Sandpile Model

• The dissipative Abelian sandpile model is defined by non-conservative topplings via sinks or bulk diagonal shifts, resulting in finite, well-posed stationary configurations.
• It employs modifications of the graph Laplacian and Green function analysis to relate avalanche dynamics to mass loss and finite correlation lengths.
• Numerical simulations and analytic techniques, including spanning-tree bijections and logarithmic field correlators, reveal dual scaling regimes and critical threshold transitions.

The dissipative Abelian sandpile model refers to Abelian sandpile dynamics in which topplings are not fully conservative because grains are lost either to a sink or root, at open boundaries, or in the bulk through a positive excess on the diagonal of the toppling matrix. In the standard finite-graph formulation, dissipation is structurally necessary: if there are no dissipative sites, then the toppling matrix is singular, $\det \Delta=0$, and the usual recurrent stationary measure is absent; with dissipation, stabilization terminates, recurrent configurations form a finite Abelian group, and one obtains a well-defined driven stationary state (Ruelle, 2013, Járai, 2014). Bulk-dissipative families go further by making every toppling lose mass, thereby producing a noncritical, massive deformation of the critical Abelian sandpile (Katori, 2015).

1. Foundational formulations of dissipation

In the standard finite-volume Abelian sandpile, one works on a finite graph $G=(V\cup\{s\},E)$ with a distinguished sink $s$. A configuration is stable when every non-sink vertex has height below its threshold, and a toppling sends one grain along each incident edge; grains arriving at the sink are lost. In reduced-Laplacian form, toppling $x$ subtracts the $x$-th row of $\Delta_G'$, where $\Delta_G'$ is the graph Laplacian with the sink removed. Open boundaries are therefore dissipative because boundary sites have edges to the sink, whereas closed boundaries are conservative relative to the finite domain (Járai, 2014, Ruelle, 2013).

A second formulation places dissipation directly in the bulk by increasing the diagonal entries of the toppling matrix. In the “massive sandpile model,” every site is made dissipative by taking $\Delta_{i,i}=z_i+t$, so each toppling loses $t$ grains. A more explicit integer family is the toroidal dissipative Abelian sandpile with parameters $n,m\in\mathbb N$: a site is unstable at $G=(V\cup\{s\},E)$0 grains, a toppling sends $G=(V\cup\{s\},E)$1 grains to each of its $G=(V\cup\{s\},E)$2 neighbors, and exactly $G=(V\cup\{s\},E)$3 grains dissipate from the system. In the rescaled height convention of that model, the dissipation rate is $G=(V\cup\{s\},E)$4, and the threshold is $G=(V\cup\{s\},E)$5 (Ruelle, 2013, Katori, 2015).

These formulations should be distinguished. The sink-based model is the canonical finite ASM used for burning tests, spanning-tree bijections, and infinite-volume approximation. Bulk-dissipative models are off-critical from the outset and remain well-defined even on periodic lattices, because every toppling strictly decreases total mass (Katori, 2015).

2. Abelian algebra, recurrent states, and Green functions

Dissipation does not destroy Abelianity. On a finite graph with sink, legal topplings commute, stabilization is independent of toppling order, and the addition operators $G=(V\cup\{s\},E)$6 commute as well. The recurrent configurations $G=(V\cup\{s\},E)$7 form an Abelian group isomorphic to

$G=(V\cup\{s\},E)$8

and the number of recurrent states is

$G=(V\cup\{s\},E)$9

The stationary distribution of the driven model is uniform on $s$0, and Dhar’s burning test characterizes recurrence by ampleness on every nonempty subset. The same sink-based formalism yields the Majumdar–Dhar bijection between recurrent configurations and spanning trees rooted at the sink (Járai, 2014).

The same structural picture persists in the review treatment of dissipation as a diagonal perturbation of the toppling matrix. Making a bulk site dissipative means increasing a diagonal entry, for example

$s$1

so that toppling at $s$2 loses one extra grain to the root. In the spanning-tree language, such a site becomes a new sink through which arrows may flow to the root. Local dissipation insertions, boundary dissipation, and uniform bulk dissipation are therefore different implementations of the same algebraic idea: a positive diagonal contribution to $s$3 (Ruelle, 2013).

The Green function $s$4 controls avalanche observables. In finite volume, Dhar’s formula identifies the expected number of topplings with matrix elements of the inverse toppling matrix. On $s$5 with a prescribed dissipative set $s$6, the toppling matrix

$s$7

produces a killed-random-walk representation of $s$8, and summability of $s$9 is a sufficient criterion for non-criticality. In one dimension this correspondence becomes exact: criticality is equivalent to divergence of the expected stopping time of a trapped random walk (Redig et al., 2017, Rezzouk, 3 Sep 2025).

3. Bulk dissipation, mass generation, and finite correlation length

The toroidal family with parameters $x$0 gives an exact bulk-dissipative ASM in any dimension $x$1. Its avalanche propagator and the height-$x$2 correlation both decay exponentially in infinite volume, with correlation length

$x$3

As $x$4, the model approaches the conservative limit and

$x$5

For the propagator $x$6, the asymptotic form is of Ornstein–Zernike type, and the connected height-$x$7 correlation decays with exponent $x$8, reflecting its quadratic dependence on the off-diagonal Green function (Katori, 2015).

A complementary off-critical construction studies a dissipative ASM on a lattice of coordination number $x$9, mainly on the triangular lattice with $x$0. There the local toppling threshold is raised from $x$1 to $x$2, and the excess $x$3 grains are lost on each toppling. In field-theory language the paper parameterizes dissipation by a mass

$x$4

To realize small dissipation in simulations, the model is implemented with

$x$5

The continuum Green function becomes massive,

$x$6

with asymptotic decay $x$7, so the correlation length scales as $x$8. The measured cutoff radius $x$9 obeys $\Delta_G'$0, wave exponents remain close to critical values in the ultraviolet regime, and avalanche exponents shift with mass, showing stronger sensitivity to off-criticality (Najafi et al., 2012).

The massive-sandpile review formulates the same phenomenon with a uniform diagonal shift $\Delta_G'$1, for which the inverse toppling matrix behaves as $\Delta_G'$2 and the correlation length scales as $\Delta_G'$3. This suggests compatibility between the two parameterizations under the identification $\Delta_G'$4 (Ruelle, 2013).

4. Two-dimensional logarithmic and geometric descriptions

In two dimensions, the critical Abelian sandpile is a strong candidate lattice realization of logarithmic conformal invariance with $\Delta_G'$5. Dissipation is central to that interpretation. Local bulk dissipation corresponds to a weight-zero logarithmic field $\Delta_G'$6 with

$\Delta_G'$7

and transformation law

$\Delta_G'$8

In this sense $\Delta_G'$9 is the logarithmic partner of the identity. The same review emphasizes that plane correlators of height fields are properly interpreted with an insertion $\Delta_G'$0, because the sink required in finite volume does not disappear conceptually in the infinite-plane limit (Ruelle, 2013).

The dissipative ASM also has a two-regime geometric description in terms of Schramm–Loewner evolution. Critical avalanche frontiers are SLE$\Delta_G'$1, and numerical extraction of the Loewner driving function in the dissipative model shows two linear regimes in $\Delta_G'$2: a short-distance ultraviolet regime with $\Delta_G'$3, and a large-distance infrared regime with $\Delta_G'$4, identified with $\Delta_G'$5 of self-avoiding walk. The simulations were carried out on a $\Delta_G'$6 triangular lattice with $\Delta_G'$7 independent samples and $\Delta_G'$8. The paper distinguishes waves and avalanches sharply: waves are single-fractal objects, avalanches are multi-fractal, and dissipation leaves wave exponents nearly unchanged in the ultraviolet while altering avalanche exponents more visibly. Within that convention, the ultraviolet point corresponds to $\Delta_G'$9, $\Delta_{i,i}=z_i+t$0, the infrared point to $\Delta_{i,i}=z_i+t$1, $\Delta_{i,i}=z_i+t$2, and the authors phrase this as a decrease in central charge from the infrared point to the ultraviolet point (Najafi et al., 2012).

5. Geometry of dissipative sites and criticality thresholds

For lattice ASMs with dissipation only on a subset $\Delta_{i,i}=z_i+t$3, the geometry of $\Delta_{i,i}=z_i+t$4 is decisive. A sufficient criterion for non-criticality is summability of the Green function, equivalently finiteness of the expected survival time of the associated random walk that is killed at dissipative sites. The paper gives several sufficient conditions: $\Delta_{i,i}=z_i+t$5; uniformly bounded distance from every site to $\Delta_{i,i}=z_i+t$6; and more probabilistic conditions expressed through local times or hitting times of $\Delta_{i,i}=z_i+t$7. It also shows that geometry can defeat naive density heuristics: in $\Delta_{i,i}=z_i+t$8, making the entire $\Delta_{i,i}=z_i+t$9-axis dissipative still leaves the model critical, while a zero-density family of horizontal dissipative lines can be non-critical if

$t$0

holds for their spacings (Redig et al., 2017).

In one dimension, the relation between dissipative ASM and trapped random walk becomes exact and sharp. On $t$1, dissipative sites at positions $t$2 correspond to soft traps that kill the walker with probability $t$3. Writing $t$4, the model is non-critical if the expected survival time $t$5 is finite and critical otherwise. For the recursive family

$t$6

there is a phase transition at

$t$7

The intervals then grow super-exponentially, and the result shows that zero density of dissipative sites is not by itself enough to preserve criticality (Redig et al., 1 Jul 2025).

A later full-line analysis on $t$8 gives a finer three-regime picture. If dissipative sites are $t$9 on a half-line reduction, define recursive sequences

$n,m\in\mathbb N$0

$n,m\in\mathbb N$1

Then the model is critical iff $n,m\in\mathbb N$2. This yields a sub-geometric regime that is always non-critical when $n,m\in\mathbb N$3 with $n,m\in\mathbb N$4, a super-double-exponential regime that is always critical, and an intermediate coexistence zone where both phases occur. In that intermediate range, the ratio $n,m\in\mathbb N$5 becomes the practical threshold parameter. The same analysis also exhibits non-monotone behavior: moving traps farther apart can shorten, rather than lengthen, the expected stopping time (Rezzouk, 3 Sep 2025).

6. Stationary density, threshold density, and the fixed-energy comparison

A persistent question in dissipative sandpile theory is whether the stationary density $n,m\in\mathbb N$6 of the driven dissipative ASM equals the threshold density $n,m\in\mathbb N$7 of the corresponding fixed-energy sandpile. The comparative study of driven dissipative and conservative models shows that this “density conjecture” fails on many graphs. The paper gives exact counterexamples on the bracelet graph, where

$n,m\in\mathbb N$8

and on the flower graph, where

$n,m\in\mathbb N$9

It also gives simulation-based discrepancies on $G=(V\cup\{s\},E)$00, with $G=(V\cup\{s\},E)$01 and $G=(V\cup\{s\},E)$02, and on ladder and regular-tree examples. At the same time, $G=(V\cup\{s\},E)$03 still marks a dynamical transition: for bracelet and flower graphs, the final density $G=(V\cup\{s\},E)$04 equals $G=(V\cup\{s\},E)$05 below $G=(V\cup\{s\},E)$06, develops a kink at $G=(V\cup\{s\},E)$07, and continues to evolve beyond that point even after a constant fraction of sand has been lost at the sink (Fey et al., 2010).

A related nearly closed geometry is the square lattice with a single sink. There the mean density $G=(V\cup\{s\},E)$08 initially grows linearly under random additions, then changes sharply at a density $G=(V\cup\{s\},E)$09 that depends on the initial uniform height $G=(V\cup\{s\},E)$10, and finally relaxes to the stationary density

$G=(V\cup\{s\},E)$11

For $G=(V\cup\{s\},E)$12, the paper finds numerically that

$G=(V\cup\{s\},E)$13

where $G=(V\cup\{s\},E)$14 is the fixed-energy threshold density for the same initial background. Only the empty initial state $G=(V\cup\{s\},E)$15 gives $G=(V\cup\{s\},E)$16 within numerical accuracy; for $G=(V\cup\{s\},E)$17, $G=(V\cup\{s\},E)$18. This isolates a common misconception: equality between driven dissipative and fixed-energy densities is not a general property even when dissipation is extremely weak (Poghosyan et al., 2011).

7. Generalizations, directed variants, and adjacent dissipative models

Several models extend the dissipative ASM while preserving only part of the classical structure. Sportiello’s LN-ASM is explicitly dissipative in the bulk through a parameter $G=(V\cup\{s\},E)$19: each toppling loses $G=(V\cup\{s\},E)$20 total mass. It preserves Abelianity under explicit inequalities relating the threshold functions $G=(V\cup\{s\},E)$21 and kernels $G=(V\cup\{s\},E)$22, but it is not the standard Dhar-style dissipative ASM, because the instability rule is neighborhood-dependent,

$G=(V\cup\{s\},E)$23

the heights are continuous, and the transport kernel can be nonlocal. The model nevertheless retains a uniform stationary measure on recurrent states and a generating function given by the Kirchhoff matrix-tree formula (Sportiello, 2012).

A different stochastic extension is defined on a finite graph with sink by allowing each edge transmission in a toppling to occur independently with probability $G=(V\cup\{s\},E)$24. For $G=(V\cup\{s\},E)$25 the model is the ordinary ASM; for $G=(V\cup\{s\},E)$26 it remains Abelian in the sense of order-independent stabilization after preassigning toppling randomness, but the recurrent set becomes larger than the deterministic recurrent set. Recurrent configurations admit an orientation characterization,

$G=(V\cup\{s\},E)$27

and are counted by the “lacking polynomial” $G=(V\cup\{s\},E)$28, which satisfies a deletion–contraction recurrence resembling that of the Tutte polynomial (Chan et al., 2012).

Directed and boundary-dissipative models give exact algebraic control in special geometries. On a directed cylinder with periodic transverse direction and dissipation only at the right boundary, recurrent configurations form the finite Abelian group

$G=(V\cup\{s\},E)$29

with

$G=(V\cup\{s\},E)$30

The same work reduces the group to a transverse matrix problem and relates cyclic decomposition directly to the period of deterministic driving (Quadir et al., 15 May 2026). In a different direction, recent results on time transport correlations in driven-dissipative Abelian sandpiles show that the integrated number of dissipated particles has bounded central moments and that, under a mixing assumption,

$G=(V\cup\{s\},E)$31

so dissipation is mostly anticorrelated in time (Lallemant, 21 Dec 2025).

Other nearby models require more caution in classification. A dissipative BTW-type model on small-world networks uses annealed dissipation during grain transfer and exhibits coexistence of BTW-like and mean-field scaling, but the work does not formally prove Abelianity (Bhaumik et al., 2013). A directed dissipative Abelian sandpile with open or absorbing boundaries can be mapped exactly to a continuous-time free-fermion problem; in that setting, the probability that $G=(V\cup\{s\},E)$32 marked grains preserve their order decays as $G=(V\cup\{s\},E)$33 on the infinite line and as $G=(V\cup\{s\},E)$34 on the half-line, with exact correspondences to the Heisenberg XX chain and random-turns vicious walkers (Bogoliubov et al., 2011). These variants enlarge the dissipative ASM landscape, but they do not eliminate the distinction between the classical sink-based ASM, bulk-dissipative massive deformations, and more general Abelian or Abelian-like sandpile systems.