An arrival-gated interpolation from IDLA to the abelian sandpile

Definitions and mathematical statement

Jordan Ellenberg's source question asks for a stochastic sandpile interpolation between "DLA [16]" and the sandpile model. The supplied reference identification is Diaconis--Fulton internal DLA (IDLA), not external diffusion-limited aggregation. The original does not specify graph, boundary, interpolation parameter, or scaling objective.

The reported construction is on a finite directed multigraph with a sink $s$, a directed path from every nonsink vertex to $s$, and nonsink outdegree $d_v\geq2$, counted with multiplicity. Start with empty heights $h_v=0$. At each vertex fix independent arrival-indexed gate instructions: with probability $1-\theta$ use an explorer gate, and with probability $\theta$ use a degree-threshold sandpile gate, $0\leq\theta\leq1$. Explorer instructions also include an independent uniform outgoing-arc choice. Each incoming chip is processed by one instruction in order at its vertex:

• Explorer gate: if $h_v=0$, retain the chip and set $h_v=1$; otherwise keep the height and send one chip along its instructed outgoing arc.
• Sandpile gate: if $h_v<d_v-1$, retain the chip and increase height by one; if $h_v=d_v-1$, set height to zero and send one chip along every outgoing arc.

Chips arriving at $s$ disappear. Input is any fixed finite multiset of chips at nonsink vertices, and legal executions process available incoming chips. The heights remain in $\{0,\ldots,d_v-1\}$.

The aid reports pathwise abelianness: after the instruction stacks are fixed, the terminal state on the almost-sure terminating event is independent of legal processing order. It also reports almost-sure finite-volume termination and conservation of input mass as final retained mass plus sink loss. At $\theta=0$ the law from the empty state is exactly graph IDLA, with first vacancy retaining a walker and later arrivals continuing it; at $\theta=1$ stabilization is the classical abelian sandpile. For each fixed finite input, the terminal-height law $\mathcal L_\theta$ satisfies \[ \|\mathcal L_\theta-\mathcal L_{\theta_0}\|_{\mathrm{TV}}\longrightarrow0 \qquad(\theta\to\theta_0,\ \theta_0\in[0,1]). \] The empty or initially singly occupied convention matters for the literal IDLA endpoint; arbitrary larger initial heights are not silently called ordinary IDLA.

The separate stationary example has one nonsink vertex with $d\geq2$ arcs, all ending at the sink, and successive single-chip inputs. On heights $0,\ldots,d-1$, zero always moves to one; a positive height stays put with probability $1-\theta$, and with probability $\theta$ increases by one, with $d-1$ returning to zero. For $0<\theta\leq1$, the unique stationary law is \[ \pi_\theta(0)=\frac\theta{\theta+d-1},\qquad \pi_\theta(h)=\frac1{\theta+d-1}\quad(1\leq h\leq d-1). \] For $0<\theta<1$ it is also the limiting law from the empty state. At $\theta=1$ the chain is periodic, so stationarity must not be confused with convergence of every iterate. At $\theta=0$, starting empty gives height one forever after the first input, although other positive heights are also absorbing states.

Consequently for $d\geq3$, \[ \lim_{\theta\downarrow0}\lim_{N\to\infty}\mathcal L(H_N^\theta) =\operatorname{Uniform}\{1,\ldots,d-1\} \neq\delta_1 =\lim_{N\to\infty}\lim_{\theta\downarrow0}\mathcal L(H_N^\theta). \] The noncommutation is compatible with fixed-finite-input total-variation continuity. No infinite-volume aggregation limit, external-DLA endpoint, or universality classification is supplied.

Short English statement

The supplied model mixes arrival-by-arrival explorer and sandpile gates on a finite graph with a sink. It has exact internal-DLA and abelian-sandpile endpoints, order-independent stabilization, and finite-input continuity. A one-vertex example shows that this continuity does not make the long-time limit commute with the IDLA endpoint.

Sources

• Local registry: permanent UnsolvedMath ID $20000999$, exact public code catalogue entry; aim-combinatorics-notes.json, notes[123].
• AIM workshop Generalizations of chip-firing and the critical group, Problem 34, attributed to Jordan Ellenberg: https://aimath.org/pastworkshops/chipfiringproblems.pdf. This URL is supplied provenance; no PDF access is claimed.
• The source cites DLA [16]; the supplied research summary identifies that reference as Diaconis--Fulton internal DLA but provides no complete bibliography entry.
• Supplied research summary, "An arrival-gated interpolation from IDLA to the abelian sandpile," attempt 1. Original remarks and literature fields are empty; no additional authoritative external theorem citations are supplied.