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arxiv: 2605.19419 · v1 · pith:ZMVMLKGZnew · submitted 2026-05-19 · 🧮 math.PR

Tail exponents of the three-dimensional uniform spanning tree and Abelian sandpile

Xinyi Li , Runsheng Liu , Daisuke Shiraishi This is my paper

Pith reviewed 2026-05-20 02:55 UTC · model grok-4.3

classification 🧮 math.PR
keywords uniform spanning treeAbelian sandpiletail exponentsthree dimensionsavalanche exponents0-wired forestpower-law tailsself-organized criticality
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The pith

Three-dimensional uniform spanning trees fix Abelian sandpile avalanche tails at exponents 1 for radius and 1/3 for size and topplings.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes sharp tail exponents, up to subpolynomial errors, for the past of the origin in the three-dimensional uniform spanning tree and for the 0-tree of the 0-wired uniform spanning forest. These exponents transfer through the known link to the Abelian sandpile model to show that avalanche-cluster radius obeys a tail exponent of 1 while avalanche-cluster size and total topplings each obey a tail exponent of 1/3. A sympathetic reader would care because the results supply the first precise leading power-law description of avalanche statistics in three-dimensional sandpiles, where only weaker bounds were available before. The work therefore connects the local geometry of spanning trees directly to the scaling of large events in a canonical model of self-organized criticality.

Core claim

We obtain sharp tail exponents, up to subpolynomial errors, for the past of the origin in the three-dimensional UST and for the 0-tree of the 0-wired uniform spanning forest. As a principal application, we prove the corresponding three-dimensional Abelian sandpile avalanche exponents: the avalanche-cluster radius has tail exponent 1, while both the avalanche-cluster size and the total number of topplings have tail exponent 1/3. These results identify the leading power-law behaviour of three-dimensional sandpile avalanches and improve previously known bounds.

What carries the argument

The correspondence between the uniform spanning tree (and 0-wired forest) and the Abelian sandpile model that maps tree paths and clusters to avalanche clusters, sizes, and toppling counts.

Load-bearing premise

The established connection between the uniform spanning tree and the Abelian sandpile model extends to three dimensions in a manner that transfers the tail exponents directly.

What would settle it

Large-scale numerical simulation of the Abelian sandpile on a three-dimensional grid that measures the empirical decay rates of avalanche radius, size, and toppling number and tests whether they match the claimed exponents 1 and 1/3 up to subpolynomial corrections.

read the original abstract

We study the local geometry of the three-dimensional uniform spanning tree and its connection with the Abelian sandpile model. We obtain sharp tail exponents, up to subpolynomial errors, for the past of the origin in the three-dimensional UST and for the $0$-tree of the $0$-wired uniform spanning forest. As a principal application, we prove the corresponding three-dimensional Abelian sandpile avalanche exponents: the avalanche-cluster radius has tail exponent $1$, while both the avalanche-cluster size and the total number of topplings have tail exponent $1/3$. These results identify the leading power-law behaviour of three-dimensional sandpile avalanches and improve previously known bounds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper proves sharp tail exponents (up to subpolynomial errors) for the past of the origin in the three-dimensional uniform spanning tree and for the 0-tree of the 0-wired uniform spanning forest. As the principal application, it derives the corresponding exponents for three-dimensional Abelian sandpile avalanches: avalanche-cluster radius with tail exponent 1, and both avalanche-cluster size and total number of topplings with tail exponent 1/3. These results improve prior bounds and identify the leading power-law behavior.

Significance. If the derivations for the UST tails are rigorous and the UST-sandpile correspondence transfers the sharp exponents to d=3 without additional error terms, the work would establish the first precise power-law tails for 3D sandpile avalanches. The explicit identification of exponents 1 and 1/3, together with the use of external model connections, strengthens the contribution provided the 3D transfer is fully justified.

major comments (1)
  1. [principal application / abstract] The principal application step (invoked in the abstract and developed after the UST results) transfers tail exponents from the 3D UST/0-wired forest directly to sandpile avalanche radius, size, and toppling counts. The classical UST-sandpile correspondence is cited, but the manuscript does not appear to supply a self-contained 3D argument that controls subpolynomial errors or accounts for the different connectivity and scaling of the 3D UST; this transfer is load-bearing for the sandpile claims and requires explicit justification or error bounds.
minor comments (2)
  1. [introduction] Notation for the 0-wired forest and the precise definition of the 'past of the origin' should be introduced with a short diagram or reference to standard conventions to aid readability.
  2. [main theorems] The statement of 'up to subpolynomial errors' would benefit from an explicit definition or reference to the precise form of the error term (e.g., o(log n) or similar) in the main theorems.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for stronger justification in the principal application to the Abelian sandpile model. We address this point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The principal application step (invoked in the abstract and developed after the UST results) transfers tail exponents from the 3D UST/0-wired forest directly to sandpile avalanche radius, size, and toppling counts. The classical UST-sandpile correspondence is cited, but the manuscript does not appear to supply a self-contained 3D argument that controls subpolynomial errors or accounts for the different connectivity and scaling of the 3D UST; this transfer is load-bearing for the sandpile claims and requires explicit justification or error bounds.

    Authors: We agree that the transfer step requires explicit justification in three dimensions. The manuscript relies on the classical UST-sandpile correspondence, but does not provide a fully self-contained argument controlling the subpolynomial errors or the 3D-specific connectivity and scaling. In the revised version we will insert a dedicated subsection that derives the necessary error bounds, showing that the subpolynomial factors inherited from the UST tails remain subpolynomial after the correspondence and therefore do not alter the leading exponents (radius tail 1, size and toppling tails 1/3). This addition will make the sandpile claims rigorous without changing the stated results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent analysis and external model connections

full rationale

The paper derives tail exponents for the 3D uniform spanning tree and 0-wired forest via direct geometric and probabilistic arguments on the local structure, then invokes a pre-existing UST-sandpile correspondence (from prior literature) as an application step to obtain the avalanche exponents. No equations or claims reduce by construction to fitted parameters, self-definitions, or self-citation chains that presuppose the target results; the central UST proofs stand independently of the sandpile transfer, and the connection is treated as an established external fact rather than re-derived from the current outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard probabilistic constructions of spanning trees and sandpiles on Z^3 with no explicit free parameters, invented entities, or ad-hoc axioms listed.

axioms (1)
  • standard math Standard axioms of probability on infinite graphs and measure-theoretic constructions of uniform spanning trees and Abelian sandpiles.
    Invoked to define the models and their local geometry in three dimensions.

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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