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Internal DLA on mated-CRT maps

Ahmed Bou-Rabee, Ewain Gwynne

Abstract

We prove a shape theorem for internal diffusion limited aggregation on mated-CRT maps, a family of random planar maps which approximate Liouville quantum gravity (LQG) surfaces. The limit is an LQG harmonic ball, which we constructed in a companion paper. We also prove an analogous result for the divisible sandpile.

Internal DLA on mated-CRT maps

Abstract

We prove a shape theorem for internal diffusion limited aggregation on mated-CRT maps, a family of random planar maps which approximate Liouville quantum gravity (LQG) surfaces. The limit is an LQG harmonic ball, which we constructed in a companion paper. We also prove an analogous result for the divisible sandpile.
Paper Structure (45 sections, 55 theorems, 338 equations, 8 figures)

This paper contains 45 sections, 55 theorems, 338 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Internal DLA
  4. Liouville quantum gravity
  5. Random planar maps and the mated-CRT map
  6. SLE/LQG embedding
  7. Harmonic balls
  8. Main results
  9. Internal DLA
  10. Divisible sandpile
  11. Open problems
  12. Paper and proof outline
  13. Notation and conventions
  14. Preliminaries
  15. Gaussian free field
  16. ...and 30 more sections

Key Result

Theorem 1.4

Let $\{\Lambda_t\}_{t\geq 0}$ be the growing family of $\gamma$-LQG harmonic balls with $\mu_h(\Lambda_t) = t$, as discussed just after Definition def:harmonic-ball, with $h$ the field corresponding to the $\gamma$-quantum cone, as in Definition def:sle-lqg-embedding. For each $\delta > 0$ and $t > where $B_{\delta}^{\pm}$ are as in eq:outer-inner-neighborhoods.

Figures (8)

  • Figure 1: Left: IDLA on the Tutte embedding of a finite $\gamma = \sqrt{2}$ mated-CRT map with boundary, with $10^4$ vertices. Walkers are added until the first time one of them hits a boundary vertex. Cells are colored according to the first time one of their vertices is hit by a random walk. Right: Illustration of the graph upon which IDLA is run. The finite mated-CRT map with boundary can be constructed from a pair of conditioned Brownian motions via a similar procedure as in Definition \ref{['def:mated-crt-map']}. Away from the boundary, it locally looks like the infinite mated-CRT map considered in this paper, but its Tutte embedding is easier to define and simulate. See Remark \ref{['remark:tutte']} for further discussion.
  • Figure 2: Left: A sample construction of the mated-CRT map. We draw the graphs of $C-L$ and $R$ in the interval $[0, 12 {\eps}]$ where $C$ is a large constant chosen so that the graphs do not intersect. Each vertical strip containing the interval $[a-{\eps}, a]$ for $a \in {\eps} \mathbb{Z}$ corresponds to a vertex in the mated-CRT map. The adjacency condition \ref{['eq:edge-map']} for $L$ (resp. $R$) corresponds to two vertices $a,b \in {\eps} \mathbb{Z}$ being adjacent if there is a horizontal line segment above the graph of $C-L$ (resp. below the graph of $R$) which intersects the graph only in the vertical strips which contain the intervals $[a-{\eps}, a]$ and $[b-{\eps},b]$. In the figure, we have illustrated, for each pair $(a,b)$ for which this adjacency condition holds, the lowest (resp. highest) such horizontal line segment in green. Right: A planar embedding of the mated-CRT map on the left, under which it is a triangulation. Edges arising from the adjacency condition for $L$ (resp. $R$) which do not join consecutive vertices are shown in red (resp. blue). Edges joining consecutive vertices are shown in black. A similar illustration appeared as gms-harmonic.
  • Figure 3: Top left: A space-filling SLE curve $\eta$ for $\kappa \geq 8$ divided into cells $\eta([a - {\eps}, a])$ for a collection of $a \in {\eps} \mathbb{Z}$. Top right: The same curve with a blue path showing the order in which cells are traversed by $\eta$. Bottom left: A point in each cell, corresponding to a vertex in ${\Geps}$, is displayed in red and red edges are drawn to adjacent neighbors. Bottom right: The same as bottom left but with the black edges removed --- this illustrates the embedding of ${\Geps}$ into $\mathbb{C}$. A similar figure has appeared previously as gms-harmonic.
  • Figure 4: A close up of the interface in Figure \ref{['fig:IDLA-tutte-embedding']}.
  • Figure 5: A set $D\subset \mathbb{C}$ and the set of mated-CRT map cells corresponding to the vertices in ${\Geps}(D)$, as in \ref{['eq:cells-domain-restriction']}. The boundary of the domain $\partial D$ is drawn in red and the cells of ${\Geps}(D)$ are filled in pink. A point $x \in \mathbb{C}$ and the cell containing it, \ref{['eq:sle-cell']}, is drawn in light blue. A similar figure appeared in gms-harmonic.
  • ...and 3 more figures

Theorems & Definitions (97)

  • Definition 1.1: Mated-CRT map
  • Definition 1.2: SLE/LQG embedding of the mated-CRT map duplantier2014liouville
  • Definition 1.3: Harmonic ball
  • Theorem 1.4
  • Remark 1.5: Convergence under the Tutte embedding
  • Theorem 1.6
  • Definition 2.1: Circle average embedding of the $\gamma$-quantum cone
  • Theorem 2.3: Combination of Theorems 1.1 and 5.5 and Lemma 3.2 in bou2022harmonic
  • Lemma 2.4
  • proof
  • ...and 87 more