Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The local limit of rooted directed animals on the square lattice

Olivier Hénard, Édouard Maurel-Segala, Arvind Singh

Abstract

We consider the local limit of finite uniformly distributed directed animals on the square lattice viewed from the root. Two constructions of the resulting uniform infinite directed animal are given: one as a heap of dominoes, constructed by letting gravity act on a right-continuous random walk and one as a Markov process, obtained by slicing the animal horizontally. We look at geometric properties of this local limit and prove, in particular, that it consists of a single vertex at infinitely many (random) levels. Several martingales are found in connection with the confinement of the infinite directed animal on the non-negative coordinates.

The local limit of rooted directed animals on the square lattice

Abstract

We consider the local limit of finite uniformly distributed directed animals on the square lattice viewed from the root. Two constructions of the resulting uniform infinite directed animal are given: one as a heap of dominoes, constructed by letting gravity act on a right-continuous random walk and one as a Markov process, obtained by slicing the animal horizontally. We look at geometric properties of this local limit and prove, in particular, that it consists of a single vertex at infinitely many (random) levels. Several martingales are found in connection with the confinement of the infinite directed animal on the non-negative coordinates.
Paper Structure (24 sections, 34 theorems, 173 equations, 16 figures)

This paper contains 24 sections, 34 theorems, 173 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Additional comments
  4. Encoding of an animal by a skip-free path
  5. Definitions and notations for directed animals
  6. Ordering of the vertices of a directed animal
  7. Mapping between simple animals and skip free paths
  8. Local limits of directed animals
  9. Push-forward measures and the animal walk S
  10. The Boltzmann half-pyramid (BHP)
  11. The local limit of finite pyramids (UIP)
  12. The local limit of finite half-pyramids
  13. Conditioning the shaved walk to stay non-positive
  14. Limit of non-positive pyramids (UIP-).
  15. Limit of non-negative pyramids (UIP+)
  16. ...and 9 more sections

Key Result

Theorem 1

Figures (16)

  • Figure 1: Example of a directed animal with a single source at the origin (i.e a pyramid). The height levels are drawn in red with layer 5 is highlighted in green in the rotated picture (b)
  • Figure 2: Directed animal (with 2 sources), pyramid and half-pyramid in $\mathbb{Z}\diamond\mathbb{N}$
  • Figure 3: Representation of an animal as a heap of pieces and the "pushing up" operation at a vertex. We have $\mathbf{a} \lhd \mathbf{b}$ even though $\mathbf{b}$ is not a direct descendant of $\mathbf{a}$.
  • Figure 4: An animal ordered with $\preccurlyeq$ (a) and then with the mirror order $\widetilde{\preccurlyeq}$ (b). Notice that we have $a \preccurlyeq b$ in Figure (a) but the order of these two vertices become reversed if we add a vertex at position $c$ (and we then get $b \preccurlyeq c \preccurlyeq a$).
  • Figure 5: Examples of infinite animal ordering. Animals (a) and (b) are simple while animals (c), (d) and (e) are not. Notice that (c) is the mirror image of (b) w.r.t. the $y$-axis yet it is not simple for $\preccurlyeq$ (but it is simple for the mirrored order $\widetilde{\preccurlyeq}$).
  • ...and 11 more figures

Theorems & Definitions (87)

  • Theorem : Theorems \ref{['thm:loclimitUIP']} and Theorem \ref{['thm:loclimit_UIHP']}
  • Theorem : Theorem \ref{['thm:marginals']} and Corollary \ref{['cor:Markov']}
  • Theorem : Theorem \ref{['thm:saucissonnage']}
  • Remark 1
  • lemma 1
  • proof
  • lemma 2: Compatibility of the ordering when dropping dominoes
  • proof
  • Definition 1: Simplicity of a directed animal
  • Remark 2
  • ...and 77 more