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First passage percolation preserves sublinearly Morse boundaries

Sagnik Jana, Yulan Qing

Abstract

Sublinearly Morse directions in proper geodesic spaces are defined by sublinearly Morse stability. In this paper we offer an alternative characterization for sublinearly Morse geodesic lines via middle recurrence. We then study first passage percolation (FPP) on proper geodesic graphs of bounded degree. We associate an i.i.d. collection of random passage times to each edge. Under suitable conditions on the passage time distribution, we prove that sublinearly Morse boundaries are invariant under first passage percolation.

First passage percolation preserves sublinearly Morse boundaries

Abstract

Sublinearly Morse directions in proper geodesic spaces are defined by sublinearly Morse stability. In this paper we offer an alternative characterization for sublinearly Morse geodesic lines via middle recurrence. We then study first passage percolation (FPP) on proper geodesic graphs of bounded degree. We associate an i.i.d. collection of random passage times to each edge. Under suitable conditions on the passage time distribution, we prove that sublinearly Morse boundaries are invariant under first passage percolation.

Paper Structure

This paper contains 10 sections, 21 theorems, 92 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. preliminaries
  4. Quasi-isometry and quasi-isometric embeddings
  5. Sublinearly Morse geodesic lines
  6. A coarse cone topology on $\partial_{\kappa} X$
  7. First passage percolation
  8. middle-recurrence of sublinear lines
  9. Sublinearly Morse directions under first passage percolations
  10. Construction of the bijective map between sublinearly Morse boundaries

Key Result

Theorem A

Let $X$ be a graph with uniformly bounded degree and $\partial_s X \neq \emptyset$. Assume that the edge length distribution has finite expectation and also $\nu(\{0\}) = 0$. Then let $\omega$ be induced graphs with weight functions. Then for almost every $\omega$, such the first passage percolation

Figures (4)

  • Figure 1: A $\kappa$-neighbourhood of a geodesic ray $b$ with multiplicative constant ${\sf n}$.
  • Figure 2: $\textbf{b} \in \mathcal{U}_{\kappa}({\sf a}, {\sf r})$ because the quasi-geodesics of $\textbf{b}$ such as $\beta, \beta_{0}$ stay inside the associated $(\kappa, {\sf m}_{\alpha_{0}}({\sf q}, {\sf Q}))$-neighborhood of $\alpha_{0}$ (as in Definition \ref{['Def:Neighborhood']}), up to distance ${\sf r}$.
  • Figure 3: if $a',b'$ are $\kappa-$separated then a sub path ${\sf p}'$ intersects a $\kappa'$ neighborhood of $\gamma.$
  • Figure 4: $f_\omega^{\star}$ is an open map.

Theorems & Definitions (56)

  • Theorem A
  • Definition 1.2
  • Theorem B
  • Definition 2.1: Quasi Isometric embedding
  • Definition 2.2: Quasi-geodesics
  • Lemma 2.3
  • proof
  • Definition 2.4: $\kappa$--neighborhood
  • Definition 2.5: $\kappa$-Morse I, $\kappa$-Morse II
  • Remark 2.6
  • ...and 46 more