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Asymptotics for the growth of the infinite-parent Spatial Lambda-Fleming-Viot model

Apolline Louvet, Matthew I. Roberts

Abstract

The infinite-parent spatial Lambda-Fleming-Viot (SLFV) process is a model of random growth, in which a set evolves by the addition of balls according to points of an underlying Poisson point process, and which was recently introduced to study genetic diversity in spatially expanding populations. In this article, we give asymptotics for the location and depth of the moving interface, and identify the exact asymptotic scale of the transverse fluctuations of geodesics. Our proofs are based on a new representation of the infinite-parent SLFV in terms of chains of reproduction events, and on the study of the properties of a typical geodesic. Moreover, we show that our representation coincides with the alternative definitions of the process considered in the literature, subject to a simple condition on the initial state. Our results represent a novel development in the study of stochastic growth models, and also have consequences for the study of genetic diversity in expanding populations.

Asymptotics for the growth of the infinite-parent Spatial Lambda-Fleming-Viot model

Abstract

The infinite-parent spatial Lambda-Fleming-Viot (SLFV) process is a model of random growth, in which a set evolves by the addition of balls according to points of an underlying Poisson point process, and which was recently introduced to study genetic diversity in spatially expanding populations. In this article, we give asymptotics for the location and depth of the moving interface, and identify the exact asymptotic scale of the transverse fluctuations of geodesics. Our proofs are based on a new representation of the infinite-parent SLFV in terms of chains of reproduction events, and on the study of the properties of a typical geodesic. Moreover, we show that our representation coincides with the alternative definitions of the process considered in the literature, subject to a simple condition on the initial state. Our results represent a novel development in the study of stochastic growth models, and also have consequences for the study of genetic diversity in expanding populations.
Paper Structure (31 sections, 54 theorems, 337 equations, 2 figures)

This paper contains 31 sections, 54 theorems, 337 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The model: the infinite-parent SLFV as a set-valued process
  3. Main results
  4. Transverse fluctuations of geodesics
  5. The shape of the expansion
  6. Equivalence of definitions of infinite-parent SLFV processes
  7. Notation
  8. Motivation and related work
  9. Layout of article
  10. Chains of reproduction events
  11. Definition and first properties of chains of reproduction events
  12. Using a slow coverage strategy to upper bound coverage times
  13. Tree representation of chains of reproduction events
  14. Fluctuations of geodesics: precise statements and proofs of Theorems \ref{['thm:geodesics']} and \ref{['thm:geodesic_plane_to_point']}
  15. Definition of geodesics from point to set and precise statement of Theorem \ref{['thm:geodesics']}
  16. ...and 16 more sections

Key Result

Theorem 1.3

Consider the $\infty$-parent SLFV started from $E=\{(0,0)\}$ and run until the first hitting time of $\mathcal{H}^x$. Say that any path of reproduction events leading from the origin to $\mathcal{H}^x$ at this time is a geodesic from $0$ to $\mathcal{H}^x$. Then: (i) For all $\varepsilon>0$, there e (ii) For all $\delta>0$ and $\beta\in(1/2,1]$, there exists $c>0$ such that for all sufficiently la

Figures (2)

  • Figure 1: A section of the growing interface of an $\infty$-parent SLFV process started from the lower half-plane. The paler regions show the process at later times.
  • Figure 2: A two-type $\infty$-parent SLFV process started from a ball with a 50:50 mix of green (pale) and purple (dark) particles grows a finite number of macroscopic sectors, as seen in experiments with bacteria hallatschek2007genetic. The left-hand image shows the process at time $0$, and the two images to the right show the emergence of macroscopic regions of a single type as time increases.

Theorems & Definitions (115)

  • Definition 1.1: $\infty$-parent ancestral skeleton
  • Definition 1.2: $\infty$-parent SLFV process
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Lemma 1.9
  • Theorem 1.10
  • ...and 105 more