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Genealogy in critical generations of a diffusive random walk in random environment on trees

Alexis Kagan

Abstract

We consider the range $R^{(n)}$, the tree made up of visited vertices by a diffusive null-recurrent randomly biased walk $\mathbb{X}$ on a Galton-Watson tree $\mathbb{T}$ up to the $n$-th return time to its root and we consider the following genealogy problem: pick two vertices uniformly at random in a generation of order $n$ in the tree $R^{(n)}$. Where does the coalescence occur? it turns out that the coalescence happens either in the recent past or in the remote past.

Genealogy in critical generations of a diffusive random walk in random environment on trees

Abstract

We consider the range , the tree made up of visited vertices by a diffusive null-recurrent randomly biased walk on a Galton-Watson tree up to the -th return time to its root and we consider the following genealogy problem: pick two vertices uniformly at random in a generation of order in the tree . Where does the coalescence occur? it turns out that the coalescence happens either in the recent past or in the remote past.

Paper Structure

This paper contains 11 sections, 8 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Randomly biased random walk on a Galton-Watson tree
  3. The range of the random walk
  4. The volume of the range in critical generations
  5. A genealogy problem for critical generations
  6. Range and genealogy in the sub-diffusive case
  7. Proofs of the results
  8. Preliminary results
  9. Proof of Theorem \ref{['ThMultiMar']}
  10. Proof of Theorem \ref{['GenealogyDiffCritic']}
  11. Proofs of \ref{['RemSmallGenerations21']}, \ref{['RemSmallGenerations22']} and \ref{['RemarkGen1Excu']}

Key Result

Theorem 1.2

Assume that the Assumptions Assumption1 and Assumption2 hold. We have, in law under $\mathbb{P}$ for the space of càdlàg functions $D([0,\infty),\mathbb{R}^3)$

Theorems & Definitions (10)

  • Theorem 1.2
  • Theorem 1.3: Two vertices picked in generation $\lfloor bn\rfloor$ of $\mathcal{R}^{(n)}$
  • Remark 1
  • Remark 2: Two vertices picked in generation $\lfloor bn\rfloor$ of $\mathcal{R}^{(1)}$
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6