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Generalized range of slow random walks on trees

Pierre Andreoletti, Alexis Kagan

Abstract

In this work, we are interested in the set of visited vertices of a tree $\mathbb{T}$ by a randomly biased random walk $\mathbb{X}:=(X_n,n \in \mathbb{N})$. The aim is to study a generalized range, that is to say the volume of the trace of $\mathbb{X}$ with both constraints on the trajectories of $\mathbb{X}$ and on the trajectories of the underlying branching random potential $\mathbb{V}:=(V(x), x \in \mathbb{T})$. Focusing on slow regime's random walks (see [HS16b], [AC18]), we prove a general result and detail examples. These examples exhibit many different behaviors for a wide variety of ranges, showing the interactions between the trajectories of $\mathbb{X}$ and the ones of $\mathbb{V}$.

Generalized range of slow random walks on trees

Abstract

In this work, we are interested in the set of visited vertices of a tree by a randomly biased random walk . The aim is to study a generalized range, that is to say the volume of the trace of with both constraints on the trajectories of and on the trajectories of the underlying branching random potential . Focusing on slow regime's random walks (see [HS16b], [AC18]), we prove a general result and detail examples. These examples exhibit many different behaviors for a wide variety of ranges, showing the interactions between the trajectories of and the ones of .

Paper Structure

This paper contains 17 sections, 20 theorems, 327 equations.

Table of Contents

  1. Introduction
  2. First results : examples
  3. A general result (informal statement)
  4. A general result (full statement)
  5. Proof of the theorems
  6. Preliminary material
  7. Proofs of Theorems \ref{['thm1']} to \ref{['thm4']}
  8. Proof of Theorem \ref{['Prop2']}
  9. Proof of Proposition \ref{['Prop1']}
  10. Preliminary
  11. Lower bound for $\mathcal{R}_{T^n}(g_n,\mathbf{f}^n)$
  12. Lower bound for $R(\mathbf{f}^n)$
  13. Control of the second moment $\mathbf{E}[Z_n^2]$
  14. Upper bound for $\mathcal{R}_{T^n}(g_n,\mathbf{f}^n)$
  15. Technical estimates for one-dimensional random walk
  16. ...and 2 more sections

Key Result

Theorem 1.1

Assume hyp0, hyp0+ and hyp1 hold. If for any $n$ and $k$, $f^{n,k}(t_1,t_2, \cdots,t_k)= \mathds{1}_{\{t_k \geq (\log n)^{\alpha}\}}$ with $\alpha\in(1,2)$ and if $g_n(t)=\mathds{1}_{\{t \geq n^{b}\}}$ with $b\in[0,1)$, then where $\log ^+ x = \log (\max(1,x))$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1
  • Theorem 1.5: Informal statement
  • Proposition 1
  • Theorem 1.5: Full statement
  • Lemma 2.1: Many-to-one Lemma
  • Remark 2
  • ...and 29 more