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Central limit theorem for a random walk on Galton-Watson trees with random conductances

Tabea Glatzel, Jan Nagel

Abstract

We show a central limit theorem for random walk on a Galton-Watson tree, when the edges of the tree are assigned randomly uniformly elliptic conductances. When a positive fraction of edges is assigned a small conductance $\varepsilon$, we study the behavior of the limiting variance as $\varepsilon\to 0$. Provided that the tree formed by larger conductances is supercritical, the variance is nonvanishing as $\varepsilon\to 0$, which implies that the slowdown induced by the $\varepsilon$-edges is not too strong. The proof utilizes a specific regeneration structure, which leads to escape estimates uniform in $\varepsilon$.

Central limit theorem for a random walk on Galton-Watson trees with random conductances

Abstract

We show a central limit theorem for random walk on a Galton-Watson tree, when the edges of the tree are assigned randomly uniformly elliptic conductances. When a positive fraction of edges is assigned a small conductance , we study the behavior of the limiting variance as . Provided that the tree formed by larger conductances is supercritical, the variance is nonvanishing as , which implies that the slowdown induced by the -edges is not too strong. The proof utilizes a specific regeneration structure, which leads to escape estimates uniform in .

Paper Structure

This paper contains 24 sections, 11 theorems, 180 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model and Results
  3. Regeneration structure
  4. Proofs of auxiliary results
  5. Proof of Lemma \ref{['lem:annealedescapeprob']}
  6. Proof of Lemma \ref{['lem:momentboundregdist']}
  7. Uniform bound for $\mathbb{E}[H_{i+1}^{2q}\mathds 1_{\{R_i<\infty\}}]$
  8. Uniform bound for $\mathbb{E}[\tilde{N}_i^{2q}\mathds 1_{\{R_i<\infty\}}]$
  9. Quenched return probability
  10. $B_r(\delta,\beta, L)^c$ has exponentially small probability
  11. Proof of Lemma \ref{['lem:quenchedindirectescapeprob']}
  12. The indirect escape probability in the case that the $L$-th generation of $T_1^{\operatorname{Bb}}$ contains zero vertices
  13. The indirect escape probability in the case that the $L$-th generation of $T_1^{\operatorname{Bb}}$ contains one vertex
  14. The indirect escape probability in the case that the $L$-th generation of $T_1^{\operatorname{Bb}}$ contains more than one vertex
  15. Proof of Lemma \ref{['lem:momentboundregtimes']}
  16. ...and 9 more sections

Key Result

Theorem 2.1

There exists a constant $\sigma^2(\varepsilon)=\sigma^2(\varepsilon,\alpha,\mu_1)> 0$, such that under $\mathbb{P}$, with $( B_t)_{t\in [0,1]}$ a standard Brownian Motion.

Figures (4)

  • Figure 1: $T(v)$ is the subtree formed by $v$ and all its descendants, $T^*(v)$ (indicated by the thick edges) is the subtree composed of $T(v)$ and $v^*$.
  • Figure 2: Edges with conductance larger $\varepsilon$ are indicated by solid lines; edges with conductance $\varepsilon$ are indicated by dotted lines; vertices in the sixth generation $G_6(T)$ are marked by dots on the dashed line. $T_1(\rho)$ is the subtree formed by the edges with conductance one containing the root (indicated by the thick lines). In the sixth generation $G_6(T)$ the vertices $z_1,\,z_2,\,z_3$ are in $T_1(\rho)$. The event $A$ contains a vertex $v\in T$ if every vertex on the path from $v^{k_\varepsilon}$ to $v^*$ has degree 2, where $(v^{k_\varepsilon-1},v^{k_\varepsilon})$ is the first $\varepsilon$-edge on the path from $\rho^*$ to $v$. In the sixth generation $G_6(T)$ the vertices $v_1,\,v_2$ are in $A$. The ancestors of $v_i$, which must have degree 2 for $v_i$ being in $A$, are circled.
  • Figure 3: Environment $\tilde{\omega}_\rho$
  • Figure 4: Modified environment $\hat{\omega}$

Theorems & Definitions (14)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1
  • Lemma 3.2: Annealed escape probability
  • Lemma 3.3: Existence of regeneration times
  • Proposition 3.4: Stationarity and independence
  • Lemma 3.5: Moment bounds on regeneration distances
  • Lemma 3.6: Moment bounds on regeneration times
  • ...and 4 more