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Multifractal spectrum of branching random walks on free groups

Shuwen Lai, Heng Ma, Longmin Wang

TL;DR

This work analyzes the multifractal structure of BRWs on free groups in the transient regime $r\in(1,R]$, identifying a detailed spectrum for the limit set $\Lambda_r$ on $\partial\mathbb{F}$. By connecting the BRW trace to large-deviation principles for the word-length of the underlying random walk, the authors derive explicit formulas for the local fractal dimensions: for $\alpha>0$ in the admissible range $I(r)$, $\dim_{\mathrm{H}} \Lambda_r(\alpha)=\frac{\ln r - L^{*}(\alpha)}{\alpha}$, and there exists a unique $\alpha(r)$ with $\dim_{\mathrm{H}} \Lambda_r=\dim_{\mathrm{H}} \Lambda_r(\alpha(r))$; moreover, in the subcritical subcase ($r<R$) one has $\alpha(r)>0$, while at the critical point ($r=R$) the phase transition yields $\alpha(R)=0$. The proofs combine covering arguments with large-deviation estimates, a refined energy (Frostman) method to secure lower bounds on the level-set dimensions, and a zero-one law to ensure almost-sure determinism of the spectra. The results illuminate an energy–entropy trade-off governing the fractal geometry of BRW traces and establish a robust multifractal framework for BRWs on free groups, with implications for more general hyperbolic settings.

Abstract

A symmetric branching random walk (BRW) on a free group $\mathbb{F}$ is transient if and only if the mean offspring number $r$ does not exceed $R$, the reciprocal of the spectral radius of the underlying random walk. In this regime, the limit set $Λ_r$ -- consisting of all ends of $\mathbb{F}$ to which the BRW's particle trajectories converge -- is a proper random subset of the boundary $\partial \mathbb{F}$. Hueter and Lalley (2000) determined the Hausdorff dimension of $Λ_r$ and proved that $\dim_{\mathrm{H}} Λ_r \le (1/2)\dim_{\mathrm{H}} \partial \mathbb{F}$, with equality possible only when $r = R$. In this paper, we further extend this study by conducting a multifractal analysis of the limit set $Λ_r$. We obtain the Hausdorff dimensions of the subfractals $Λ_r(α) \subset Λ_r$, which consist of all ends of $\mathbb{F}$ approached by particle trajectories escaping at rate $α\in [0,1]$. Notably, there exists a unique $α(r) \in [0,1]$ such that \[ \dim_{\mathrm{H}} Λ_r = \dim_{\mathrm{H}} Λ_r(α(r)). \] Moreover, an interesting phase transition occurs: $α(r) > 0$ for $r < R$ while $α(R) = 0$.

Multifractal spectrum of branching random walks on free groups

TL;DR

This work analyzes the multifractal structure of BRWs on free groups in the transient regime , identifying a detailed spectrum for the limit set on . By connecting the BRW trace to large-deviation principles for the word-length of the underlying random walk, the authors derive explicit formulas for the local fractal dimensions: for in the admissible range , , and there exists a unique with ; moreover, in the subcritical subcase () one has , while at the critical point () the phase transition yields . The proofs combine covering arguments with large-deviation estimates, a refined energy (Frostman) method to secure lower bounds on the level-set dimensions, and a zero-one law to ensure almost-sure determinism of the spectra. The results illuminate an energy–entropy trade-off governing the fractal geometry of BRW traces and establish a robust multifractal framework for BRWs on free groups, with implications for more general hyperbolic settings.

Abstract

A symmetric branching random walk (BRW) on a free group is transient if and only if the mean offspring number does not exceed , the reciprocal of the spectral radius of the underlying random walk. In this regime, the limit set -- consisting of all ends of to which the BRW's particle trajectories converge -- is a proper random subset of the boundary . Hueter and Lalley (2000) determined the Hausdorff dimension of and proved that , with equality possible only when . In this paper, we further extend this study by conducting a multifractal analysis of the limit set . We obtain the Hausdorff dimensions of the subfractals , which consist of all ends of approached by particle trajectories escaping at rate . Notably, there exists a unique such that Moreover, an interesting phase transition occurs: for while .
Paper Structure (27 sections, 31 theorems, 315 equations, 1 figure)

This paper contains 27 sections, 31 theorems, 315 equations, 1 figure.

Table of Contents

  1. Introduction and Main Results
  2. Related work
  3. Overview of the proofs
  4. Preliminaries
  5. Free groups and trees
  6. Random walks on free groups
  7. Branching random walks on free groups
  8. Hausdorff dimension
  9. Local limit theorems for RWs
  10. Large deviations for word length of RWs
  11. Asymptotics of a free energy
  12. Proof of the Upper Bound
  13. Upper bound for $\dim_{\mathrm{H}} E_{r}(\alpha, \beta)$
  14. Upper bound for $\Lambda_{r}(\alpha, \beta)$
  15. Level Sets of BRWs on Free Groups
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $r \in (1,R]$. Almost surely for any $\alpha \in [0,1]$, $\Lambda_r(\alpha)$ is nonempty if and only if $\alpha \in I(r)$. In this case, the Hausdorff dimension of $\Lambda_r(\alpha)$ is given by Here, $\alpha=0$ is permissible only when $r=R$, in which case the expression $\frac{\ln R - L^{*}(0)}{0}$ should be interpreted as $\lim_{\alpha \downarrow 0} \frac{L^{*}(0) - L^{*}(\alpha)}{\alpha}

Figures (1)

  • Figure 1: Illustration for $\alpha(r)$ in subcritical case $1 < r < R$ (left) and critical case $r = R$ (right).

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: LS97
  • Proposition 2.2: Lal91
  • Remark 2.3
  • Lemma 2.4
  • Proposition 2.5: L93
  • Lemma 2.6: HL00
  • Lemma 2.7
  • ...and 49 more