Multifractal spectrum of branching random walks on free groups
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917061995790336 |
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| author | Lai, Shuwen Ma, Heng Wang, Longmin |
| author_facet | Lai, Shuwen Ma, Heng Wang, Longmin |
| contents | 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 $α(R) = 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01346 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multifractal spectrum of branching random walks on free groups Lai, Shuwen Ma, Heng Wang, Longmin Probability 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 $α(R) = 0$. |
| title | Multifractal spectrum of branching random walks on free groups |
| topic | Probability |
| url | https://arxiv.org/abs/2409.01346 |