Collisions of random walks on comb graphs with a planar base
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909755444822016 |
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| author | De Ambroggio, Umberto Nitzschner, Maximilian Scali, Carlo |
| author_facet | De Ambroggio, Umberto Nitzschner, Maximilian Scali, Carlo |
| contents | In this article we study collisions of two independent random walks on comb graphs $\mathrm{Comb}(\tilde{G},f)$ for a large class of recurrent planar graphs $\tilde{G}$ and profile functions $f$, the latter governing the length of vertical segments (called "teeth") attached to vertices of the base graph $\tilde{G}$. We prove that the number of collisions of two random walks starting from the same site undergoes a phase transition depending on the growth of $f$. As a benchmark example, we show that for $\mathrm{Comb}(\mathbb{Z}^2,f_γ)$ with $f_γ(z) = \log^γ(\|z\|_\infty\vee 1)$ and $\|\cdot \|_\infty$ denoting the supremum norm, two independent random walks started at the origin collide finitely often almost surely if $γ> 1$, answering a question of Barlow, Peres, and Sousi, see arXiv:1003.3255, who established that infinitely many collisions occur almost surely if $γ\leq 1$. We furthermore establish phase transitions in the cases where the base graph $\tilde{G}$ is pre-fractal, or a typical realization of a supercritical cluster of planar Bernoulli bond percolation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_19814 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Collisions of random walks on comb graphs with a planar base De Ambroggio, Umberto Nitzschner, Maximilian Scali, Carlo Probability 60J10, 05C81, 60J35 In this article we study collisions of two independent random walks on comb graphs $\mathrm{Comb}(\tilde{G},f)$ for a large class of recurrent planar graphs $\tilde{G}$ and profile functions $f$, the latter governing the length of vertical segments (called "teeth") attached to vertices of the base graph $\tilde{G}$. We prove that the number of collisions of two random walks starting from the same site undergoes a phase transition depending on the growth of $f$. As a benchmark example, we show that for $\mathrm{Comb}(\mathbb{Z}^2,f_γ)$ with $f_γ(z) = \log^γ(\|z\|_\infty\vee 1)$ and $\|\cdot \|_\infty$ denoting the supremum norm, two independent random walks started at the origin collide finitely often almost surely if $γ> 1$, answering a question of Barlow, Peres, and Sousi, see arXiv:1003.3255, who established that infinitely many collisions occur almost surely if $γ\leq 1$. We furthermore establish phase transitions in the cases where the base graph $\tilde{G}$ is pre-fractal, or a typical realization of a supercritical cluster of planar Bernoulli bond percolation. |
| title | Collisions of random walks on comb graphs with a planar base |
| topic | Probability 60J10, 05C81, 60J35 |
| url | https://arxiv.org/abs/2508.19814 |