Tightness of the maximum of branching random walk in random environment and zero-crossings of solutions to discrete parabolic differential equations
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| Format: | Preprint |
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2026
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| _version_ | 1866915728274227200 |
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| author | Černý, Jiří Dalessi, Flavio |
| author_facet | Černý, Jiří Dalessi, Flavio |
| contents | We study branching random walk on $\mathbb{Z}$ in a bounded i.i.d. random environment. For this process, we prove that, for almost every realization of the environment, the distributions of the maximally displaced particle (re-centered around their medians) are tight. This extends the result of arXiv:2408.01555 , where tightness was established in the annealed sense, and of arXiv:2212.12390 , where a similar quenched result was proved for branching Brownian motion in random environment. Our proof relies on studying certain discrete-space linear PDEs and establishing that the number of zero-crossings of their solutions is non-increasing in time. We observe that our technique requires no additional assumptions on the environment, in contrast to arXiv:2408.01555 , arXiv:2212.12390 . |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_09214 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tightness of the maximum of branching random walk in random environment and zero-crossings of solutions to discrete parabolic differential equations Černý, Jiří Dalessi, Flavio Probability 60J80 We study branching random walk on $\mathbb{Z}$ in a bounded i.i.d. random environment. For this process, we prove that, for almost every realization of the environment, the distributions of the maximally displaced particle (re-centered around their medians) are tight. This extends the result of arXiv:2408.01555 , where tightness was established in the annealed sense, and of arXiv:2212.12390 , where a similar quenched result was proved for branching Brownian motion in random environment. Our proof relies on studying certain discrete-space linear PDEs and establishing that the number of zero-crossings of their solutions is non-increasing in time. We observe that our technique requires no additional assumptions on the environment, in contrast to arXiv:2408.01555 , arXiv:2212.12390 . |
| title | Tightness of the maximum of branching random walk in random environment and zero-crossings of solutions to discrete parabolic differential equations |
| topic | Probability 60J80 |
| url | https://arxiv.org/abs/2601.09214 |