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arXiv · 2601.09214

Tightness of the maximum of branching random walk in random environment and zero-crossings of solutions to discrete parabolic differential equations

Jiří Černý·Flavio Dalessi
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Abstract

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 .

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Jiří Černý, Flavio Dalessi. 2026-01-14. Tightness of the maximum of branching random walk in random environment and zero-crossings of solutions to discrete parabolic differential equations. https://arxiv.org/abs/2601.09214

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