{
  "annotation_id": "38898d6e-cd8b-464b-b93a-4ae4620776d9",
  "date_created": "2026-02-17T05:53:19.856000Z",
  "date_modified": "2026-02-17T05:53:19.856000Z",
  "file_hash": "8b58fb6b4e0051d2362ed3fae2870cbc5b5a31880eaa7e7bde4eac10ea2ebe85",
  "private": false,
  "record": {
    "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 .",
    "arxiv_id": "2601.09214",
    "authors": [
      "Ji\u0159\u00ed \u010cern\u00fd",
      "Flavio Dalessi"
    ],
    "categories": [
      "math.PR"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Tightness of the maximum of branching random walk in random environment and zero-crossings of solutions to discrete parabolic differential equations",
    "url": "https://arxiv.org/abs/2601.09214",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "279b1ad2-38c8-42b6-ba8b-f08a63d35edb",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}