{
  "abstract": "The Poisson approximation theorem states that when $m$ balls are tossed uniformly and randomly into $n$ bins, the distribution of the bin loads can sometimes be approximated by $n$ independent Poisson random variables of mean $m/n$. We show that when the balls are tossed via a random $\\mathbb{F}_2$-linear map, the load distribution is still governed by this Poisson distribution, albeit in a (necessarily) weaker manner. This allows us to show that a random linear map behaves like a fully random map in the following ways: 1. $\\textbf{Expected loads.}$ We show that the expected value of the $r$th largest bin is governed by the $r$th largest value among $n$ independent Poisson random variables of mean $m/n$. Specializing to the canonical setting of $m=n$ and $r=1$, our bounds show that the expected maximum load when $n$ balls are hashed into $n$ bins by a random $\\mathbb{F}_2$-linear map is no more than the corresponding expectation for a fully random map $\\textit{plus}$ $\\frac{3}2 + o(1)$. Specializing to $m\\gg n\\log n$, we show that the expected max load is $(1+o(1))m/n$ in this regime. 2. $\\textbf{Concentration.}$ When $m\\gg n\\log (1/ρ)$ balls are tossed into $n$ bins randomly, we expect with $0.99$ probability that $\\textit{every}$ density-$ρ$ subset of bins will contain a $(1\\pm o(1) )ρ$-fraction of the balls. We show that this remains the case if the $m$ balls are tossed using a random linear map, and our bound matches the $o(1)$-factor produced by the fully random case. This gives a stronger version of the Leftover Hash Lemma for linear maps. An implication of this is that all bins have load $m/n \\pm O(\\sqrt{(m/n)\\log n})$ with probability $0.99$. We obtain these results by showing that the average of a log-potential on the bin loads is upper bounded by the cumulant generating function of a Poisson random variable of mean $m/n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34750",
  "author_total": 1,
  "authors": [
    {
      "name": "Vinayak M. Kumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a0554q21zw9jc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34750",
  "fulltext_url": null,
  "id": "a0554q21zw9jc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34750",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "1702|Algorithms and Data Compression"
    ]
  ],
  "openalex": "W7223781679",
  "pdf_url": "https://arxiv.org/pdf/2610.34750",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a0554q21zw9jc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34750",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34750"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Linear Hashing Is Poissonesque",
  "title_text": "Linear Hashing Is Poissonesque",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/a0554q21zw9jc",
  "venue": "arXiv",
  "versions": []
}