{
  "abstract": "In graphical two-choice allocation, each arriving ball is assigned to one endpoint of a random edge. We study rules whose decision is a monotone function of the two endpoint loads, allowing edge-dependent thresholds and fresh randomization. Such a rule has an exact unit-discrepancy coupling: adding one ball to the initial state produces one tagged discrepancy at every later time. We represent the tag by conditional-expectation projections on the marked edge space and obtain diffusive displacement bounds. A transport-volume inequality then converts slow propagation of influence into lower bounds for the load gap. On the cycle with $n$ vertices, from every initial distribution and at every physical time $t\\ge 1/n$, the expected gap is at least a constant times $\\min\\{\\sqrt n,t^{1/4}\\}$, and the gap exceeds this scale with probability at least $1/8$. After exactly $k\\ge1$ allocations, the corresponding scale is $\\min\\{\\sqrt n,(k/n)^{1/4}\\}$. No stationarity, symmetry, recurrence, or moment assumption is used. A smoothed threshold rule in the same class has expected gap $O(\\sqrt n\\log n)$ up to any fixed polynomial time horizon, so the saturated cycle bound is sharp within the class up to a logarithmic factor. The general inequality also yields a lower bound of order $\\sqrt{L/K}$ on the $L\\times K$ rectangular torus $C_L\\square C_K$; combined with a strategy-independent logarithmic bound, this gives order $\\sqrt{L/K}+\\log(LK)$. These results separate endpoint-local rules from global-information strategies that achieve polylogarithmic gaps on cycles. All numbered results are verified in Lean.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38726",
  "author_total": 1,
  "authors": [
    {
      "name": "Obinna Okechukwu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/exkzd2gns215x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38726",
  "fulltext_url": null,
  "id": "exkzd2gns215x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38726",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "1705|Optimization and Search Problems"
    ]
  ],
  "openalex": "W7215108553",
  "pdf_url": "https://arxiv.org/pdf/2609.38726",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/exkzd2gns215x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.38726",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38726"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Local decisions, diffusive influence, and lower bounds for graphical balanced allocation",
  "title_text": "Local decisions, diffusive influence, and lower bounds for graphical balanced allocation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/exkzd2gns215x",
  "venue": "arXiv",
  "versions": []
}