{"paper":{"title":"On the dependence of the zero-free region of a partition function on the external field","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Alexander Barvinok","submitted_at":"2026-08-04T13:56:28Z","abstract_excerpt":"Let $\\{0, 1\\}^n$ be the Boolean cube, endowed with the probability product measure, where ${\\Bbb P}(1)=p$ and ${\\Bbb P}(0)=q$ with $0 < p \\leq q=1-p$. Let $\\phi_i: \\{0, 1\\}^n \\longrightarrow {\\Bbb C}$ be $1$-Lipschitz functions in the Hamming metric, such that each $\\phi_i$ depends on at most $r$ coordinates of $x \\in \\{0, 1\\}^n$, where $rp \\geq 12$. For $j=1, \\ldots, n$, let $I_j $ be the set of indices $i$ such that $\\phi_i$ depends on the $j$-th coordinate. We prove that $E\\thinspace \\exp\\left\\{ \\sum_{i=1}^m \\lambda_i \\phi_i \\right\\} \\ne 0$ provided $\\lambda_i \\in {\\Bbb C}$ satisfy $\\sum_{i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.03687","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.03687/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}