{"paper":{"title":"Gaussian Approximation of the Distribution of Strongly Repelling Particles on the Unit Circle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Alexander Soshnikov, Yuanyuan Xu","submitted_at":"2017-10-31T05:12:08Z","abstract_excerpt":"In this paper, we consider a strongly-repelling model of $n$ ordered particles $\\{e^{i \\theta_j}\\}_{j=0}^{n-1}$ with the density $p({\\theta_0},\\cdots, \\theta_{n-1})=\\frac{1}{Z_n} \\exp \\left\\{-\\frac{\\beta}{2}\\sum_{j \\neq k} \\sin^{-2} \\left( \\frac{\\theta_j-\\theta_k}{2}\\right)\\right\\}$, $\\beta>0$. Let $\\theta_j=\\frac{2 \\pi j}{n}+\\frac{x_j}{n^2}+const$ such that $\\sum_{j=0}^{n-1}x_j=0$. Define $\\zeta_n \\left( \\frac{2 \\pi j}{n}\\right) =\\frac{x_j}{\\sqrt{n}}$ and extend $\\zeta_n$ piecewise linearly to $[0, 2 \\pi]$. We prove the functional convergence of $\\zeta_n(t)$ to $\\zeta(t)=\\sqrt{\\frac{2}{\\beta}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.11328","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}