{"paper":{"title":"Critical points of multidimensional random Fourier series: central limits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.PR","authors_text":"Liviu I. Nicolaescu","submitted_at":"2015-11-16T14:17:14Z","abstract_excerpt":"We investigate certain families $X^\\hbar$, $0<\\hbar \\ll 1$, of Gaussian random smooth functions on the $m$-dimensional torus $\\mathbb{T}^m_\\hbar:=\\mathbb{R}^m/(\\hbar^{-1}\\mathbb{Z} )^m$. We show tha,t for any cube $B\\subset \\mathbb{R}^m$ of size $<1/2$ and centered at the origin, the number of critical points of $X^\\hbar$ in the region $\\hbar^{-1}B/(\\hbar^{-1}\\mathbb{Z} )^m\\subset\\mathbb{T}^m_\\hbar$ has mean $\\sim c_1\\hbar^{-m}$, variance $\\sim c_2\\hbar^{-m/2}$, $c_1,c_2>0$, and satisfies a central limit theorem as $\\hbar\\searrow 0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.04965","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"}