{"paper":{"title":"Median and mean of the Supremum of $L^2$ normalized random holmorphic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.PR","authors_text":"Renjie Feng, Steve Zelditch","submitted_at":"2013-03-17T20:03:21Z","abstract_excerpt":"We prove that the expected value and median of the supremum of $L^2$ normalized random holomorphic fields of degree $n$ on $m$-dimensional K\\\"ahler manifolds are asymptotically of order $\\sqrt{m\\log n}$. This improves the prior result of Shiffman-Zelditch (arXiv:math/0303335) that the upper bound of the media is of order $\\sqrt{\\log n}$\n  The estimates are based on the entropy methods of Dudley and Sudakov combined with a precise analysis of the relevant pseudo-metric and its covering numbers, which can be precisely evaluated using off-diagonal asymptotics of Bergman kernels. Recent work of th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.4096","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":""},"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"}