{"paper":{"title":"Asymptotics for the Expected Maximum of Random Walks and L\\'evy Flights with a Constant Drift","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"cond-mat.stat-mech","authors_text":"Gregory Schehr, Philippe Mounaix, Satya N. Majumdar","submitted_at":"2018-05-31T14:25:20Z","abstract_excerpt":"In this paper, we study the large $n$ asymptotics of the expected maximum of an $n$-step random walk/L\\'evy flight (characterized by a L\\'evy index $1<\\mu\\leq 2$) on a line, in the presence of a constant drift $c$. For $0<\\mu\\leq 1$, the expected maximum is infinite, even for finite values of $n$. For $1<\\mu\\leq 2$, we obtain all the non-vanishing terms in the asymptotic expansion of the expected maximum for large $n$. For $c<0$ and $\\mu =2$, the expected maximum approaches a non-trivial constant as $n$ gets large, while for $1<\\mu < 2$, it grows as a power law $\\sim n^{2-\\mu}$. For $c>0$, the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.12489","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"}