{"paper":{"title":"Exact Statistics of the Gap and Time Interval Between the First Two Maxima of Random Walks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.dis-nn","math.PR","q-fin.ST"],"primary_cat":"cond-mat.stat-mech","authors_text":"Gregory Schehr, Philippe Mounaix, Satya N. Majumdar","submitted_at":"2013-03-19T14:14:14Z","abstract_excerpt":"We investigate the statistics of the gap, G_n, between the two rightmost positions of a Markovian one-dimensional random walker (RW) after n time steps and of the duration, L_n, which separates the occurrence of these two extremal positions. The distribution of the jumps \\eta_i's of the RW, f(\\eta), is symmetric and its Fourier transform has the small k behavior 1-\\hat{f}(k)\\sim| k|^\\mu with 0 < \\mu \\leq 2. We compute the joint probability density function (pdf) P_n(g,l) of G_n and L_n and show that, when n \\to \\infty, it approaches a limiting pdf p(g,l). The corresponding marginal pdf of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.4607","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"}