{"ID":2832585,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.05756","arxiv_id":"2512.05756","title":"A phase transition in Erdős-Barak random graphs","abstract":"We study monotone paths in Erdős-Rényi random graphs on numbered vertices. Benjamini \u0026 Tzalik established a phase transition at $p = \\frac{\\log n}{n}$ for this model. We refine the critical value to $p = \\frac{\\log n - \\log \\log n }{n}$ and identify the critical window of order $Θ(1/n)$.","short_abstract":"We study monotone paths in Erdős-Rényi random graphs on numbered vertices. Benjamini \u0026 Tzalik established a phase transition at $p = \\frac{\\log n}{n}$ for this model. We refine the critical value to $p = \\frac{\\log n - \\log \\log n }{n}$ and identify the critical window of order $Θ(1/n)$.","url_abs":"https://arxiv.org/abs/2512.05756","url_pdf":"https://arxiv.org/pdf/2512.05756v2","authors":"[\"Gilles Blanchard\",\"Nicolas Curien\",\"Klara Krause\",\"Alexander Reisach\"]","published":"2025-12-05T14:42:25Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.CO\",\"math.ST\"]","methods":"[]","has_code":false}