https://hal-ens-lyon.archives-ouvertes.fr/ensl-01651022v2Mallein, BastienBastienMalleinLAGA - Laboratoire Analyse, Géométrie et Applications - UP8 - Université Paris 8 Vincennes-Saint-Denis - CNRS - Centre National de la Recherche Scientifique - Université Sorbonne Paris NordIG - Institut Galilée - Université Sorbonne Paris NordRamassamy, SanjaySanjayRamassamyDepartment of Mathematics - Brown UniversityBarak-Erdös graphs and the infinite-bin modelHAL CCSD2021[MATH.MATH-PR] Mathematics [math]/Probability [math.PR][MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]Ramassamy, Sanjay2022-10-27 09:16:382022-10-29 03:58:452022-10-28 08:58:07enJournal articleshttps://hal-ens-lyon.archives-ouvertes.fr/ensl-01651022v2/document10.1214/20-AIHP1141https://hal-ens-lyon.archives-ouvertes.fr/ensl-01651022v1application/pdf2A Barak-Erdös graph is a directed acyclic version of the Erdös-Rényi random graph. It is obtained by performing independent bond percolation with parameter p on the complete graph with vertices {1, ..., n}, in which the edge between two vertices i < j is directed from i to j. The length of the longest path in this graph grows linearly with the number of vertices, at rate C(p). In this article, we use a coupling between Barak-Erdös graphs and infinite-bin models to provide explicit estimates on C(p). More precisely, we prove that the front of an infinite-bin model grows at linear speed, and that this speed can be obtained as the sum of a series. Using these results, we prove the analyticity of C for p > 1/2, and compute its power series expansion. We also obtain the first two terms of the asymptotic expansion of C as p → 0, using a coupling with branching random walks with selection.