We show that the diameter diam(Gn) of a random labelled connected planar graph withnvertices is equal ton1/4+o(1), in probability. More precisely, there exists a constantc> 0 such that P((Gₙ)∈(n1/4-,n1/4+))≥ 1-exp(-nᶜ) for ε small enough andn ≥ n0(ε). We prove similar statements for 2-connected and 3-connected planar graphs and maps.
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