On the growth of random planar maps with a prescribed degree sequence

2019/02/12 by Marzouk, Cyril
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1902.04539

Abstract

For non-negative integers (d_n(k))_{k ≥ 1} such that ∑_{k ≥ 1} d_n(k) = n, we sample a bipartite planar map with n faces uniformly at random amongst those which have d_n(k) faces of degree 2k for every k ≥ 1 and we study its asymptotic behaviour as n → ∞. We prove that the diameter of such maps grow like σ_n^1/2, where σ_n² = ∑_{k ≥ 1} k (k-1) d_n(k) is a global variance term. More precisely, we prove that the vertex-set of these maps equipped with the graph distance divided by σ_n^1/2 and the uniform probability measure always admits subsequential limits in the Gromov-Hausdorff-Prokhorov topology. Our proof relies on a bijection with random labelled trees; we are able to prove that the label process is always tight when suitably rescaled, even if the underlying tree is not tight for the Gromov-Hausdorff topology. We also rely on a new spinal decomposition which is of independent interest. Finally this paper also serves as a toolbox for a companion paper in which we discuss more precisely Brownian limits of such maps.

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