{'fraction_non_alphanumeric': 0.03391959798994975, 'fraction_numerical': 0.016331658291457288, 'mean_word_length': 4.9037037037037035, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the distance-profile of random rooted plane graphs

We study the distance-profile of the random rooted plane graph Gn with n edges (by a plane graph we mean a planar map with no loops nor multiple edges). Our main result is that the profile and radius of Gn (with respect to the root-vertex), rescaled by (2n), converge to explicit distributions related to the Brownian snake. A crucial ingredient of our proof is a bijection we have recently introduced between rooted outer-triangular plane graphs and rooted eulerian triangulations, combined with ingredients from Chassaing and Schaeffer (2004), Bousquet-Mélou and Schaeffer (2000), and Addario-Berry and Albenque (2013). We also show that the result for plane graphs implies similar results for random rooted loopless maps and general maps.