Explosion and linear transit times in infinite trees

Abstract

Let T be an infinite rooted tree with weights w e assigned to its edges. Denote by m n (T ) the minimum weight of a path from the root to a node of the nth generation. We consider the possible behaviour of m n (T ) with focus on the two following cases: we say T is explosive if lim n→∞ m n (T ) < ∞ , and say that T exhibits linear growth if lim inf n→∞ m n (T ) n > 0 . B Omid AminiWe consider a class of infinite randomly weighted trees related to the Poisson-weighted infinite tree, and determine precisely which trees in this class have linear growth almost surely. We then apply this characterization to obtain new results concerning the event of explosion in infinite randomly weighted spherically-symmetric trees, answering a question of Pemantle and Peres (Ann Probab 22 (1), [180][181][182][183][184][185][186][187][188][189][190][191][192][193][194] 1994). As a further application, we consider the random real tree generated by attaching sticks of deterministic decreasing lengths, and determine for which sequences of lengths the tree has finite height almost surely.Mathematics Subject Classification 60K35 · 60C05 · 60G55 · 60J80 · 05C80

Asymptotics of heights in random trees constructed by aggregation

“…This model has been recently studied by Curien and Haas [8] and Amini et al [4]. The paper [8] gives necessary and sufficient conditions on the sequence (a n ) for T to be compact (equivalently bounded) and studies its Hausdorff dimension.…”

“…The issue of finding an exact condition on (a n ) for T to be bounded is still open. However, Amini et al [4] obtained an exact condition for T to be bounded, provided that (a n ) is non-increasing. In that case, almost surely, T is bounded if and only if i≥1 i −1 a i < ∞.…”

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To each sequence (a n ) of positive real numbers we associate a growing sequence (T n ) of continuous trees built recursively by gluing at step n a segment of length a n on a uniform point of the pre-existing tree, starting from a segment T 1 of length a 1 . Previous works [4,8] on that model focus on the influence of (a n ) on the compactness and Hausdorff dimension of the limiting tree. Here we consider the cases where the sequence (a n ) is regularly varying with a non-negative index, so that the sequence (T n ) exploses.

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Random gluing of metric spaces

“…where the (H k ) k≥1 are i.i.d., independent of the (U k ) k≥1 and have the law of H, see (4). Under H d (iii) the random variable H has a finite second moment, and an application of Kolmogorov's three series theorem tells us that the almost sure convergence of k≥1…”

“…This model was introduced and studied by Curien and Haas in [7], who proved that when λ n = n −α+o (1) for some α > 0, the tree obtained is a.s. compact and has Hausdorff dimension (1 ∨ α −1 ). In [4], Amini et. al.…”