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SPINAL PARTITIONS AND INVARIANCE UNDER REROOTING OF CONTINUUM RANDOM TREES
"... We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a twoparameter Poisson–Dirichlet family of continuous fragmentation trees ..."
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We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a twoparameter Poisson–Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform rerooting. 1. Introduction. Starting from a rooted combinatorial tree T[n] with n leaves labeled by [n] ={1,...,n}, we call the path from the root to the leaf labeled 1 the spine of T[n]. Deleting each edge along the spine of T[n] defines a graph whose connected components we call bushes. If, as well as cutting each edge on the spine, we cut each edge connected to a spinal vertex, each bush is further decomposed
Weak convergence of random pmappings and the exploration process of inhomogeneous continuum random trees
 PROBAB. THEORY RELAT. FIELDS
, 2005
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The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin’s local time identity
, 2004
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Regenerative tree growth: Markovian embedding of fragmenters, bifurcators and bead splitting processes ∗
, 2013
"... Some, but not all processes of the form Mt = exp(−ξt) for a purejump subordinator ξ with Laplace exponent Φ arise as residual mass processes of particle 1 (tagged particle) in an exchangeable fragmentation processes. We introduce the notion of a Markovian embedding of M in an exchangeable fragmenta ..."
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Cited by 1 (1 self)
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Some, but not all processes of the form Mt = exp(−ξt) for a purejump subordinator ξ with Laplace exponent Φ arise as residual mass processes of particle 1 (tagged particle) in an exchangeable fragmentation processes. We introduce the notion of a Markovian embedding of M in an exchangeable fragmentation process and show that for each Φ, there is a unique binary dislocation measure ν such that M has a Markovian embedding in an associated exchangeable fragmentation process. The identification of the Laplace exponent Φ ∗ of its tagged particle process M ∗ gives rise to a symmetrisation operation Φ ↦ → Φ ∗ , which we investigate in a general study of pairs (M,M ∗ ) that coincide up to a junction time and then evolve independently. We call M a fragmenter and (M,M ∗ ) a bifurcator. For all Φ and α> 0, we can represent a fragmenter M as an interval R1 = [0, ∫ ∞ 0 Mα t dt] equipped with a purely atomic probability measure µ1 capturing the jump sizes of Mt after an αselfsimilar timechange. We call (R1,µ1) an (α,Φ)string of beads. We study binary tree growth processes that in the nth step sample a bead from µn and build (Rn+1,µn+1) by splitting the bead into a new string of beads, a rescaled independent copy of (R1,µ1) that we tie to the position of the sampled bead. We show that all such bead splitting processes converge almost surely to an αselfsimilar CRT, in the GromovHausdorffProhorov sense.
Period Lengths for Iterated Functions. (Preliminary Version)
, 2008
"... Let Ωn be the n nelement set consisting of functions that have [n] as both domain and codomain. Since Ωn is finite, it is clear by the pigeonhole principle that, for any f ∈ Ωn, the sequence of compositional iterates f, f (2) , f (3) , f (4)... must eventually repeat. Let T(f) be the period of this ..."
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Let Ωn be the n nelement set consisting of functions that have [n] as both domain and codomain. Since Ωn is finite, it is clear by the pigeonhole principle that, for any f ∈ Ωn, the sequence of compositional iterates f, f (2) , f (3) , f (4)... must eventually repeat. Let T(f) be the period of this eventually periodic sequence of functions, i.e. the least positive integer T such that, for all m ≥ n, f (m+T) = f (m). A closely related number B(f) = the product of the lengths of the cycles of f, has previously been used as an approximation for T. This paper proves that the average values of these two quantities are quite different. The expected value of T is 1 n n