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Cutting down trees with a Markov chainsaw
, 2011
"... We provide simplified proofs for the asymptotic distribution of the number of cuts required to cut down a Galton–Watson tree with critical, finitevariance offspring distribution, conditioned to have total progeny n. Our proof is based on a coupling which yields a precise, nonasymptotic distributio ..."
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We provide simplified proofs for the asymptotic distribution of the number of cuts required to cut down a Galton–Watson tree with critical, finitevariance offspring distribution, conditioned to have total progeny n. Our proof is based on a coupling which yields a precise, nonasymptotic distributional result for the case of uniformly random rooted labeled trees (or, equivalently, Poisson Galton–Watson trees conditioned on their size). Our approach also provides a new, random reversible transformation between Brownian excursion and Brownian bridge. 1
THE VERTICAL PROFILE OF EMBEDDED TREES
, 2012
"... Consider a rooted binary tree with n nodes. Assign with the root the abscissa 0, and with the left (resp. right) child of a node of abscissa i the abscissa i−1 (resp. i +1). We prove that the number of binary trees of size n having exactly ni nodes at abscissa i, for ℓ ≤ i ≤ r (with n = ∑ ini), is ..."
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Consider a rooted binary tree with n nodes. Assign with the root the abscissa 0, and with the left (resp. right) child of a node of abscissa i the abscissa i−1 (resp. i +1). We prove that the number of binary trees of size n having exactly ni nodes at abscissa i, for ℓ ≤ i ≤ r (with n = ∑ ini), is n0 nℓnr n−1 +n1 n0 −1 ℓ≤i≤r i=0
FURTHER EXAMPLES WITH MOMENTS OF GAMMA TYPE
"... This is an appendix to [9] containing further examples. See [9] for notation and for examples and equations referred to below by numbers. See also the further references in [9, Addendum]. This appendix will probably be updated in the future. Appendix A. Further examples Example A.1 (Rayleigh distrib ..."
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This is an appendix to [9] containing further examples. See [9] for notation and for examples and equations referred to below by numbers. See also the further references in [9, Addendum]. This appendix will probably be updated in the future. Appendix A. Further examples Example A.1 (Rayleigh distribution). The Rayleigh distribution R is the chi distribution χ(2), with density xe−x2 /2. This is a special case of Example 3.6, and we have E R s = 2 s/2 Γ(s/2 + 1), −2 < Re s < ∞. (A.1)
A NOTE ON NATURALLY EMBEDDED TERNARY TREES
"... Abstract. In this note we consider ternary trees naturally embedded in the plane in a deterministic way. The root has position zero, or in other words label zero, and the three children of a node with position j ∈ Z have positions j − 1, j, and j + 1. We derive the generating function of embedded te ..."
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Abstract. In this note we consider ternary trees naturally embedded in the plane in a deterministic way. The root has position zero, or in other words label zero, and the three children of a node with position j ∈ Z have positions j − 1, j, and j + 1. We derive the generating function of embedded ternary trees where all internal nodes have labels less than or equal to j, with j ∈ N. Furthermore, we study the generating function of the number of ternary trees of size n with a given number of internal nodes with label j. Moreover, we discuss generalizations of this counting problem to several labels at the same time. We also study a refinement of the depth of the external node of rank s, with 0 ≤ s ≤ 2n, by keeping track of the left, center, and right steps on the unique path from the root to the external node. The 2n + 1 external nodes of a ternary tree are ranked from the left to the right according to an inorder traversal of the tree. Finally, we discuss generalizations of the considered enumeration problems to embedded dary trees. 1.
Large unicellular maps in high genus
"... We study the geometry of a random unicellular map which is uniformly distributed on the set of all unicellular maps whose genus size is proportional to the number of edges. We prove that the distance between two uniformly selected vertices of such a map is of order log n and the diameter is also of ..."
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We study the geometry of a random unicellular map which is uniformly distributed on the set of all unicellular maps whose genus size is proportional to the number of edges. We prove that the distance between two uniformly selected vertices of such a map is of order log n and the diameter is also of order log n with high probability. We further prove a quantitative version of the result that the map is locally planar with high probability. The main ingredient of the proofs is an exploration procedure which uses a bijection due to Chapuy, Feray and Fusy ([14]).
ON EMBEDDED TREES AND LATTICE PATHS
"... Abstract. Bouttier, Di Francesco and Guitter introduced a method for solving certain classes of algebraic recurrence relations arising the context of embedded trees and map enumeration. The aim of this note is to apply this method to three problems. First, we discuss a general family of embedded bin ..."
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Abstract. Bouttier, Di Francesco and Guitter introduced a method for solving certain classes of algebraic recurrence relations arising the context of embedded trees and map enumeration. The aim of this note is to apply this method to three problems. First, we discuss a general family of embedded binary trees, trying to unify and summarize several enumeration results for binary tree families, and also to add new results. Second, we discuss the family of embedded dary trees, embedded in the plane in a natural way. Third, we show that several enumeration problems concerning simple families of lattice paths can be solved without using the kernel method by regarding simple families of lattice paths as degenerated families of embedded trees. 1.
The range of treeindexed random walk in low dimensions
, 2014
"... We study the range Rn of a random walk on the ddimensional lattice Zd indexed by a random tree with n vertices. Under the assumption that the random walk is centered and has finite fourth moments, we prove in dimension d ≤ 3 that n−d/4Rn converges in distribution to the Lebesgue measure of the supp ..."
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We study the range Rn of a random walk on the ddimensional lattice Zd indexed by a random tree with n vertices. Under the assumption that the random walk is centered and has finite fourth moments, we prove in dimension d ≤ 3 that n−d/4Rn converges in distribution to the Lebesgue measure of the support of the integrated superBrownian excursion (ISE). An auxiliary result shows that the suitably rescaled local times of the treeindexed random walk converge in distribution to the density process of ISE. We obtain similar results for the range of critical branching random walk in Zd, d ≤ 3. As an intermediate estimate, we get exact asymptotics for the probability that a critical branching random walk starting with a single particle at the origin hits a distant point. The results of the present article complement those derived in higher dimensions in our earlier work.