On trees with a given degree sequence

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1 On trees with a given degree sequence Osvaldo ANGTUNCIO-HERNÁNDEZ and Gerónimo URIBE BRAVO Instituto de Matemáticas Universidad Nacional Autónoma de México 3rd Bath-UNAM-CIMAT workshop

2 Our discrete tree variety Trees as graphs: A tree is a graph in which any two vertices can be joined by a unique path. A rooted tree is a graph with a distinguished element ρ called the root. The genealogical partial order is defined as: u v if u belongs to the path joining ρ to v. Plane trees: A plane tree is a tree in which a total order has been given which is adapted to the genealogical partial order: u v implies u v.

3 Trees with a given degree sequence Let (V, E, ρ, ) be a plane tree. Write V = {v 0,..., v n 1 } where ρ = v 0 < < v n 1 and c i = # {j i : {i, j} E}. Degree sequence The degree sequence N 0, N 1,... is obtained by setting N i = # {j : c j = i}. It is characterized by: i N i = 1 + i in i. Every such sequence arises from a plane tree. Tree with a given degree sequence Let s = (N 0, N 1,...) be a degree sequence. We will be interested in uniform trees from the set of plane trees having degree sequence s.

4 Examples Root

5 Examples Root

6 Examples Root

7 Conditioned Galton-Watson trees Let U = { } {u 1 u n : n 1, u i Z + } be the set of canonical labels of plane trees. Let µ = (µ k, k N) be an offspring distribution and (ξ u, u U ) be iid with law µ. Galton-Watson trees Θ = { } {uj U : j ξ u }. Conditioned Galton-Watson trees Θ n d = Θ conditioned on having n vertices Proposition Θ conditioned on having degree sequence s is uniform on trees with degree sequence s.

8 Coding walks for plane trees Let τ be a plane tree. Order its individuals as = u 0 < < u n 1. Define x i = k u0 (τ) + + k ui 1 (τ) i. x is called the depth-first walk of the tree. x satisfies: x i 0 for i n 1 and x n = 1. Conversely, any such walk gives rise to a tree.

9 Coding walks for CGW trees Let µ be an offspring distribution and let µ i = µ i+1. Let S be a random walk conditioned to visit 1 at time n. Let V be the Vervaat transform of S: If ρ is uniform on the set of times that S visits its minimum, consider V i = S ρ+i mod n S ρ Then random tree whose DFW is V is a CGW(µ, n).

10 Generating trees with a given degree sequence Let s = (N 0, N 1,...) be a degree sequence of size n = i N i. Let c = (c 1,..., c n ) be an associated child sequence: N j = # {i : c i = j}. Let C be obtained from c by a uniform random permutation and define: X i = C C i i. Let V be the Vervaat transform of X. Then V is the depth-first walk (or breadth first walk) of a uniform tree with degree sequence s.

11 Exchangeable increment processes Let β = (β 1, β 2,...) be such that βi 2 < and let σ [0, ). Let b be a Brownian bridge and U 1, U 2,... uniform on (0, 1); everything independent. EI process An EI process directed by (σ, β) admits the representation X t = σb t + i β i [1 Ui t t]. Simulation

12 Convergence of Depth-First Walks Let (s n ) n be a sequence of degree sequences, s n = (N0 n, Nn 1,...). Let cn be the unique non-increasing associated child sequence. Let X n be the DFW of a uniform tree with degree sequence s n. Assume the technical hypotheses: 1. s n = i Nn i as n. 2. There exists a non-negative sequence (b n, n 1), b n, M N { }, σ [0, ) and β l 2 such that 1 M (i 1) 2 Ni n σ 2 + b n i i=1 β 2 i c i b n β i 3. Either σ > 0 or β i =. Then, X n s n /b n converges on Skorohod space to the Vervaat transform of X, where X is a process with exchangeable increments on [0, 1] directed by σ and β.

13 The number k u (t) is interpreted as the number of children of u in t. The height We denote function by A the set of ofaallplane rooted ordered tree trees. In what follows, we see each vertex of the tree t as an individual of a population whose t is the family tree. The cardinality #(t) of t is the total progeny. (1,2,1) (1,2,2) (1,1) (1,2) (1,3) C s s ζ(t) 2 1 h t(n) n #(t) 1 Tree t Contour function C s Height function h t(n) Figure 1 Figure: From Random trees and applications by Jean-François Le Gall We will now explain how trees can be coded by discrete functions. We first introduce the (discrete) height function associated with a tree t. Let us denote by u 0 =, u 1, u 2,..., u #(t) 1 the elements of t listed in lexicographical order. The height function (h t (n); 0 n < #(t)) is defined by

14 The main open problem For a sequence of degree sequences (s n ), let τ n be uniform on trees with degree sequence s n. Find conditions on (s n ) such that τ n has a scaling limit. The scaling limit should be a continuum tree: a metric space in which two elements can be joined by a unique path.

15 A first step For a sequence of degree sequences (s n ) of sizes s n, let τ n be uniform on trees with degree sequence s n. Let U be uniform on {0,..., s n 1} and independent of τ n. Find conditions on (s n ) such that the height of individual U in τ n has a scaling limit.

16 A related problem for CGW The DFW of a µ n -CGW, say V n, is the Vervaat transform of a bridge from 0 to 1 of the associated random walk. Find conditions on µ n for the convergence of these bridges and of their Vervaat transformation. The problem is relevant since the height process of the limit should code the limiting tree.

17 A subproblem Let µ n be a sequence of degree distributions. Let θ n be a µ n -GW tree of size s n. Find conditions on µ n and s n such that the sequence of trees θ n has a limit.

Shifting processes with cyclically exchangeable increments at random

Shifting processes with cyclically exchangeable increments at random Gerónimo URIBE BRAVO (Collaboration with Loïc CHAUMONT) Instituto de Matemáticas UNAM Universidad Nacional Autónoma de México www.matem.unam.mx/geronimo