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SUMMARY:Limits of (randomly) growing Schröder trees and exchangeability -
Julian Gerstenberg (Leibniz Universität Hannover)
DTSTART;VALUE=DATE-TIME:20181008T120000
DTEND;VALUE=DATE-TIME:20181008T130000
UID:https://talks.ox.ac.uk/talks/id/16131e1b-de82-4227-ad4a-ab46c536a9a0/
DESCRIPTION:We consider finite rooted ordered trees in which every interna
l node has at least two children\, sometimes called Schröder trees\; the
size |t| of such a tree t is the number of its leaves. An important concep
t with trees is that of inducing subtrees. Given a tree t of size k and a
larger tree t' of size n\\geq k we define 0 \\leq \\theta(t\,t')\\leq 1 to
be the probability of obtaining t as a randomly induced subtree of size k
in t'. One can think of \\theta(t\,t') to be the _density of the pattern
t in t'_. In this talk we consider two closely related questions concernin
g the nature of \\theta:\n1. A sequences of trees (t_n)_n with |t_n|\\righ
tarrow\\infty is called \\theta-convergent\, if \\theta(t\,t_n) converges
for every fixed tree t. The limit of (t_n)_n is the function t\\mapsto \\l
im_n\\theta(t\,t_n). What limits exist? \n2. A Markov chain (X_n)_n with X
_n being a random tree of size n is called a \\theta-chain if P(X_k=t|X_n=
t')=\\theta(t\,t') for all k \\leq n. What \\theta-chains exist?\n\nSimila
r questions have been treated for many different types of discrete structu
res (words\, permutations\, graphs \\dots)\; binary Schröder trees (Catal
an trees) are considered in [1]. We present a De Finetti-type representati
on for \\theta-chains and a homeomorphic description of limits of \\theta-
convergent sequences involving certain tree-like compact subsets of the sq
uare [0\,1]^2. Questions and results are closely linked to the study of ex
changeable hierarchies\, see [2]. \n\n[1] Evans\, Grübel and Wakolbinger.
"Doob-Martin boundary of Rémy's tree growth chain". The Annals of Probab
ility\, 2017.\n[2] Forman\, Haulk and Pitman. "A representation of exchang
eable hierarchies by sampling from random real trees". Prob.Theory and rel
ated Fields\, 2017.\n[3] Gerstenberg. "Exchangeable interval hypergraphs a
nd limits of ordered discrete structures". arXiv\, 2018.\nSpeakers:\nJulia
n Gerstenberg (Leibniz Universität Hannover)
LOCATION:Mathematical Institute (L4)\, Woodstock Road OX2 6GG
TZID:Europe/London
URL:https://talks.ox.ac.uk/talks/id/16131e1b-de82-4227-ad4a-ab46c536a9a0/
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DESCRIPTION:Talk:Limits of (randomly) growing Schröder trees and exchange
ability - Julian Gerstenberg (Leibniz Universität Hannover)
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