59 research outputs found
Poset limits can be totally ordered
S. Janson [Poset limits and exchangeable random posets, Combinatorica 31 (2011), 529-563] defined limits of finite posets in parallel to the emerging theory of limits of dense graphs.
We prove that each poset limit can be represented as a kernel on the unit interval with the standard order, thus answering an open question of Janson. We provide two proofs: real-analytic and combinatorial. The combinatorial proof is based on a Szemerédi-type Regularity Lemma for posets which may be of independent interest.
Also, as a by-product of the analytic proof, we show that every atomless ordered probability space admits a measure-preserving and almost order-preserving map to the unit interval
Limits of interval orders and semiorders
We study poset limits given by sequences of finite interval orders or, as a
special case, finite semiorders. In the interval order case, we show that every
such limit can be represented by a probability measure on the space of closed
subintervals of [0,1], and we define a subset of such measures that yield a
unique representation. In the semiorder case, we similarly find unique
representations by a class of distribution functions.Comment: 18 page
A note on limits of sequences of binary trees
We discuss a notion of convergence for binary trees that is based on subtree
sizes. In analogy to recent developments in the theory of graphs, posets and
permutations we investigate some general aspects of the topology, such as a
characterization of the set of possible limits and its structure as a metric
space. For random trees the subtree size topology arises in the context of
algorithms for searching and sorting when applied to random input, resulting in
a sequence of nested trees. For these we obtain a structural result based on a
local version of exchangeability. This in turn leads to a central limit
theorem, with possibly mixed asymptotic normality
Limits of Structures and the Example of Tree-Semilattices
The notion of left convergent sequences of graphs introduced by Lov\' asz et
al. (in relation with homomorphism densities for fixed patterns and
Szemer\'edi's regularity lemma) got increasingly studied over the past
years. Recently, Ne\v set\v ril and Ossona de Mendez introduced a general
framework for convergence of sequences of structures. In particular, the
authors introduced the notion of -convergence, which is a natural
generalization of left-convergence. In this paper, we initiate study of
-convergence for structures with functional symbols by focusing on the
particular case of tree semi-lattices. We fully characterize the limit objects
and give an application to the study of left convergence of -partite
cographs, a generalization of cographs
Record-dependent measures on the symmetric groups
A probability measure P[subscript n] on the symmetric group S[subscript n] is said to be record-dependent if P[subscript n]( σ ) depends only on the set of records of a permutation σ ∈ S[subscript n]. A sequence P = ( P n )[subscript n ∈ N] of consistent record-dependent measures determines a random order on N. In this paper we describe the extreme elements of the convex set of such P. This problem turns out to be related to the study of asymptotic behavior of permutation-valued growth processes, to random extensions of partial orders, and to the measures on the Young-Fibonacci lattice
- …