An asymptotic formula for the maximum size of an h-family in products of partially ordered sets

AbstractAn h-family of a partially ordered set P is a subset of P such that no h + 1 elements of the h-family lie on any single chain. Let S1, S2,… be a sequence of partially ordered sets which are not antichains and have cardinality less than a given finite value. Let Pn be the direct product of S1,…, Sn. An asymptotic formula of the maximum size of an h-family in Pn is given, where h=o(n) and n → ∞

Weighted dependency graphs

The theory of dependency graphs is a powerful toolbox to prove asymptotic normality of sums of random variables. In this article, we introduce a more general notion of weighted dependency graphs and give normality criteria in this context. We also provide generic tools to prove that some weighted graph is a weighted dependency graph for a given family of random variables. To illustrate the power of the theory, we give applications to the following objects: uniform random pair partitions, the random graph model $G(n,M)$, uniform random permutations, the symmetric simple exclusion process and multilinear statistics on Markov chains. The application to random permutations gives a bivariate extension of a functional central limit theorem of Janson and Barbour. On Markov chains, we answer positively an open question of Bourdon and Vall\'ee on the asymptotic normality of subword counts in random texts generated by a Markovian source.Comment: 57 pages. Third version: minor modifications, after review proces

Coherent random permutations with record statistics

Random permutations with distribution conditionally uniform given the set of record values can be generated in a unified way, coherently for all values of $n$. Our central example is a two-parameter family of random permutations that are conditionally uniform given the counts of upper and lower records. This family interpolates between two versions of Ewens' distribution. We discuss characterisations of the conditionally uniform permutations, their asymptotic properties, constructions and relations to random partitions.Comment: 17 page

Towards a Definition of Locality in a Manifoldlike Causal Set

It is a common misconception that spacetime discreteness necessarily implies a violation of local Lorentz invariance. In fact, in the causal set approach to quantum gravity, Lorentz invariance follows from the specific implementation of the discreteness hypothesis. However, this comes at the cost of locality. In particular, it is difficult to define a "local" region in a manifoldlike causal set, i.e., one that corresponds to an approximately flat spacetime region. Following up on suggestions from previous work, we bridge this lacuna by proposing a definition of locality based on the abundance of m-element order-intervals as a function of m in a causal set. We obtain analytic expressions for the expectation value of this function for an ensemble of causal set that faithfully embeds into an Alexandrov interval in d-dimensional Minkowski spacetime and use it to define local regions in a manifoldlike causal set. We use this to argue that evidence of local regions is a necessary condition for manifoldlikeness in a causal set. This in addition provides a new continuum dimension estimator. We perform extensive simulations which support our claims.Comment: 35 pages, 17 figure