Johan Tykesson

Connectedness of Poisson cylinders in Euclidean space

We consider the Poisson cylinder model in R-d, d >= 3. We show that given any two cylinders c(1) and c(2) in the process, there is a sequence of at most d - 2 other cylinders creating a connection between c(1) and c(2). In particular, this shows that the union of the cylinders is a connected set, answering a question appearing in (Probab. Theory Related Fields 154 (2012) 165-191). We also show that there are cylinders in the process that are not connected by a sequence of at most d - 3 other cylinders. Thus, the diameter of the cluster of cylinders equals d - 2.

Differential Item Functioning in the National Tests in Religious Education in Sweden

The Mantel-Haenszel method is used to investigate whether there are items in the national tests in religious education from 2013 exhibiting differential item functioning (DIF) between groups of students. DIF in an item means that the item functions differently between two groups, after adjusting for the two groups’ overall abilities. Two comparisons are made: between boys and girls and between native speakers and pupils with Swedish as their second language. The results of the analysis lead, for example, to the speculation that closed format items exhibiting DIF are more likely to favour boys than girls and the reverse speculation holds for items of open format. Having data from only two tests, these speculations need to be investigated further with data from later tests.

Poisson cylinders in hyperbolic space

We consider the Poisson cylinder model in d-dimensional hyperbolic space. We show that in contrast to the Euclidean case, there is a phase transition in the connectivity of the collection of cylinders as the intensity parameter varies. We also show that for any non-trivial intensity, the diameter of the collection of cylinders is infinite.

Asymptotics of visibility in the hyperbolic plane

At each point of a Poisson point process of intensity λ in the hyperbolic plane, center a ball of bounded random radius. Consider the probability P r that, from a fixed point, there is some direction in which one can reach distance r without hitting any ball. It is known (see Benjamini, Jonasson, Schramm and Tykesson (2009)) that if λ is strictly smaller than a critical intensity λ gv thenP r does not go to 0 as r → ∞. The main result in this note shows that in the case λ=λ gv, the probability of reaching a distance larger than r decays essentially polynomially, while if λ>λ gv, the decay is exponential. We also extend these results to various related models and we finally obtain asymptotic results in several situations.