Sequences and Limits

Limits of permutation sequences

A permutation sequence (σn)n∈N is said to be convergent if, for every fixed permutation τ , the density of occurrences of τ in the elements of the sequence converges. We prove that such a convergent sequence has a natural limit object, namely a Lebesgue measurable function Z : [0, 1] → [0, 1] with the additional properties that, for every fixed x ∈ [0, 1], the restriction Z(x, ·) is a cumulativ...

Limits of dense graph sequences

We show that if a sequence of dense graphs Gn has the property that for every fixed graph F , the density of copies of F in Gn tends to a limit, then there is a natural “limit object”, namely a symmetric measurable function W : [0, 1] → [0, 1]. This limit object determines all the limits of subgraph densities. Conversely, every such function arises as a limit object. We also characterize graph ...

Sequences of Knots and Their Limits

Hyperfinite knots, or limits of equivalence classes of knots induced by a knot invariant taking values in a metric space, were introduced in a previous article by the author. In this article, we present new examples of hyperfinite knots stemming from sequences of torus knots.

Generalized Covering Groups and Direct Limits

Limits of randomly grown graph sequences

3 Convergent graph sequences and their limits 8 3.1 Growing uniform attachment graphs . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Growing ranked attachment graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Growing prefix attachment graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 Preferential attachment graph on n fixed nodes . . . . . . . . . . . . ....