English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

It is well-known that the height profile of a critical conditionedGalton-Watson tree with finite offspring variance converges, after a suitablenormalization, to the local time of a standard Brownian excursion. In thiswork, we study the distance profile, defined as the profile of all distancesbetween pairs of vertices. We show that after a proper rescaling the distanceprofile converges to a continuous random function that can be described as thedensity of distances between random points in the Brownian continuum randomtree.  We show that this limiting function a.s. is H\"older continuous of any order$\alpha<1$, and that it is a.e. differentiable. We note that it cannot bedifferentiable at $0$, but leave as open questions whether it is Lipschitz, andwhether is continuously differentiable on the half-line $(0,\infty)$.  The distance profile is naturally defined also for unrooted trees contrary tothe height profile that is designed for rooted trees. This is used in ourproof, and we prove the corresponding convergence result for the distanceprofile of random unrooted simply generated trees. As a minor purpose of thepresent work, we also formalize the notion of unrooted simply generated treesand include some simple results relating them to rooted simply generated trees,which might be of independent interest.

The distance profile of rooted and unrooted simply generated trees