Christina Goldschmidt

The scaling limit of the minimum spanning tree of the complete graph

Assign i.i.d. standard exponential edge weights to the edges of the complete graph K_n, and let M_n be the resulting minimum spanning tree. We show that M_n converges in the local weak sense (also called Aldous-Steele or Benjamini-Schramm convergence), to a random infinite tree M. The tree M may be viewed as the component containing the root in the wired minimum spanning forest of the Poisson-weighted infinite tree (PWIT). We describe a Markov process construction of M starting from the invasion percolation cluster on the PWIT. We then show that M has cubic volume growth, up to lower order fluctuations for which we provide explicit bounds. Our volume growth estimates confirm recent predictions from the physics literature, and contrast with the behaviour of invasion percolation on the PWIT and on regular trees, which exhibit quadratic volume growth.

Coagulation–fragmentation duality, Poisson–Dirichlet distributions and random recursive trees

In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the two-parameter family of Poisson--Dirichlet distributions

Preservation of log-concavity on summation

\operatorname {PD}(\alpha,\theta)

Dual Random Fragmentation and Coagulation and an Application to the Genealogy of Yule Processes

that take values in this space. We introduce families of random fragmentation and coagulation operators

Critical random hypergraphs: The emergence of a giant set of identifiable vertices

\mathrm {Frag}_{\alpha}

Behavior near the extinction time in self-similar fragmentations I: The stable case

and

Random Recursive Trees and the Bolthausen-Sznitman Coalesent

\mathrm {Coag}_{\alpha,\theta}

The continuum limit of critical random graphs

, respectively, with the following property: if the input to

Quantum Heisenberg models and their probabilistic representations

\mathrm {Frag}_{\alpha}

Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent

has