Christina Goldschmidt

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We imagine a coalescent process as modelling the genealogy of a sample from a population which is subject to neutral mutation. We work under the assumptions of the infinitely many alleles model so that, in particular, every mutation gives rise to a completely new type in the population. Mutations occur as a Poisson process of rate ρ along the branches of(More)
We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n+ λn−4/3, for some fixed λ ∈ R. Then, as a metric space with the graph distance rescaled by n−1/3, the sequence of connected components G(n, p) converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and(More)
We consider the Erdős–Rényi random graph G(n, p) inside the critical window, where p = 1/n + λn−4/3 for some λ ∈ R. We proved in [1] that considering the connected components of G(n, p) as a sequence of metric spaces with the graph distance rescaled by n−1/3 and letting n→∞ yields a non-trivial sequence of limit metric spaces C = (C1,C2, . . . ). These(More)
We consider a model for random hypergraphs with identifiability, an analogue of connectedness. This model has a phase transition in the proportion of identifiable vertices when the underlying random graph becomes critical. The phase transition takes various forms, depending on the values of the parameters controlling the different types of hyperedges. It(More)
The stable fragmentation with index of self-similarity α ∈ [−1/2, 0) is derived by looking at the masses of the subtrees formed by discarding the parts of a (1 + α)–stable continuum random tree below height t, for t ≥ 0. We give a detailed limiting description of the distribution of such a fragmentation, (F (t), t ≥ 0), as it approaches its time of(More)
These notes give a mathematical introduction to two seemingly unrelated topics: (i) quantum spin systems and their cycle and loop representations, due to Tóth and Aizenman-Nachtergaele; (ii) coagulation-fragmentation stochastic processes. These topics are nonetheless related, as we argue that the lengths of cycles and loops satisfy an effective(More)