

Lecture 3: What is the Universal Scaling Limit of Random Interface Growth, and What Does It Tell Us?
De Ivan Corwin


Coulomb gas approach to conformal field theory and lattice models of 2D statistical physics
De Stanislav Smirnov
De Henrik Hult
Apparaît dans la collection : Heavy Tails, Long-Range Dependence, and Beyond / Queues lourdes, dépendance de long terme et au-delà
The Kingman coalescent is a fundamental process in population genetics modelling the ancestry of a sample of individuals backwards in time. In this paper, weak convergence is proved for a sequence of Markov chains consisting of two components related to the Kingman coalescent, under a parent dependent d-alleles mutation scheme, as the sample size, grows to infinity. The first component is the normalised d-dimensional jump chain of the block counting processes of the Kingman coalescent. The second component is a d^2-dimensional process counting the number of mutations between types occurring in the Kingman coalescent. Time is scaled by the sample size. The limiting process consists of a deterministic d-dimensional component, describing the limit of the block counting jump chain, and d^2 independent Poisson processes with state-dependent intensities, exploding at the origin, describing the limit of the number of mutations. The weak convergence result is first proved, using a generator approach, in the setting of parent independent mutations. A change of measure argument is used to extend the weak convergence result to include parent dependent mutations.