Limiting shapes of birth-and-death processes on young diagrams

We consider a family of birth processes and birth-and-death processes on Youngdiagrams of integer partitions of n. This family incorporates three famous modelsfrom very dierent elds: Rost's totally asymmetric particle model (in discretetime), Simon's urban growth model, and Moran's innite alleles model.We study stationary distributions and limit shapes as n tends to innity, andpresent a number of results and conjectures.Key words: birth process, birth-and-death process, limit shape, Youngdiagram, random growth model2000 MSC: 05A17, 60G5