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We investigate a class of Young diagrams growing via the addition of unitcells and satisfying the constraint that the height difference between adjacentcolumns $\geq r$. In the long time limit, appropriately re-scaled Youngdiagrams approach a limit shape that we compute for each integer $r\geq 0$. Wealso determine limit shapes of `diffusively' growing Young diagrams satisfyingthe same constraint and evolving through the addition and removal of cells thatproceed with equal rates.

Stochastic Dynamics of Growing Young Diagrams and Their Limit Shapes