Integer partitions and exclusion statistics: Limit shapes and the largest part of Young diagrams

We compute the limit shapes of the Young diagrams of the minimal difference $p$ partitions and provide a simple physical interpretation for the limit shapes. We also calculate the asymptotic distribution of the largest part of the Young diagram and show that the scaled distribution has a Gumbel form for all $p$. This Gumbel statistics for the largest part remains unchanged even for general partitions of the form $E=\sum_i n_i i^{1/\nu}$ with $\nu>0$ where $n_i$ is the number of times the part $i$ appears.