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

Alain Comtet ^1, 2, Satya N. Majumdar ¹, Stephane Ouvry ¹, Sanjib Sabhapandit ¹

Journal of statistical mechanics-theory and experiment (2007) P10001

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.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
  CNRS : UMR8626 – Université Paris XI – Paris Sud
• 2. Unite mixte de service de l’institut Henri Poincaré (UMSIHP),
  CNRS : UMS839 – Université Paris VI – Pierre et Marie Curie