This talk is about a modification of the moment method applied to extract limiting distributions of the first, second, and so on rows of randomly evolving partitions. We will proceed slowly, first by describing the usual moment method together with a modification using orthogonal polynomials. Here, the central limit theorem and Wigner's semicircle law will be presented as simple examples of how the passage to the modified moments works out.
We then give a brief survey of recent developments in random matrix theory regarding universality in the fluctuations of extreme eigenvalues of random matrix ensembles.
After that, we describe the stochastic dynamical systems on the set of partitions, which is natural with respect to the Plancherel measure. Our main result shows that the limiting dynamics of the first, second, and so on rows is given by the Airy line ensemble. Our proof highlights the similarity between random partitions and random matrices.
We plan to discuss some further directions. This is based on a joint work with Sasha Sodin.