Asymptotic representation theory of symmetric groups

We will discuss certain problems of asymptotic representation theory. We will show how local and global statistical properties of Young diagrams with respect to certain probability measures can be used to obtain logarithmic, order-sharp bounds for the dimensions of certain representations of finite symmetric groups. In particular we will discuss the asymptotics of the dimensions of the isotypic components of the tensor representations of the symmetric groups with respect to the Schur-Weyl measures, which are essentially a one parameter deformation of the family of Plancherel measures on Young diagrams.