Big symmetric groups, and related problems in Analysis, Probability and Combinatorics

We discuss the asymptotics of functions on the symmetric groups $S_n$ and their duals, in the limit of large degree $n\to\infty$. In particular, we deal with:
- maximal dimensions of irreducible representations;
- the limiting measure for uniform distributions on $S_n$;
- the asymptotics of Plancherel measures on the dual sets $\widehat{S}_n$ of irreducible representations of $S_n$;
- approximations of characters of the infinite symmetric group $S_\infty$.
Combinatorial problems related to big permutations frequently exhibit unexpected connections with analysis. In this direction, we mention the Martin boundary construction for abstract graphs, the asymptotics of interlacing roots of orthogonal polynomials, the typical spectra of big hermitian matrices.