Character Theory of Symmetric Groups, Analysis of Long Relators, and Random Walks

Character Theory of Symmetric Groups, Subgroup Growth of Fuchsian Groups, and Random Walks

Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks

Fuchsian groups (acting as isometries of the hyperbolic plane) occur naturally in geometry, combinatorial group theory, and other contexts. We use character-theoretic and probabilistic methods to study the spaces of homomorphisms from Fuchsian groups to symmetric groups. We obtain a wide variety of applications, ranging from counting branched coverings of Riemann surfaces, to subgroup growth an...

A PRELUDE TO THE THEORY OF RANDOM WALKS IN RANDOM ENVIRONMENTS

A random walk on a lattice is one of the most fundamental models in probability theory. When the random walk is inhomogenous and its inhomogeniety comes from an ergodic stationary process, the walk is called a random walk in a random environment (RWRE). The basic questions such as the law of large numbers (LLN), the central limit theorem (CLT), and the large deviation principle (LDP) are ...

Characters of Symmetric Groups: Sharp Bounds and Applications

We provide new estimates on character values of symmetric groups which hold for all characters and which are in some sense best possible. It follows from our general bound that if a permutation σ ∈ Sn has at most n cycles of length < m, then |χ(σ)| ≤ χ(1) for all irreducible characters χ of Sn. This is a far reaching generalization of a result of Fomin and Lulov. We then use our various charact...

Random Walks on Dicyclic Group

In this paper I work out the rate of convergence of a non-symmetric random walk on the dicyclic group (Dicn) and that of its symmetric analogue on the same group. The analysis is via group representation techniques from Diaconis (1988) and closely follows the analysis of random walk on cyclic group in Chapter 3C of Diaconis. I find that while the mixing times (the time for the random walk to ge...