Asymptotics for skew standard Young tableaux via bounds for characters

We are interested in the asymptotics of the number of standard Young tableaux $f^{\lambda \mathrm{/}\mu }$ of a given skew shape $\lambda \mathrm{/}\mu$. We mainly restrict ourselves to the case where both diagrams are balanced, but investigate all growth regimes of $\mathrm{\mid }\mu \mathrm{\mid }$ compared to $\mathrm{\mid }\lambda \mathrm{\mid }$, from $\mathrm{\mid }\mu \mathrm{\mid }$ fixed to $\mathrm{\mid }\mu \mathrm{\mid }$ of order $\mathrm{\mid }\lambda \mathrm{\mid }$. When $\mathrm{\mid }\mu \mathrm{\mid }=o(\mathrm{\mid }\lambda {\mathrm{\mid }}^{1\mathrm{/}3})$, we get an asymptotic expansion to any order. When $\mathrm{\mid }\mu \mathrm{\mid }=o(\mathrm{\mid }\lambda {\mathrm{\mid }}^{1\mathrm{/}2})$, we get a sharp upper bound. For bigger $\mathrm{\mid }\mu \mathrm{\mid }$, we prove a weaker bound and give a conjecture on what we believe to be the correct order of magnitude. Our results are obtained by expressing $f^{\lambda \mathrm{/}\mu }$ in terms of irreducible character values of the symmetric group and applying known upper bounds on characters.