  In this article, we study the large $n$ asymptotic expansions of $n\times n$
Toeplitz determinants whose symbols are indicator functions of unions of
arc-intervals of the unit circle. In particular, we use an Hermitian matrix
model reformulation of the problem to provide a rigorous derivation of the
general form of the large $n$ expansion when the symbol is an indicator
function of either a single arc-interval or several arc-intervals with a
discrete rotational symmetry. Moreover, we prove that the coefficients in the
expansions can be reconstructed, up to some constants, from the Eynard-Orantin
topological recursion applied to some explicit spectral curves. In addition,
when the symbol is an indicator function of a single arc-interval, we provide
the corresponding normalizing constants using a Selberg integral and illustrate
the theoretical results with numeric simulations up to order
$o\left(\frac{1}{n^4}\right)$. We also briefly discuss the situation when the
number of arc-intervals increases with $n$, as well as more general Toeplitz
determinants to which we may apply the present strategy.
