  Bordered and framed Toeplitz/Hankel determinants have the same structure as
Toeplitz/Hankel determinants except in small number of matrix rows and/or
columns. We review these structured determinants and their connections to
orthogonal polynomials, collecting well-known and perhaps less well-known
results. We present some applications for these structured determinants to
ensembles of non-intersecting paths and the six-vertex model, with an eye
towards asymptotic analysis. We also prove some asymptotic formulae for the
probability of non-intersection for an ensemble of continuous time random walks
for certain choices of starting and ending points as the number of random
walkers tends to infinity.
