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Bordered and framed Toeplitz/Hankel determinants have the same structure asToeplitz/Hankel determinants except in small number of matrix rows and/orcolumns. We review these structured determinants and their connections toorthogonal polynomials, collecting well-known and perhaps less well-knownresults. We present some applications for these structured determinants toensembles of non-intersecting paths and the six-vertex model, with an eyetowards asymptotic analysis. We also prove some asymptotic formulae for theprobability of non-intersection for an ensemble of continuous time random walksfor certain choices of starting and ending points as the number of randomwalkers tends to infinity.

Bordered and Framed Toeplitz and Hankel Determinants with Applications to Integrable Probability