  Based on a relationship with continuous-time random walks discovered by
Igl\'oi, Turban, and Rieger [Phys. Rev. E {\bf 59}, 1465 (1999)], we derive
exact lower and upper bounds on the lowest energy gap of open transverse-field
Ising chains, which are explicit in the parameters and are generally valid for
arbitrary sets of possibly random couplings and fields. In the homogeneous
chain and in the random chain with uncorrelated parameters, both the lower and
upper bounds are found to show the same finite-size scaling in the
ferromagnetic phase and at the critical point, demonstrating the ability of
these bounds to infer the correct finite-size scaling of the critical gap.
Applying the bounds to random transverse-field Ising chains with coupling-field
correlations, a model which is relevant for adiabatic quantum computing, the
finite-size scaling of the gap is shown to be related to that of sums of
independent random variables. We determine the critical dynamical exponent of
the model and reveal the existence of logarithmic corrections at special
points.
