  The influence of nonmagnetic doping on the thermodynamic properties of
two-leg S = 1/2 spin ladders is studied in this paper. It is shown that, for a
weak interchain coupling, the problem can be mapped onto a model of random mass
Dirac (Majorana) fermions. We investigate in detail the structure of the
fermionic states localized at an individual mass kink (zero modes) in the
framework of a generalized Dirac model. The low-temperature thermodynamic
properties are dominated by these zero modes. We use the single-fermion density
of states, known to exhibit the Dyson singularity in the zero-energy limit, to
construct the thermodynamics of the spin ladder. In particular, we find that
the magnetic susceptibility $\chi$ diverges at $T\to 0$ as $1/T\ln^2(1/T)$, and
the specific heat behaves as $C\propto 1/\ln^3(1/T)$. The predictions on
magnetic susceptibility are consistent with the most recent results of quantum
Monte Carlo simulations on doped ladders with randomly distributed impurities.
We also calculate the average staggered magnetic susceptibility induced in the
system by such defects.
