  We consider a one-dimensional simple random walk surviving among a field of
static soft traps : each time it meets a trap the walk is killed with
probability 1--e --$\beta$ , where $\beta$ is a positive and fixed parameter.
The positions of the traps are sampled independently from the walk and
according to a renewal process. The increments between consecutive traps, or
gaps, are assumed to have a power-law decaying tail with exponent $\gamma$ > 0.
We prove convergence in law for the properly rescaled logarithm of the quenched
survival probability as time goes to infinity. The normalization exponent is
$\gamma$/($\gamma$ + 2), while the limiting law writes as a variational formula
with both universal and non-universal features. The latter involves (i) a
Poisson point process that emerges as the universal scaling limit of the
properly rescaled gaps and (ii) a function of the parameter $\beta$ that we
call asymptotic cost of crossing per trap and that may, in principle, depend on
the details of the gap distribution. Our proof suggests a confinement strategy
of the walk in a single large gap. This model may also be seen as a (1 +
1)-directed polymer among many repulsive interfaces, in which case $\beta$
corresponds to the strength of repulsion, the survival probability to the
partition function and its logarithm to the finite-volume free energy. Along
the way we prove a stochastic monotonicity property for the hitting time of the
killed random walk with respect to the non-killed one, that could be of
interest in other contexts, see Proposition 3.5.
