  In critical systems, the effect of a localized perturbation affects points
that are arbitrarily far from the perturbation location. In this paper, we
study the effect of localized perturbations on the solution of the random dimer
problem in $2D$. By means of an accurate numerical analysis, we show that a
local perturbation of the optimal covering induces an excitation whose size is
extensive with finite probability. We compute the fractal dimension of the
excitations and scaling exponents. In particular, excitations in random dimer
problems on non-bipartite lattices have the same statistical properties of
domain walls in the $2D$ spin glass. Excitations produced in bipartite
lattices, instead, are compatible with a loop-erased self-avoiding random walk
process. In both cases, we find evidence of conformal invariance of the
excitations that is compatible with $\mathrm{SLE}_\kappa$ with parameter
$\kappa$ depending on the bipartiteness of the underlying lattice only.
