  We prove a connection between the Green's function of the fractional Anderson
model and the two point function of a self-avoiding random walk with long range
jumps, adapting a strategy proposed by Schenker in 2015. This connection allows
us to exploit results from the theory of self-avoiding random walks to improve
previous bounds known for the fractional Anderson model at strong disorder. In
particular, we enlarge the range of the disorder parameter where spectral
localization occurs. Moreover we prove that the decay of Green's function at
strong disorder for any $0<\alpha <1$ is arbitrarily close to the decay of the
massive resolvent of the corresponding fractional Laplacian, in agreement with
the case of the standard Anderson model $\alpha =1$. We also derive upper and
lower bounds for the resolvent of the discrete fractional Laplacian with
arbitrary mass $m\geq 0,$ that are of independent interest.
