  In this work we are concerned with the crossover between strong disorder (SD)
and weak disorder (WD) behaviors in three well-known problems that involve
minimal paths: directed polymers (directed paths with fixed starting point and
length), optimal paths (undirected paths with fixed end-to-end/spanning
distance) and undirected polymers (undirected paths with fixed starting point
and length). We present a unified theoretical framework from which we can
easily deduce the scaling of the crossover point of each problem. Our theory is
based on the fact that the SD limit behavior of these systems is closely
related to the corresponding percolation problem. As a result, the properties
of those minimal paths are completely controlled by the so-called red bonds of
percolation theory. Our model is first addressed numerically and then
approximated by a ``two-term'' approach. This approach provides us with an
analytical expression that seems to be reasonably accurate. The results are in
perfect agreement with our simulations and with most of the results reported in
related works. Our research also lead us to propose this crossover point as a
universal measure of the disorder strength in each case. Interestingly, that
measure depends on both the statistical properties of the disorder and the
topological properties of the network.
