  In this work we develop a new theory to analyse the process of routing in
large-scale ad-hoc wireless networks. We use a path integral formulation to
examine the properties of the paths generated by different routing strategies
in these kinds of networks. Using this theoretical framework, we calculate the
statistical distribution of the distances between any source to any destination
in the network, hence we are able to deduce a length parameter that is unique
for each routing strategy. This parameter, defined as the {\it effective
radius}, effectively encodes the routing information required by a node.
Analysing the aforementioned statistical distribution for different routing
strategies, we obtain a threefold result for practical Large-Scale Wireless
Ad-Hoc Networks: 1) We obtain the distribution of the lengths of all the paths
in a network for any given routing strategy, 2) We are able to identify "good"
routing strategies depending on the evolution of its effective radius as the
number of nodes, $N$, increases to infinity, 3) For any routing strategy with
finite effective radius, we demonstrate that, in a large-scale network, is
equivalent to a random routing strategy and that its transport capacity scales
as $\Theta(\sqrt{N})$ bit-meters per second, thus retrieving the scaling law
that Gupta and Kumar (2000) obtained as the limit for single-route large-scale
wireless networks.
