  We consider an Erdos-Renyi random graph consisting of N vertices connected by
randomly and independently drawing an edge between every pair of them with
probability c/N so that at N->infinity one obtains a graph of finite mean
degree c. In this regime, we study the distribution of resistance distances
between the vertices of this graph and develop an auxiliary field
representation for this quantity in the spirit of statistical field theory.
Using this representation, a saddle point evaluation of the resistance distance
distribution is possible at N->infinity in terms of an 1/c expansion. The
leading order of this expansion captures the results of numerical simulations
very well down to rather small values of c; for example, it recovers the
empirical distribution at c=4 or 6 with an overlap of around 90%. At large
values of c, the distribution tends to a Gaussian of mean 2/c and standard
deviation sqrt{2/c^3}. At small values of c, the distribution is skewed toward
larger values, as captured by our saddle point analysis, and many fine features
appear in addition to the main peak, including subleading peaks that can be
traced back to resistance distances between vertices of specific low degrees
and the rest of the graph. We develop a more refined saddle point scheme that
extracts the corresponding degree-differentiated resistance distance
distributions. We then use this approach to recover analytically the most
apparent of the subleading peaks that originates from vertices of degree 1.
Rather intuitively, this subleading peak turns out to be a copy of the main
peak, shifted by one unit of resistance distance and scaled down by the
probability for a vertex to have degree 1. We comment on a possible lack of
smoothness in the true N->infinity distribution suggested by the numerics.
