  We consider the simple random walk on the Euclidean lattice, in three
dimensions and higher, conditioned to visit fewer sites than expected, when the
deviation from the mean scales like the mean. The associated large deviation
principle was first derived in 2001 by van den Berg, Bolthausen and den
Hollander in the continuous setting, that is for the volume of a Wiener
sausage, and later taken up by Phetpradap in the discrete setting. One of the
key ideas in their work is to condition the range of the random walk to a
certain skeleton, that is a sub-sequence of the random walk path taken along an
appropriate mesoscopic scale. In this paper we prove that (i) the rate function
obtained by van den Berg, Bolthausen and den Hollander has a unique minimizer
over the set of probability measures modulo shifts, at least for deviations of
the range well below the mean, and (ii) the empirical measure of the skeleton
converges under the conditioned law, in a certain manner, to this minimizer. To
this end we use an adaptation of the topology recently introduced by Mukherjee
and Varadhan to compactify the space of probability measures.
