  We prove a large deviation principle for the sum of n independent
heavy-tailed random variables, which are subject to a moving cut-off boundary
at location n. Conditional on the sum being large at scale n, we show that a
finite number of summands take values near the cut-off boundary, while the
remaining variables still obey the law of large numbers. This generalises the
well-known single-big-jump principle for random variables without cut-off to a
situation where just the minimal necessary number of jumps occur. As an
application, we consider a random graph with vertex set given by the lattice
points of a torus with sidelength 2N+1. Every vertex is the centre of a ball
with random radius sampled from a heavy-tailed distribution. Oriented edges are
drawn from the central vertex to all other vertices in this ball. When this
graph is conditioned on having an exceptionally large number of edges we use
our main result to show that, as N goes to infinity, the excess outdegrees
condense in a fixed, finite number of randomly scattered vertices of
macroscopic outdegree. By contrast, no condensation occurs for the indegrees of
the vertices, which all remain microscopic in size.
