  We introduce the multiplicative coalescent with linear deletion, a
continuous-time Markov process describing the evolution of a collection of
blocks. Any two blocks of sizes $x$ and $y$ merge at rate $xy$, and any block
of size $x$ is deleted with rate $\lambda x$ (where $\lambda\geq 0$ is a fixed
parameter). This process arises for example in connection with a variety of
random-graph models which exhibit self-organised criticality. We focus on
results describing states of the process in terms of collections of excursion
lengths of random functions. For the case $\lambda=0$ (the coalescent without
deletion) we revisit and generalise previous works by authors including Aldous,
Limic, Armendariz, Uribe Bravo, and Broutin and Marckert, in which the
coalescence is related to a "tilt" of a random function, which increases with
time; for $\lambda>0$ we find a novel representation in which this tilt is
complemented by a "shift" mechanism which produces the deletion of blocks. We
describe and illustrate other representations which, like the tilt-and-shift
representation, are "rigid", in the sense that the coalescent process is
constructed as a projection of some process which has all of its randomness in
its initial state. We explain some applications of these constructions to
models including mean-field forest-fire and frozen-percolation processes.
