We consider the cluster algebra associated to the Q-system for Ar as a
tool for relating Q-system solutions to all possible sets of initial data. We
show that the conserved quantities of the Q-system are partition functions
for hard particles on particular target graphs with weights, which are
determined by the choice of initial data. This allows us to interpret the
simplest solutions of the Q-system as generating functions for Viennot's heaps
on these target graphs, and equivalently as generating functions of weighted
paths on suitable dual target graphs. The generating functions take the form of
finite continued fractions. In this setting, the cluster mutations correspond
to local rearrangements of the fractions which leave their final value
unchanged. Finally, the general solutions of the Q-system are interpreted as
partition functions for strongly non-intersecting families of lattice paths on
target lattices. This expresses all cluster variables as manifestly positive
Laurent polynomials of any initial data, thus proving the cluster positivity
conjecture for the ArQ-system. We also give an alternative formulation in
terms of domino tilings of deformed Aztec diamonds with defects.Comment: 106 pages, 38 figure