  In order to study the invariant measures of discrete KdV- and Toda-type
systems, this article focusses on models, discretely indexed in space and time,
whose dynamics are deterministic and defined locally via lattice equations. A
detailed balance criterion is presented that, amongst the measures that
describe spatially independent and identically/alternately distributed
configurations, characterizes those that are temporally invariant in
distribution. A condition for establishing ergodicity of the dynamics is also
given. These results are applied to various examples of discrete integrable
systems, namely the ultra-discrete and discrete KdV equations, for which it is
shown that the relevant invariant measures are of exponential/geometric and
generalized inverse Gaussian form, respectively, as well as the ultra-discrete
and discrete Toda lattice equations, for which the relevant invariant measures
are found to be of exponential/geometric and gamma form. Ergodicity is
demonstrated in the case of the KdV-type models. Links between the invariant
measures of the different systems are presented, as are connections with
stochastic integrable models and iterated random functions. Furthermore, a
number of conjectures concerning the characterization of standard distributions
are posed.
