  We construct a minimal two-chain random walk model and study the information
that fluctuations of the flux and higher cumulants can reveal about the model:
its structure, parameters, and whether it operates under nonequilibrium
conditions. The two coupled chains allow for both horizontal and cyclic
transport. We capture these processes by deriving the cumulant generating
function of the system, which characterizes both horizontal and cyclic
transport in the long time limit. First, we show that either the horizontal or
the cyclic currents, along with their higher-order cumulants, can be used to
unravel the intrinsic structure and parameters of the model. Second, we
investigate the "zero current" situation, in which the {\it horizontal} current
vanishes. We find that fluctuations of the horizontal current reveal the
nonequilibrium condition at intermediate bias, while the cyclic current remains
nonzero throughout. We also show that in nonequilibrium scenarios close to the
zero {\it horizontal} current limit, the entropy production rate is more
tightly lower-bounded by the relative noise of the {\it cyclic} current, and
vice versa. Finally, simulations of transport before the steady state sets in
allow for the extraction of the interchain hopping rate. Our study,
illustrating the information concealed in fluctuations, could see applications
in chemical networks, cellular processes, and charge and energy transport
materials.
