  The cycle current is a crucial quantity in stochastic thermodynamics. The
absolute and net cycle currents of a Markovian system can be defined in the
loop-erased (LE) or spanning tree (ST) manner. Here we make a comparative study
between the large deviations and fluctuation theorems for LE and ST currents,
i.e. cycle currents defined in the LE and ST manners. First, we derive the
exact joint distribution and large deviation rate function for the LE currents
of a system with a cyclic topology and also obtain the exact rate function for
the ST currents of a general system. The relationship between the rate
functions for LE and ST currents is clarified and the analytical results are
applied to examine the fluctuations in the product rate of a three-step
reversible enzyme reaction. Furthermore, we examine various types of
fluctuation theorems satisfied by LE and ST currents and clarify their ranges
of applicability. We show that both the absolute and net LE currents satisfy
the strong form of all types of fluctuation theorems. In contrast, the absolute
ST currents do not satisfy fluctuation theorems, while the net ST currents only
satisfy the weak form of fluctuation theorems under the periodic boundary
condition.
