  Considering the large deviations of activity and current in the Asymmetric
Simple Exclusion Process (ASEP), we show that there exists a non-trivial
correspondence between the joint scaled cumulant generating functions of
activity and current of two ASEPs with different parameters. This mapping is
obtained by applying a similarity transform on the deformed Markov matrix of
the source model in order to obtain the deformed Markov matrix of the target
model. We first derive this correspondence for periodic boundary conditions,
and show in the diffusive scaling limit (corresponding to the Weakly Asymmetric
Simple Exclusion Processes, or WASEP) how the mapping is expressed in the
language of Macroscopic Fluctuation Theory (MFT). As an interesting specific
case, we map the large deviations of current in the ASEP to the large
deviations of activity in the SSEP, thereby uncovering a regime of
Kardar-Parisi-Zhang in the distribution of activity in the SSEP. At large
activity, particle configurations exhibit hyperuniformity [Jack et al., PRL 114
060601 (2015)]. Using results from quantum spin chain theory, we characterize
the hyperuniform regime by evaluating the small wavenumber asymptotic behavior
of the structure factor at half-filling. Conversely, we formulate from the MFT
results a conjecture for a correlation function in spin chains at any fixed
total magnetization (in the thermodynamic limit). In addition, we generalize
the mapping to the case of two open ASEPs with boundary reservoirs, and we
apply it in the WASEP limit in the MFT formalism. This mapping also allows us
to find a symmetry-breaking dynamical phase transition (DPT) in the WASEP
conditioned by activity, from the prior knowledge of a DPT in the WASEP
conditioned by the current.
