  Fluctuations around critical behavior of a holographic charged plasmas are
investigated by studying quasi-normal modes of the corresponding black branes
in 5D Einstein-Maxwell-Dilaton gravity. The near horizon geometry of black
branes approaches the well-known 2D charged string black hole in the critical
limit, for which a world-sheet description is available, and the corresponding
quasi-normal modes can be obtained analytically from the reflection amplitude
of the 2D black hole geometry. We find two distinct set of modes: a purely
imaginary ``decoupled'' set, directly following from the reflection amplitude,
and a ``non-decoupled'' set that was already identified in the neutral
holographic plasma in \cite{Betzios:2018kwn}. In the extremal limit, the former
set of imaginary quasi-normal modes coalesce on a branch cut starting from the
the origin, signaling breakdown of hydrodynamic approximation. We further
complete the black brane geometry with a slice of AdS near the boundary, to
allow for a holographic construction, and find another set of modes localized
in the UV. Finally, we develop an alternative WKB method to obtain the
quasi-normal modes in the critical limit and apply this method to study the
spectrum of hyperscaling-violating Lifshitz black branes. The critical limit of
the plasma we consider in this paper is in one-to-one correspondence with the
large D limit of Einstein's gravity which allows for an alternative interesting
interpretation of our findings.
