  The entropic barrier, studied by Bubeck and Eldan (Proc. Mach. Learn.
Research, 2015), is a self-concordant barrier with asymptotically optimal
self-concordance parameter. In this paper, we study the tropicalization of the
central path associated with the entropic barrier, i.e., the logarithmic limit
of this central path for a parametric family of linear programs defined over
the field of Puiseux series. Our main result is that the tropicalization of the
entropic central path is a piecewise linear curve which coincides with the
tropicalization of the logarithmic central path studied by Allamigeon et al.
(SIAM J. Applied Alg. Geom., 2018). One consequence is that the number of
linear pieces in the tropical entropic central path can be exponential in the
dimension and the number of inequalities defining the linear program.
