  Temporal correlation for randomly growing interfaces in the KPZ universality
class is a topic of recent interest. Most of the works so far have been
concentrated on the zero temperature model of exponential last passage
percolation, and three special initial conditions, namely droplet, flat and
stationary. We focus on studying the time correlation problem for generic
random initial conditions with diffusive growth. We formulate our results in
terms of the positive temperature exactly solvable model of the inverse-gamma
polymer and obtain up to constant upper and lower bounds for the correlation
between the free energy of two polymers whose endpoints are close together or
far apart. Our proofs apply almost verbatim to the zero temperature set-up of
exponential LPP and are valid for a broad class of initial conditions. Our work
complements and completes the partial results obtained in (Ferrari-Occelli'19),
following the conjectures of (Ferrari-Spohn'16). Moreover, our arguments rely
on the one-point moderate deviation estimates which have recently been obtained
using stationary polymer techniques and thus do not depend on complicated exact
formulae.
