Time correlations in KPZ models with diffusive initial conditions

Temporal correlation for randomly growing interfaces in the KPZ universality class is a topic of recent interest. Following the conjectures in (SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016) Paper No. 074, 23), most of the works so far have been concentrated on the 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. While these generalize the stationary initial conditions, the transition from stationary to diffusive initial conditions fundamentally disrupts the integrability of the model, necessitating the development of new analytical techniques. We formulate our results in terms of the positive temperature exactly solvable model of the inverse-gamma polymer and obtain upper and lower bounds, up to a constant, for the correlation between the free energy of two polymers whose endpoints are close together or far apart. Although the exponents governing the time correlation decay that we obtain are the same as in the droplet initial condition recently studied in (Comm. Math. Phys. 405 (2024) Paper No. 163, 72), the mechanism producing the correlations is different when the endpoints of the polymers are far apart in time. We obtain new results that have not been covered previously in either zero- or positive-temperature KPZ models, complementing the results obtained in (Math. Phys. Anal. Geom. 22 (2019) Paper No. 1, 33), where the exponent was only obtained in the case of close endpoints and for a narrower class of initial conditions in the zero temperature exponential LPP model. 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.