Time correlations in the inverse-gamma polymer with flat initial condition

Temporal correlations in the KPZ universality class have gained significant attention, following the conjectures in [Ferr-Spoh'16]. Building on prior work in the zero temperature setting [Basu-Gang-Zhan'21], we address the time correlation problem with flat initial conditions in the positive temperature regime. Our study focuses on the inverse-gamma polymer, where we establish an upper bound for the correlation between two free energies whose endpoints are far apart in time. In contrast to the previous work [Basu-Gang-Zhan'21], our work not only extends the result to positive temperatures but also eliminates the reliance on integrable probability inputs related to the Airy process. This advancement allows us to address local scales, where the short time remains fixed while the large time grows arbitrarily, a scenario beyond the reach of the Airy scaling limit.