Intermediate disorder limits for multi-layer semi-discrete directed polymers

Understand

We show that the partition function of the multi-layer semi-discrete directed polymer converges in the intermediate disorder regime to the partition function for the multi-layer continuum polymer introduced by O'Connell and Warren.

• This verifies, modulo a previously hidden constant, an outstanding conjecture proposed by Corwin and Hammond.
• A consequence is the identification of the KPZ line ensemble as logarithms of ratios of consecutive layers of the continuum partition function.
• Other properties of the continuum partition function, such as continuity, strict positivity and contour integral formulas to compute mixed moments, are also identified from this convergence result.