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 in . 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.
Supported by NSF grant DMS-1209165, the MacCracken Fellowship from New York University, and the NSERC postdoctoral fellowship.
The author is grateful to Ivan Corwin for suggesting the problem as well as for discussions and advice during the writing process. The author also thanks Jeremy Quastel for helpful discussions and for making an early draft of a related result  available. I am also grateful to the anonymous referees for their careful reading and comments which greatly improved the presentation of the article.
"Intermediate disorder limits for multi-layer semi-discrete directed polymers." Electron. J. Probab. 26 1 - 50, 2021. https://doi.org/10.1214/21-EJP614