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We consider two directed polymer models in the Kardar-Parisi-Zhang (KPZ)universality class: the O'Connell-Yor semi-discrete directed polymer withboundary sources and the continuum directed random polymer with (m,n)-spikedboundary perturbations. The free energy of the continuum polymer is theHopf-Cole solution of the KPZ equation with the corresponding (m,n)-spikedinitial condition. This new initial condition is constructed using twosemi-discrete polymer models with independent bulk randomness and coupledboundary sources. We prove that the limiting fluctuations of the free energiesrescaled by the 1/3rd power of time in both polymer models converge to theBorodin-Peche type deformations of the GUE Tracy-Widom distribution.

Borodin-P\'ech\'e fluctuations of the free energy in directed random polymer models