The scaling-limit conjecture for periodic polymer Markov chains

Let $N$ be fixed, let $\gamma$ denote the parameters of the coupled Markov chain, and consider the coupled chain defined by, its dual chain defined by, the system of SDEs, and its dual system. Let the KPZ equation with white noise $\xi$ be denoted by. Periodic polymer scaling-limit conjecture. There should be a suitable scaling of time and tuning of $\gamma$ under which the coupled Markov chain converges to the system of SDEs, and under the same scaling the dual Markov chain converges to the dual system of SDEs. There should likewise be a scaling in which time and $N$ tend to infinity, together with a tuning of $\gamma$, such that the coupled Markov chain converges directly to coupled solutions of the KPZ equation driven by the same white noise $\xi$.

This conjecture proposes periodic analogues of the known scaling limits from the inverse-gamma polymer to the O'Connell–Yor polymer and, ultimately, to KPZ. The source explains that finite-time explosion of the discrete polymer initial quantity complicates the proof, while suggesting that the explosion scale should be separated from the scaling relevant to the O'Connell–Yor limit.