Polymer in a Multi-Interface Medium with Weak Repulsion

Pinning phenomena for long linear polymers have been studied for a long time. In 2009 Caravenna and Pétrélis (Electron J Probab 14(70):2038–2067, 2009) investigated the effect of a periodic and repulsive multi-interface medium on a \((1+1)\)-directed polymer model, when the distance between consecutive interfaces scales with the length of the polymer and with a constant temperature. In this paper, we extend that model and consider weak repulsion, by letting both the temperature and the distance between interfaces scale with the length of the polymer. We obtain a full diagram for this model, showing the behaviour of the polymer depending on the scaling exponents associated to the repulsion and the spacing parameters. When the repulsion is not too weak compared to the interface spacing, we obtain different regimes that extend those obtained by Caravenna and Pétrélis, and either finitely or infinitely many interfaces are visited. When the two exponents match we obtain a diffusive regime with a non-trivial and temperature-dependent diffusion constant. Our key tools include the renewal approach used in the original paper as well as new sharp results on the simple random walk evolving between interfaces.