Unconventional critical behavior of polymers at sticky boundaries

We discuss the generalization of a classical problem involving an $N$-step ideal polymer adsorption at a sticky boundary (potential well of depth $U$). It is known that as $N$ approaches infinity, the path undergoes a 2nd-order localization transition at a certain value of $U_{\text{tr}}$. By considering the random walk on a half-line with a sticky boundary (Model I), we demonstrate that the order of the phase transition can be altered by adjusting the scaling of the first return probability to the boundary. Additionally, we present a model of a random path on a discrete 1D lattice with non-uniform local hopping amplitudes and a potential well at the boundary (Model II). We illustrate that one can tailor such amplitudes so that the polymer undergoes a 3rd-order phase transition.