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We discuss the generalization of a classical problem involving an $N$-stepideal polymer adsorption at a sticky boundary (potential well of depth $U$). Itis known that as $N$ approaches infinity, the path undergoes a 2nd-orderlocalization transition at a certain value of $U_{\text{tr}}$. By consideringthe random walk on a half-line with a sticky boundary (Model I), we demonstratethat the order of the phase transition can be altered by adjusting the scalingof the first return probability to the boundary. Additionally, we present amodel of a random path on a discrete 1D lattice with non-uniform local hoppingamplitudes and a potential well at the boundary (Model II). We illustrate thatone can tailor such amplitudes so that the polymer undergoes a 3rd-order phasetransition.

Unconventional critical behavior of polymers at sticky boundaries