Adsorption of self-avoiding walks at a defect

We consider the model of self-avoiding walks on the $d$-dimensional hypercubic lattice interacting with a $d^*$-dimensional defect, where $1\leq d^*<d$. Such an interaction can be attractive or repulsive, and is controlled by a Boltzmann weight $a$ associated with visits to the defect. When $d=3$ and $d^*=1$ or $2$, this can be seen as a model of long linear polymers in a good solvent, interacting with a linear filament or the interface of two liquids of different density. For all combinations of dimensions, there is a critical value $a_{\rm c}$ which separates the desorbed and adsorbed phases of the model. We prove that in all cases $a_{\rm c}=1$, confirming conjectures by a number of authors.