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We consider various lattice models of polymers: lattice trees, latticeanimals, and self-avoiding walks. The polymer interacts with a surface(hyperplane), receiving a unit energy reward for each site in the surface.There is an adsorption transition of the polymer at a critical value of$\beta$, the inverse temperature. We present a new proof of the result ofHammersley, Torrie, and Whittington (1982) that the transition occurs at astrictly positive value of $\beta$ when the surface is impenetrable, i.e. whenthe polymer is restricted to a half-space. In contrast, for a penetrablesurface, it is an open problem to prove that the transition occurs at $\beta=0$(i.e., infinite temperature). We reduce this problem to showing that thefraction of N-site polymers whose span is less than $N/\log^2 N$ is not toosmall.

Location of the Adsorption Transition for Lattice Polymers