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A planar self-avoiding walk cannot revisit its occupied sites. In the good-solvent scaling regime, the two-dimensional Flory approximation gives $R\sim b{N}^{3/4}$, larger at large $N$ than either ideal walk and the projected three-dimensional self-avoiding walk. This is a scaling statement, with microscopic prefactors dependent on the walk model.

Write $\ell$ for a segment length and $v_2$ for the excluded-area coefficient, so that the interaction term has the correct dimensions. For a coil of planar size $R$ and area of order $R^2$, the two-dimensional Flory approximation isThe first term is the entropic stretching cost and the second counts repulsive monomer contacts. Minimization gives $2R/(N{\ell }^2)-2v_2{N}^2/R^3=0$, henceFor $v_2$ of order ${\ell }^2$ this becomes $R\sim \ell N^{3/4}$. The parameter multiplying $N^2/A$ in a two-dimensional interaction energy must have dimensions of area; it cannot simultaneously denote an unsquared bond length.

For polymers restricted to an interface, overlap means roughly one coil per coil area, $n_c{R}^2\sim 1$. A chain contains $N$ monomers with typical occupied area ${\ell }^2$, so its overlap area fraction is ${\varphi }^{\ast }\sim N{\ell }^2/R^2$. With the planar good-solvent exponent $\nu =3/4$ from the two-dimensional Flory approximation, this gives ${\varphi }^{\ast }\sim N^{-1/2}$. The same exponent happens to occur for a three-dimensional ideal chain's volume fraction, but its geometric origin is different.