  We study a model of continuous-time nearest-neighbor random walk on
$\mathbb{Z}^d$ penalized by its occupation time at the origin, also known as a
homopolymer. For a fixed real parameter $\beta$ and time $t>0$, we consider the
probability measure on paths of the random walk starting from the origin whose
Radon-Nikodym derivative is proportional to the exponent of the product $\beta$
times the occupation time at the origin up to time $t$. The case $\beta>0$ was
studied previously by Cranston and Molchanov arXiv:1508.06915. We consider the
case $\beta<0$, which is intrinsically different only when the underlying walk
is recurrent, that is $d=1,2$. Our main result is a scaling limit for the
distribution of the homopolymer on the time interval $[0,t]$, as $t\to\infty$,
a result that coincides with the scaling limit for penalized Brownian motion
due to Roynette and Yor. In two dimensions, the penalizing effect is
asymptotically diminished, and the homopolymer scales to standard Brownian
motion. Our approach is based on potential analytic and martingale
approximation for the model. We also apply our main result to recover a scaling
limit for a wetting model. We study the model through analysis of resolvents.
