The scaling limits for d-dimensional random walks perturbed by an attractive force toward the origin are studied under the critical situation that the rate functional of the corresponding large deviation principle admits two minimizers. Our results extend those obtained by  from the mean-zero Gaussian to non-Gaussian setting under the absence of the wall.
Tadahisa FUNAKI. Tatsushi OTOBE. "Scaling limits for weakly pinned random walks with two large deviation minimizers." J. Math. Soc. Japan 62 (3) 1005 - 1041, July, 2010. https://doi.org/10.2969/jmsj/06231005