We study the scaling limit and prove the law of large numbers for weakly pinned Gaussian random fields under the critical situation that two possible candidates of the limits exist at the level of large deviation principle. This paper extends the results of ,  for one dimensional fields to higher dimensions: $d\ge3 $, at least if the strength of pinning is sufficiently large.
"Scaling limits for weakly pinned Gaussian random fields under the presence of two possible candidates." J. Math. Soc. Japan 67 (4) 1359 - 1412, October, 2015. https://doi.org/10.2969/jmsj/06741359