We derive a moderate deviation principle for the lower tail probabilities of the length of a longest increasing subsequence in a random permutation. It refers to the regime between the lower tail large deviation regime and the central limit regime. The present article together with the upper tail moderate deviation principle in Ref. 12 yields a complete picture for the whole moderate deviation regime. Other than in Ref. 12, we can directly apply estimates by Baik, Deift, and Johansson, who obtained a (non-standard) Central Limit Theorem for the same quantity.
Löwe, M., Merkl, F., & Rolles, S. W. W. (2002). Moderate deviations for longest increasing subsequences: The lower tail. Journal of Theoretical Probability, 15(4), 1031-1047. https://doi.org/10.1023/A:1020649006254