Critical fluctuations at the central-limit threshold

Determine the limiting fluctuations of the spatial integral of the one-dimensional parabolic Anderson model with spatial white-noise potential when the box-growth parameter is exactly α=2, and establish whether a central limit theorem holds with the variance replaced by a truncated second moment.

Background

The paper proves a weak law of large numbers for α>1, a central limit theorem for α>2, and totally asymmetric α-stable limits for α∈(0,2). The value α=2 is the boundary between the Gaussian and stable regimes and is not covered by the stated central limit theorem, although the weak law does cover it.

For sums of i.i.d. random exponentials, the corresponding boundary case admits a central limit theorem after replacing the ordinary variance by a truncated second moment. The authors explicitly expect an analogous result for the spatial integral U(t), but do not establish it.

References

Theorem~\ref{thm:PAM_WLLN} covers the case $\alpha=2$, but the limiting fluctuations remain open. For sums of i.i.d.\ random exponentials, a central limit theorem still holds at the boundary between the Gaussian and stable regimes, provided the variance is replaced by a truncated second moment Theorem~2.5. We expect an analogous behavior for $U(t)$ when $\alpha=2$.

— Limit theorems for the one-dimensional parabolic Anderson model with white noise potential  (2609.19971 - Kim et al., 17 Sep 2026) in Remark following Theorem 1.3, Section 1.2 (Main results)