Uniform all-moment asymptotics for Critical 2D SHF small-ball masses

Establish that, whenever the second moment of the Critical 2D Stochastic Heat Flow mass \(\mathscr{Z}_{t}^{\vartheta}(U_r)\) assigned to a ball of radius \(r\leq\sqrt{t}\) diverges, every real moment \(p\in\mathbb{R}\) satisfies \(\mathbb{E}[\mathscr{Z}_{t}^{\vartheta}(U_r)^p]=\mathbb{E}[\mathscr{Z}_{t}^{\vartheta}(U_r)^2]^{p(p-1)/2+o(1)}\), with the error term tending to zero as the second moment diverges, uniformly in the parameters \(t\), \(\vartheta\), and \(r\).

Background

The paper studies fractional moments of the mass that the Critical 2D Stochastic Heat Flow assigns to small balls. Its main upper bound shows that, for fractional exponents $p\in(0,1)$, these moments decay whenever the second moment diverges, while a separately announced result gives the sharp small-radius exponent for fixed time and disorder parameters. The authors propose that the exponent $p(p-1)/2$ should hold for every real moment, including negative moments, and uniformly across the full parameter range rather than only in the fixed-parameter small-radius regime.

The conjecture is known for positive integer moments according to the cited work, and is supported for $p\in(0,1)$ by the paper’s estimates and the announced sharp asymptotics. Negative moments remain unresolved: the paper notes that existing results on negative tails of the Stochastic Heat Flow are insufficient to derive corresponding negative-moment information. The proposed relation is motivated by the approximate log-normality of the flow at small scales.