For 0<p<1, the p-th moment of the critical 2D SHF mass on a small ball is bounded by the second moment raised to a negative power whenever the second moment diverges, uniformly in time, disorder, and radius.
Fractional Moments of Small-Ball Masses for the Stochastic Heat Flow
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abstract
We prove the logarithmic fractional-moment exponent for the mass of a shrinking ball under the critical two-dimensional stochastic heat flow. The proof builds on the geometric block decomposition of Gu and Tsai and introduces a fractional-product change of measure that preserves block independence and allows the fractional moments to be analyzed.
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Fractional moments of the Stochastic Heat Flow and 2D Directed Polymers
For 0<p<1, the p-th moment of the critical 2D SHF mass on a small ball is bounded by the second moment raised to a negative power whenever the second moment diverges, uniformly in time, disorder, and radius.