Fractional moments of the Stochastic Heat Flow and 2D Directed Polymers

Abstract: We estimate the fractional moments of the normalized mass assigned by the Critical 2D Stochastic Heat Flow to small balls. Our results also cover the discrete case corresponding to the 2D directed polymer model and provide estimates that are uniform in all parameters. One key takeaway of our results is that the vanishing of the fractional moments is completely governed by the divergence of the second moment. We use a quite robust method, by refining the change of measure argument and introducing a novel coarse-graining procedure, reducing the proof to essentially second moment estimates (in fact, we also provide sharp second moment estimates for directed polymers, of independent interest).

• The paper establishes parameter-uniform upper bounds for fractional moments of Critical 2D Stochastic Heat Flow masses and directed-polymer partition functions, showing that divergent second moments force these quantities to vanish in probability.
• The authors combine a size-biased change-of-measure estimate with non-homogeneous geometric coarse-graining to obtain logarithmic decay rates uniformly across subcritical, critical, and supercritical disorder regimes.
• The results support a conjectured exponent of one-half p(p−1), while leaving sharp uniform estimates, negative moments, and a rigorous multifractal spectrum as important open problems.

The Critical 2D Stochastic Heat Flow (SHF), constructed by Caravenna–Sun–Zygouras as the universal scaling limit of 2D directed polymers under critical disorder scaling, is a measure-valued process that also provides the canonical solution to the ill-posed two-dimensional Stochastic Heat Equation. This paper, by Berger, Turchi and Zygouras (2608.13359), establishes upper bounds on fractional moments of the mass that the SHF assigns to small balls, together with matching estimates for discrete directed polymer partition functions that are uniform in all parameters. The central message is sharp and clean: the vanishing of fractional moments is completely governed by the divergence of the second moment.

Main results

For $p \in (0,1)$, uniformly over all times $t>0$, interpolation parameters $\vartheta \in \mathbb{R}$, and radii $r \in (0,\sqrt{t}]$, the authors prove

$$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C\, \mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^2\big]^{\,c\,p(p-1)},$$

with an explicit version involving the Volterra function $\mathcal{V}(T) = \int_0^\infty T^s/\Gamma(s+1)\,ds$:

$$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C' \Big(1 + \log\Big(1+\tfrac{t}{r^2}\Big)\,\mathcal{V}(t e^\vartheta)\Big)^{c' p(p-1)}.$$

Since $\mathcal{V}(T) \sim e^T$ as $T\to\infty$ and $\mathcal{V}(T) \sim 1/\log(1/T)$ as $t>0$0, the second moment diverges either because $t>0$1 or because $t>0$2 sufficiently fast; whenever it does, every fractional moment vanishes, so $t>0$3 in probability. This is consistent with the singularity of SHF time marginals with respect to Lebesgue measure established previously, and with recent fine-scale results showing that Lebesgue-almost-everywhere the pointwise mass concentrates near $t>0$4.

The corresponding variance estimate is two-sided: constants $t>0$5 exist such that

$t>0$6

In the discrete setting, for the polymer partition function $t>0$7 started from the uniform distribution on a ball of radius $t>0$8, with the interpolating parameter $t>0$9 (where $\vartheta \in \mathbb{R}$0 is the return-time sum and $\vartheta \in \mathbb{R}$1), the same bound holds uniformly over $\vartheta \in \mathbb{R}$2, $\vartheta \in \mathbb{R}$3, and $\vartheta \in \mathbb{R}$4 — notably including subcritical ($\vartheta \in \mathbb{R}$5) and supercritical ($\vartheta \in \mathbb{R}$6) regimes, not only the critical window. The SHF result follows by taking the scaling limit along the critical tuning.

Relation to concurrent work

A concurrent announcement by Huang obtains the sharper asymptotic $\vartheta \in \mathbb{R}$7 as $\vartheta \in \mathbb{R}$8, identifying the correct exponent $\vartheta \in \mathbb{R}$9 — first known for integer moments via log-correlated Gaussian analysis. The present paper's bound carries a non-sharp constant $r \in (0,\sqrt{t}]$0 in the exponent but is uniform in $r \in (0,\sqrt{t}]$1 and $r \in (0,\sqrt{t}]$2, whereas the sharp result fixes these parameters and sends only $r \in (0,\sqrt{t}]$3 to zero. The authors conjecture that

$r \in (0,\sqrt{t}]$4

holds for all $r \in (0,\sqrt{t}]$5, uniformly in parameters, whenever the second moment diverges. This is proven for integer $r \in (0,\sqrt{t}]$6 and supported here for $r \in (0,\sqrt{t}]$7; negative moments remain open, though approximate log-normality at small scales (a CLT for $r \in (0,\sqrt{t}]$8 with centering $r \in (0,\sqrt{t}]$9) suggests the relation, which holds exactly for mean-one log-normal variables. The paper notes honestly that existing negative-tail results are insufficient to settle this.

Proof strategy

Two methodological novelties drive the argument, both developed at the polymer level and transferred to the SHF by weak convergence.

Step 1: a streamlined change-of-measure estimate. The half-moment $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C\, \mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^2\big]^{\,c\,p(p-1)},$$0 is controlled through a size-biased tilt $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C\, \mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^2\big]^{\,c\,p(p-1)},$$1 and a proxy $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C\, \mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^2\big]^{\,c\,p(p-1)},$$2 built from strip partition functions $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C\, \mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^2\big]^{\,c\,p(p-1)},$$3 on geometrically growing scales $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C\, \mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^2\big]^{\,c\,p(p-1)},$$4. A general lemma shows that if a proxy has mean shift $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C\, \mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^2\big]^{\,c\,p(p-1)},$$5 under the tilt large relative to combined fluctuations $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C\, \mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^2\big]^{\,c\,p(p-1)},$$6, then $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C\, \mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^2\big]^{\,c\,p(p-1)},$$7; the authors remark this inequality is sharp, with equality attained for two-point distributions. Since $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C\, \mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^2\big]^{\,c\,p(p-1)},$$8 while $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C\, \mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^2\big]^{\,c\,p(p-1)},$$9, the task reduces to showing $\mathcal{V}(T) = \int_0^\infty T^s/\Gamma(s+1)\,ds$0 and $\mathcal{V}(T) = \int_0^\infty T^s/\Gamma(s+1)\,ds$1, where $\mathcal{V}(T) = \int_0^\infty T^s/\Gamma(s+1)\,ds$2. The diagonal terms of the size-biased variance are handled by hypercontractivity; the off-diagonal covariances are computed via the Wiener-chaos expansion and collision diagrams, yielding a decay like $\mathcal{V}(T) = \int_0^\infty T^s/\Gamma(s+1)\,ds$3 across scales, summable to $\mathcal{V}(T) = \int_0^\infty T^s/\Gamma(s+1)\,ds$4. This yields the key proposition: a quantitative but sub-optimal bound $\mathcal{V}(T) = \int_0^\infty T^s/\Gamma(s+1)\,ds$5, valid when $\mathcal{V}(T) = \int_0^\infty T^s/\Gamma(s+1)\,ds$6.

Step 2: non-homogeneous coarse-graining. The sub-optimal bound gives only a fixed contraction per block; iterating it requires many blocks. The crucial innovation is that coarse-graining scales are allowed to grow geometrically ($\mathcal{V}(T) = \int_0^\infty T^s/\Gamma(s+1)\,ds$7), whereas prior applications used constant scales. Choosing $\mathcal{V}(T) = \int_0^\infty T^s/\Gamma(s+1)\,ds$8 large enough makes the contraction per block smaller than $\mathcal{V}(T) = \int_0^\infty T^s/\Gamma(s+1)\,ds$9, and the half-moment then decays like $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C' \Big(1 + \log\Big(1+\tfrac{t}{r^2}\Big)\,\mathcal{V}(t e^\vartheta)\Big)^{c' p(p-1)}.$$0 in the number of scales $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C' \Big(1 + \log\Big(1+\tfrac{t}{r^2}\Big)\,\mathcal{V}(t e^\vartheta)\Big)^{c' p(p-1)}.$$1, producing the correct logarithmic rate. For $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C' \Big(1 + \log\Big(1+\tfrac{t}{r^2}\Big)\,\mathcal{V}(t e^\vartheta)\Big)^{c' p(p-1)}.$$2, the argument reduces to the critical case $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C' \Big(1 + \log\Big(1+\tfrac{t}{r^2}\Big)\,\mathcal{V}(t e^\vartheta)\Big)^{c' p(p-1)}.$$3 via monotonicity of fractional moments in $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C' \Big(1 + \log\Big(1+\tfrac{t}{r^2}\Big)\,\mathcal{V}(t e^\vartheta)\Big)^{c' p(p-1)}.$$4, and combines with an exponential bound from prior work of the authors. An interpolation lemma converts half-moment control into bounds for all $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C' \Big(1 + \log\Big(1+\tfrac{t}{r^2}\Big)\,\mathcal{V}(t e^\vartheta)\Big)^{c' p(p-1)}.$$5 via Jensen and Hölder.

Sharp second moment estimates

Supporting the main theorem, the paper proves uniform variance estimates of independent interest. The point-to-plane quantity $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C' \Big(1 + \log\Big(1+\tfrac{t}{r^2}\Big)\,\mathcal{V}(t e^\vartheta)\Big)^{c' p(p-1)}.$$6 satisfies

$$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C' \Big(1 + \log\Big(1+\tfrac{t}{r^2}\Big)\,\mathcal{V}(t e^\vartheta)\Big)^{c' p(p-1)}.$$7

uniformly across subcritical, critical and supercritical regimes — the critical-window convergence was known, but uniform control outside it required new work. The proof introduces a "soft tilt" renewal representation replacing the hard cut-off of earlier treatments, and an auxiliary interpolating parameter $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C' \Big(1 + \log\Big(1+\tfrac{t}{r^2}\Big)\,\mathcal{V}(t e^\vartheta)\Big)^{c' p(p-1)}.$$8 satisfying $$\mathbb{E}\big[\mathscr{Z}_t^\vartheta(\mathcal{U}_r)^p\big] \le C' \Big(1 + \log\Big(1+\tfrac{t}{r^2}\Big)\,\mathcal{V}(t e^\vartheta)\Big)^{c' p(p-1)}.$$9. Upper bounds follow from a one-line geometric-series estimate with well-chosen tilts; lower bounds restrict the renewal sum to windows around its mean, controlled by Chebyshev and Markov inequalities. Combining with the first-collision decomposition and local CLT estimates on $\mathcal{V}(T) \sim e^T$0 yields the full variance proposition.

Multifractal implications

The fractional moment bounds feed directly into a prospective Parisi–Frisch multifractal formalism for the SHF. Summing $\mathcal{V}(T) \sim e^T$1 over the roughly $\mathcal{V}(T) \sim e^T$2 disjoint balls in a unit square gives $\mathcal{V}(T) \sim e^T$3: the Euclidean factor $\mathcal{V}(T) \sim e^T$4 combines with a logarithmic correction whose exponent would be exactly $\mathcal{V}(T) \sim e^T$5 if the sharp form held. Establishing the full $\mathcal{V}(T) \sim e^T$6 spectrum would additionally require controlling the supremum over ball coverings, which remains beyond current techniques.

Limitations and open questions

Several caveats are explicit in the paper. First, the exponent obtained is $\mathcal{V}(T) \sim e^T$7 with an unspecified constant $\mathcal{V}(T) \sim e^T$8, so the upper bound is not sharp; closing the gap to Conjecture 1/2 would require refining the change-of-measure or coarse-graining steps. Second, the conjectured extension to negative moments $\mathcal{V}(T) \sim e^T$9 is entirely open: no technique currently available controls the lower tails of $T\to\infty$0 well enough, and the paper concedes that existing negative-tail estimates fall short. Third, the multifractal program sketched informally ignores the supremum over coverings, and no rigorous spectrum statement is claimed. Finally, whether the uniform-in-parameters approach can be pushed to recover the exact exponent $T\to\infty$1 — rather than serving as a robust but coarser alternative to the fixed-parameter analysis — is left unresolved.

Conclusion

This work delivers parameter-uniform fractional moment bounds for the Critical 2D SHF and 2D directed polymers, demonstrating quantitatively that intermittency at small scales is dictated by second-moment divergence. Technically, it contributes a sharpened change-of-measure lemma with a sharp inequality, a genuinely non-homogeneous coarse-graining scheme adapted to the model's scale separation, and uniform Volterra-function asymptotics for polymer variances valid across all disorder regimes. These tools are positioned as broadly applicable to fractional moments of disordered systems, where the classical integer-moment machinery tied to random walk collisions and the delta-Bose gas breaks down.