临界二维随机热流、聚合物与GMC的矩比较不等式
Moments Comparison Inequalities for Critical 2d Stochastic Heat Flow, Polymers and GMC
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中文总结 AI 辅助
本文利用高斯凸不等式和自举论证，为临界二维随机热流等对象建立了矩比较不等式，证明局部平均SHF的p阶矩与其二阶矩的p(p-1)/2次幂在乘法常数下互有界。
中文摘要 AI 辅助
我们利用[AC15]、[Che26]、[Gue22]和[FUMPC26]中发展的高斯凸不等式，建立了高斯乘性混沌、高斯环境中有向聚合物以及特别是临界二维随机热流（SHF）的（单侧）矩比较不等式，包括负矩的上界，以及$p\in(0,1)$时$p$阶矩的类似下界。对于在小球上平均的临界二维SHF，我们还获得了匹配（至多相差乘法常数）的互补界。我们通过受[DS10]启发的自举论证和[GT26]中引入的几何分解来建立互补界。具体而言，我们证明对于局部平均的SHF，对每个$p\in\mathbb R$，$p$阶矩在乘法常数意义下被其二阶矩的$\frac{p(p-1)}{2}$次幂上下界定。
英文摘要
We employ a Gaussian convex inequality developed in [AC15], [Che26], [Gue22], and [FUMPC26] to establish (one sided) moment comparison inequalities for Gaussian multiplicative chaos, directed polymers in Gaussian environment and, in particular, the Critical 2d Stochastic Heat Flow (SHF), including upper bounds on negative moments, as well as similar lower bounds for $p$-th moments with $p\in (0,1)$. For the Critical 2d SHF, averaged over small balls, we also obtain the matching (up to multiplicative constants) complementary bounds. We establish the complementary bounds using a bootstrap argument inspired by [DS10] and a geometric decomposition introduced in [GT26]. Specifically, we prove that for locally averaged SHF, for every $p\in\mathbb R$, the $p$-th moment is, up to multiplicative constants, bounded in both sides by the $\frac{p(p-1)}{2}$-th power of its second moment.
