AI 中文总结
该研究量化了临界二维随机热流（SHF）的奇异性与间歇性，证明其支撑集具有对数分形行为，关键利用大偏差理论与布朗运动条件分布特性完成分析。
AI 中文摘要
虽然1+1维带乘性噪声的随机热方程的解与布朗运动的指数密切相关，但二维情形呈现出额外的弱无序到强无序的转变。在[CSZ '23]中，临界二维随机热流（stochastic heat flow, SHF）被构造为临界状态下2+1维定向聚合物配分函数在对数中间无序标度下的标度极限。SHF是一种随机测度，与许多自然出现的随机测度一样，预计会表现出丰富的间歇性。[CSZ '25]证明了它相对于勒贝格测度几乎必然是奇异的。更近的，[GT '26]表明，在小球上平均的SHF的对数渐近高斯，其均值和方差都随球半径趋于零而发散。在本文中，我们证明了一个精确结果，量化了SHF支撑集的奇异性及其间歇性。特别地，我们证明几乎必然地，对于所有小的ε>0，在消失误差范围内，任意区域内点对点SHF的所有质量都集中在1/(ε² log^(1/2+o(1))(1/ε))个半径为ε的球中，每个球包含ε² log^(1/2+o(1))(1/ε)的质量，从而精确确立了其对数分形行为。证明中的一个关键要素是精细的大偏差理论，该理论通过利用SHF在准临界标度下的类高斯行为，能够获取条件分布。另一个重要观察是，对布朗运动在其端点异常大的条件下，本质上会诱导其增量均值的偏移，因此在足够小的标度下，其分布不会发生显著改变。
英文摘要
While the solution to the $1+1$ dimensional stochastic heat equation with multiplicative noise is closely related to the exponential of a Brownian motion, the two-dimensional picture exhibits an additional weak-to-strong disorder transition. In [CSZ '23], the critical two-dimensional stochastic heat flow (SHF) was constructed as the scaling limit of the partition function of $2+1$ dimensional directed polymers under the logarithmic intermediate-disorder scaling at criticality. The SHF is a random measure and, like many naturally occurring random measures, it is expected to exhibit rich intermittency. [CSZ '25] established that it is almost surely singular with respect to the Lebesgue measure. More recently, [GT '26] showed that the logarithm of the SHF averaged over small balls is asymptotically Gaussian, with both its mean and variance diverging as the ball radius tends to zero. In this paper we prove a sharp result quantifying the singularity of the support of the SHF as well as its intermittency. In particular, we show that, almost surely, for all small $\varepsilon>0$, up to a vanishing error, all the mass of the point-to-plane SHF in any domain is concentrated on ${1}/{\big(\varepsilon^2\log^{1/2+o(1)}(1/\varepsilon)\big)}$ balls of radius $\varepsilon$, each containing $\varepsilon^2{\log^{1/2+o(1)}(1/\varepsilon)}$ mass, thus precisely establishing its logarithmic fractal behavior. A key ingredient in the proof is a refined large-deviations theory, which allows access to conditional distributions, by taking advantage of the Gaussian-like behavior of the SHF at quasi-critical scales. A further useful observation that features prominently is that conditioning a Brownian motion on its endpoint being unusually large essentially induces a shift in the mean of its increments, and consequently, at small enough scales, their distributions do not alter significantly.
Comments78 pages, 2 figures. Abstract shortened to meet arXiv requirements
