具有临界空间相关性的二维定向聚合物的自由能与相变
Free energy and phase transition for 2D directed polymers with critical spatial correlations
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中文总结 AI 辅助
本文研究具有临界空间相关性的二维定向聚合物模型，确定其自由能的精确高温渐近行为并证实相关猜想，还建立了扩散重标分配分函数的相变，所用方法或可推广至其他无序模型。
中文摘要 AI 辅助
我们研究高斯环境中的二维定向聚合物模型，该模型在时间上独立且空间相关，协方差函数为$h(x)$，要么可求和，要么具有临界衰减，满足当$|x|\to\infty$时$h(x)\sim (\log |x|)^a/|x|^2$（其中$a>-1$）。我们确定了自由能的精确高温渐近行为，证实了Lacoin（Ann. Probab. 2011）的猜想，该猜想后被Cosco、Cottini和Donadini（2025）改进。我们还建立了扩散重标分配分函数的相变：低于某临界点时，它们收敛到勒贝格测度；高于临界点时，收敛到零。我们方法的关键特征是，两个结果仅通过二阶矩估计得到，且基于超临界区域的简化测度变换论证，该论证或可对其他无序模型有用。
英文摘要
We study the two-dimensional directed polymer model in a Gaussian environment which is independent in time and spatially correlated, with covariances $h(x)$ either summable or with a critical decay, satisfying $h(x) \sim (\log |x|)^a/|x|^2$ as $|x|\to\infty$ for some $a>-1$. We determine the precise high-temperature asymptotics of the free energy, confirming a conjecture of Lacoin (Ann. Probab. 2011), later refined by Cosco, Cottini and Donadini (2025). We also establish a phase transition for the diffusively rescaled partition functions: below some critical point they converge to the Lebesgue measure, while above it they converge to zero. A key feature of our approach is that both results are obtained using only second-moment estimates and are based on a simplified change-of-measure argument in the supercritical regime, that may prove useful for other disordered models.
发表机构
- Université Sorbonne Paris Nord（巴黎北索邦大学）
- Institut Universitaire de France（法国高等研究院）
- Sorbonne Université（索邦大学）
机构由 AI 辅助整理，请以论文原文为准。
