平面定向聚合物中的根吉布斯-DLR测度

中文摘要 AI 辅助

我们研究了满足额外温和假设的遍历无序分布下，$\boldsymbol{Z}^2$上定向聚合物模型中的根吉布斯-DLR测度。我们证明了极端根吉布斯-DLR测度的集合是闭且全序的，通过路径合并刻画了极端性，并表明每个完全支撑的极端根吉布斯测度可典范生成由所有格点索引的全局一致且合并的极端根吉布斯测度族，这些族同样是全序的。基于该结构，我们证明了相关Busemann过程的强存在性与强唯一性，以及平移协变Busemann上同调在逆温度、倾斜参数和随机环境联合作用下的$L^1$连续性定理。由此作为推论，在权重的有界独立同分布扰动下可微方向对应的生成极端吉布斯测度依概率连续；在正温度下，这表明生成的极端吉布斯测度依概率收敛；在零温度下，这还给出了路径空间上对应正温度根吉布斯-DLR测度的淬火子序列大偏差原理，速率函数由零温度Busemann上同调确定，且仅在该上同调生成的无限测地线上消失。

英文摘要

We study rooted Gibbs-DLR measures in the directed polymer model on $\mathbb{Z}^2$ with an ergodic disorder distribution which satisfies an additional mild hypothesis. We prove that the set of extremal rooted Gibbs-DLR measures is closed and totally ordered, characterize extremality in terms of path coalescence, and show that each fully supported extremal rooted Gibbs measure canonically generates a globally consistent and coalescing family of extremal rooted Gibbs measures indexed by all lattice sites. These families are, moreover, totally ordered. Building on this structure, we prove strong existence and strong uniqueness of the associated Busemann process, together with an $L^1$ continuity theorem for the shift-covariant Busemann cocycles which is joint in the inverse temperature, the tilt parameter, and the random environment. This yields, as a corollary, in-probability continuity of the generated extremal Gibbs measures corresponding to directions of differentiability under bounded i.i.d. perturbations of the weights. In positive temperature, it shows in-probability convergence of the generated extremal Gibbs measures. At zero temperature, it also yields quenched subsequential large deviation principles for the corresponding positive-temperature rooted Gibbs-DLR measures on path space. The rate functions are determined by a zero-temperature Busemann cocycle and vanish precisely on the infinite geodesics generated by that cocycle.

补充信息

备注

64 pages, 1 figure