论文标题
随机环境中的无向聚合物：平均场限制中的路径属性
Undirected polymers in random environment: path properties in the mean field limit
论文作者
论文摘要
我们考虑在随机环境中的无向聚合物（在端点上绑定）的问题，在大量方面的极限处，也称为Hypercube上未取向的第一通道渗透。通过对第二矩方法的多尺度改进，我们获得了对最佳路径的相当精确的几何描述，即具有最小能量的聚合物。出现的图片可以松散地总结如下。首先近似聚合物的能量沿链均匀散布。但是，聚合物的键携带的能量低于定向环境，并通过随后的几何演化到达。接近起源，聚合物以定向方式进行 - 因此，它尽可能伸展。然而，链的张力逐渐减少，当聚合物进入HyperCube的核心时，聚合物允许越来越多的后退。反击虽然增加了链的长度，但可以使聚合物连接能量有利的边缘的储层，这些储层在完全定向的方向上是无法实现的。这些储层分开的介质距离，但由于环境空间的高维质，聚合物设法通过相对于锤式指标将它们连接到它们：这是导致最佳能量/熵平衡的关键策略。一半左右，镜像设置为：聚合物张力逐渐累积，直到完全取向端点。作为推论，该方法产生了Martinsson [Ann。应用。概率。 26（2016），安。概率。 46（2018）]关于基态的领先顺序。
We consider the problem of undirected polymers (tied at the endpoints) in random environment, also known as the unoriented first passage percolation on the hypercube, in the limit of large dimensions. By means of the multiscale refinement of the second moment method we obtain a fairly precise geometrical description of optimal paths, i.e. of polymers with minimal energy. The picture which emerges can be loosely summarized as follows. The energy of the polymer is, to first approximation, uniformly spread along the strand. The polymer's bonds carry however a lower energy than in the directed setting, and are reached through the following geometrical evolution. Close to the origin, the polymer proceeds in oriented fashion -- it is thus as stretched as possible. The tension of the strand decreases however gradually, with the polymer allowing for more and more backsteps as it enters the core of the hypercube. Backsteps, although increasing the length of the strand, allow the polymer to connect reservoirs of energetically favorable edges which are otherwise unattainable in a fully directed regime. These reservoirs lie at mesoscopic distance apart, but in virtue of the high dimensional nature of the ambient space, the polymer manages to connect them through approximate geodesics with respect to the Hamming metric: this is the key strategy which leads to an optimal energy/entropy balance. Around halfway, the mirror picture sets in: the polymer tension gradually builds up again, until full orientedness close to the endpoint. The approach yields, as a corollary, a constructive proof of the result by Martinsson [Ann. Appl. Prob. 26 (2016), Ann. Prob. 46 (2018)] concerning the leading order of the ground state.
