Undirected Polymers in Random Environment: path properties in the mean field limit.

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY
N. Kistler, A. Schertzer
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引用次数: 1

Abstract

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.
随机环境中的无向聚合物:平均场极限中的路径性质。
我们考虑随机环境中的无向聚合物(连接在端点)问题,也称为超立方体上的无向第一通道渗流,在大维极限下。通过二阶矩方法的多尺度精化,我们获得了最优路径(即具有最小能量的聚合物)的相当精确的几何描述。出现的情况可以大致概括如下。聚合物的能量,首先近似地,沿着链均匀地分布。然而,聚合物的键携带的能量低于定向设置中的能量,并通过以下几何演变达到。在接近原点的地方,聚合物以定向的方式进行,从而尽可能地拉伸。然而,链的张力逐渐降低,当聚合物进入超立方体的核心时,它允许越来越多的反跳。后台阶虽然增加了链的长度,但允许聚合物连接具有能量有利边缘的储层,否则在完全定向的状态下是无法实现的。这些储层相距介观距离,但由于环境空间的高维性质,聚合物设法通过关于汉明度量的近似测地线将它们连接起来:这是导致最佳能量/熵平衡的关键策略。大约在中途,镜像开始显现:聚合物张力逐渐再次建立,直到接近终点时完全定向。作为推论,该方法得出了Martinsson[Ann.Appl.Prob.26(2016),Ann.Prob.46(2018)]关于基态主导序的结果的构造性证明。
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来源期刊
CiteScore
1.10
自引率
0.00%
发文量
48
期刊介绍: ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted. ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper. ALEA is affiliated with the Institute of Mathematical Statistics.
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