二维及二维以上带电聚合物的退火标度
Annealed scaling for a charged polymer in dimensions two and higher
摘要：
本文研究了$\mathbb{Z}^d$上的无向高分子链$d \geq 2$，其i.i.d.随机电荷附着在其组成单体上。聚合物链的每个自交叉对相互作用哈密顿量贡献能量，该能量等于相遇的两个单体的电荷的乘积。聚合物链和电荷的联合概率分布由与相互作用哈密顿量相关的吉布斯分布给出。感兴趣的对象是在聚合物链的长度$n$趋于无穷大的极限下每个单体的\emph{annealed free energy}。我们发现，有一个关键的曲线在参数平面跨越的电荷偏置和逆温度分离的\emph{extended phase}从\emph{collapsed phase}。我们推导出小的和大的电荷偏压的临界曲线的缩放和小的反温度的退火自由能的缩放，这两者都是反常的。我们发现，在一个子集的崩溃阶段的聚合物链是\emph{subdiffusive}，即，规模$(n/\log n)^{1/(d+2)}$，它像一个布朗运动的条件，留在一个球内具有确定性的半径和一个随机移动的中心。我们预计这种缩放将在整个崩溃阶段保持不变。我们进一步预期，在扩展阶段的聚合物链规模像一个弱自回避行走。证明是基于一个简单的随机游走的自相交当地时间的向下大偏差和向上大偏差的范围的详细分析。我们的缩放结果的一部分是粗糙的，我们制定的架构下，他们可以锐化。自由能的存在仍然是一个悬而未决的问题，我们能够解决的一个子集的崩溃阶段的一个子类的电荷分布。
英文摘要：
This paper considers an undirected polymer chain on $\mathbb{Z}^d$, $d \geq 2$, with i.i.d.\ random charges attached to its constituent monomers. Each self-intersection of the polymer chain contributes an energy to the interaction Hamiltonian that is equal to the product of the charges of the two monomers that meet. The joint probability distribution for the polymer chain and the charges is given by the Gibbs distribution associated with the interaction Hamiltonian. The object of interest is the \emph{annealed free energy} per monomer in the limit as the length $n$ of the polymer chain tends to infinity. We show that there is a critical curve in the parameter plane spanned by the charge bias and the inverse temperature separating an \emph{extended phase} from a \emph{collapsed phase}. We derive the scaling of the critical curve for small and for large charge bias and the scaling of the annealed free energy for small inverse temperature, which are both anomalous. We show that in a subset of the collapsed phase the polymer chain is \emph{subdiffusive}, namely, on scale $(n/\log n)^{1/(d+2)}$ it moves like a Brownian motion conditioned to stay inside a ball with a deterministic radius and a randomly shifted center. We expect this scaling to hold throughout the collapsed phase. We further expect that in the extended phase the polymer chain scales like a weakly self-avoiding walk. Proofs are based on a detailed analysis for simple random walk of the downward large deviations of the self-intersection local time and the upward large deviations of the range. Part of our scaling results are rough, and we formulate conjectures under which they can be sharpened. The existence of the free energy remains an open problem, which we are able to settle in a subset of the collapsed phase for a subclass of charge distributions.
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