脱钉相的聚合物动力学：具有对数势垒的亚稳态
Polymer dynamics in the depinned phase: metastability with logarithmic barriers
摘要：
我们考虑了（1 + 1）维聚合物在脱钉状态下的随机演化。在平衡时，该系统表现出双阱结构：聚合物（基本上）位于排斥线的上方或下方。因此，人们期望一种亚稳态行为，在两个相之间具有罕见的跳跃，并且在每个相内部具有快速热化。然而，这两个阶段之间的能量势垒在系统大小L中仅为对数，因此两个相关的时间尺度在L中仅为多项式，它们之间没有明确的分离。整个演化是由一个阶段内的扩散行为和跨越能量障碍的跳跃之间的微妙竞争。特别是通常的情况下，隧穿时间与能量势垒的指数相一致，打破了。我们的主要结果是：（i）证明系统的混合时间介于L^{5/2}和L^{5/2+2}之间;（ii）识别与聚合物的正相和负相相关的两个区域，并证明它们之间的隧穿时间的渐近指数性，其速率等于光谱间隙的一半。
英文摘要：
We consider the stochastic evolution of a (1 + 1)-dimensional polymer in the depinned regime. At equilibrium the system exhibits a double well structure: the polymer lies(essentially) either above or below the repulsive line. As a consequence one expects a metastable behavior with rare jumps between the two phases combined with a fast thermalization inside each phase. However the energy barrier between these two phases is only logarithmic in the system size L and therefore the two relevant time scales are only polynomial in L with no clear-cut separation between them. The whole evolution is governed by a subtle competition between the diffusive behavior inside one phase and the jumps across the energy barriers. In particular the usual scenario in which the tunneling time coincides with the exponential of the energy barrier breaks down. Our main results are: (i) a proof that the mixing time of the system lies between L^{5/2} and L^{5/2+2}; (ii) the identification of two regions associated with the positive and negative phase of the polymer together with the proof of the asymptotic exponentiality of the tunneling time between them with rate equal to a half of the spectral gap.
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