标题： 淬火和退火临界点在聚合物固定模型中的研究 显示英文标题

标题： Quenched and Annealed Critical Points in Polymer Pinning Models

摘要： 我们考虑一个配置由马尔可夫链路径建模的聚合物，当链在时间$n$访问特殊状态 0 时与势能$u+V_n$相互作用。无序$(V_n)$是一个独立同分布序列的固定实现。当$u$超过临界值时，聚合物被固定，即链在其时间中花费正的比例在状态 0。 我们假设在没有势的情况下马尔可夫链，长度为$n$的从 0 出发的跃迁概率具有形式$n^{-c}\phi(n)$，其中$c \geq 1$和$\phi$是缓慢变化的。 与相应的退火系统相比，在退火系统中$V_n$被有效替换为一个常数，已知对于$3/2<c<2$和$c>2$，淬火和退火的临界点在所有温度下都不同，但对于$c<3/2$仅在低温下不同。 对于高温和$3/2<c<2$，我们建立了临界点之间的差距随温度变化的精确阶数。 For the borderline case $c=3/2$ we show that the gap is positive provided $\phi(n) \to 0$ as $n \to \infty$, and for $c >3/2$ with arbitrary temperature we provide a new proof that the gap is positive, and extend it to $c=2$. 显示英文摘要

摘要： We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential $u+V_n$ which the chain encounters when it visits a special state 0 at time $n$. The disorder $(V_n)$ is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spends a positive fraction of its time at state 0, when $u$ exceeds a critical value. We assume that for the Markov chain in the absence of the potential, the probability of an excursion from 0 of length $n$ has the form $n^{-c}\phi(n)$ with $c \geq 1$ and $\phi$ slowly varying. Comparing to the corresponding annealed system, in which the $V_n$ are effectively replaced by a constant, it is known that the quenched and annealed critical points differ at all temperatures for $3/2<c<2$ and $c>2$, but only at low temperatures for $c<3/2$. For high temperatures and $3/2<c<2$ we establish the exact order of the gap between critical points, as a function of temperature. For the borderline case $c=3/2$ we show that the gap is positive provided $\phi(n) \to 0$ as $n \to \infty$, and for $c >3/2$ with arbitrary temperature we provide a new proof that the gap is positive, and extend it to $c=2$.