Quenched and Annealed Critical Points in Polymer Pinning Models

Details Quenched and Annealed Critical Points in Polymer Pinning Models Quenched and Annealed Critical Points in Polymer Pinning Models

2008-05-12

arxiv.org/abs/0805.1708v1

Mathematics

Probability

33 pages

Scientific paper

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 $3/22$, but only at low temperatures for $c<3/2$. For high temperatures and $3/23/2$ with arbitrary temperature we provide a new proof that the gap is positive, and extend it to $c=2$.

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