Quenched and Annealed Critical Points in Polymer Pinning Models

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 \({n^{-c}\varphi(n)}\) with c ≥  1 and φ slowly varying. Comparing to the corresponding annealed system, in which the V _n are effectively replaced by a constant, it was shown in 1,4,13] 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 \({\varphi(n) \to 0}\) as n → ∞, and for c > 3/2 with arbitrary temperature we provide an alternate proof of the result in 4] that the gap is positive, and extend it to c = 2.