# Paper No. 09-21

**KS Alexander and N Zygouras**

**Quenched and annealed critical points in polymer pinning models**

**Abstract:** We consider a polymer with con¯guration modeled by the path of a Markov chain, interacting with a potential u+Vn which the chain encounters when it visits a special state 0 at time n. The disorder (Vn) is a ¯xed 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'(n) with c ¸ 1 and ' slowly varying. Comparing to the corresponding annealed system, in which the Vn are effectively replaced by a constant, it was shown in [1], [4], [11] 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 '(n) ! 0 as n ! 1, 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.