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Quenched and annealed critical points in polymer pinning models
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Alexander, Kenneth S. and Zygouras, Nikos (2009) Quenched and annealed critical points in polymer pinning models. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. Working papers, Vol.2009 (No.21).

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Abstract
We consider a polymer with configuration 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 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 ncφ(n) with c ≥ 1 and φ 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 φ(n) > 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.
Item Type:  Working or Discussion Paper (Working Paper) 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Statistics 
Library of Congress Subject Headings (LCSH):  Markov processes, Critical point 
Series Name:  Working papers 
Publisher:  University of Warwick. Centre for Research in Statistical Methodology 
Place of Publication:  Coventry 
Official Date:  2009 
Volume:  Vol.2009 
Number:  No.21 
Number of Pages:  33 
Status:  Not Peer Reviewed 
Access rights to Published version:  Open Access 
Funder:  National Science Foundation (U.S.) (NSF) 
Grant number:  DMS0405915 (NSF) 
References:  [1] Alexander, K.S. (2008). The e®ect of disorder on polymer depinning transitions. Commun. Math. Phys. 
URI:  http://wrap.warwick.ac.uk/id/eprint/35209 
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