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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).

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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 
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Status:  Not Peer Reviewed 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. 279 117146. [2] Alexander, K. S. and Sidoravicius, V. (2006). Pinning of polymers and interfaces by random potentials. Ann. Appl. Probab. 16 636{669. [3] Bodineau, T. and Giacomin, G. (2004). On the localization transition of random copolymers near selective interfaces. J. Stat. Phys. 117 801{818. [4] Derrida, B., Giacomin, G., Lacoin, H. and Toninelli, F. L. (2007). Fractional moment bounds and disorder relevance for pinning models. arXiv: math.PR/0712.2515. [5] Derrida, B., Hakim, V. and Vannimenus, J. (1992). E®ect of disorder on twodimensional wetting. J. Stat. Phys. 66 1189{1213. [6] Forgacs, G., Luck, J.M., Nieuwenhuizen, Th. M. and Orland, H. (1988). Exact critical behavior of twodimensional wetting problems with quenched disorder. J. Stat. Phys. 51 29{56. [7] Giacomin, G. (2007). Random Polymer Models. Imperial College Press, London. [8] Giacomin, G. and Toninelli, F. L. (2006). Smoothing e®ect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266 (2006) 116. [9] Naidenov, A. and Nechaev, S. (2001). Adsorption of a random heteropolymer at a potential well revis ited: location of transition point and design of sequences. J. Phys. A: Math. Gen. 34 5625{5634. [10] Seneta, E. (1976). Regularly Varying Functions. Lecture Notes in Math. 508. SpringerVerlag, Berlin. [11] Toninelli, F. L. (2007). Disordered pinning models and copolymers: beyond annealed bounds. Ann. Appl. Probab., to appear. arXiv:0709.1629v1 [math.PR] [12] Toninelli, F.L. (2008). A replicacoupling approach to disordered pinning models. Commun. Math. Phys. 280, 389401. 
URI:  http://wrap.warwick.ac.uk/id/eprint/35209 
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