On the irrelevant disorder regime of pinning models
Giambattista Giacomin and Fabio Lucio Toninelli
Source: Ann. Probab. Volume 37, Number 5 (2009), 1841-1875.
Abstract
Recent results have lead to substantial progress in understanding the role of disorder in the (de)localization transition of polymer pinning models. Notably, there is an understanding of the crucial issue of disorder relevance and irrelevance that is now rigorous. In this work, we exploit interpolation and replica coupling methods to obtain sharper results on the irrelevant disorder regime of pinning models. In particular, in this regime, we compute the first order term in the expansion of the free energy close to criticality and this term coincides with the first order of the formal expansion obtained by field theory methods. We also show that the quenched and quenched averaged correlation length exponents coincide, while, in general, they are expected to be different. Interpolation and replica coupling methods in this class of models naturally lead to studying the behavior of the intersection of certain renewal sequences and one of the main tools in this work is precisely renewal theory and the study of these intersection renewals.
Primary Subjects: 60K35, 60K37, 60K05, 82B41, 82B44
Keywords: Directed polymers; pinning and wetting models; renewal theory; irrelevant disorder; Harris criterion; intersection of renewals
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1253539858
Digital Object Identifier: doi:10.1214/09-AOP454
Zentralblatt MATH identifier:
05625054
References
[1] Aizenman, M. and Wehr, J. (1990). Rounding effects of quenched randomness on first-order phase transitions. Comm. Math. Phys. 130 489–528.
Mathematical Reviews (MathSciNet):
MR1060388
Zentralblatt MATH:
0714.60090
Digital Object Identifier: doi:10.1007/BF02096933
Project Euclid: euclid.cmp/1104200601
[2] Albeverio, S. and Zhou, X. Y. (1996). Free energy and some sample path properties of a random walk with random potential. J. Statist. Phys. 83 573–622.
Mathematical Reviews (MathSciNet):
MR1386352
Zentralblatt MATH:
1081.82559
Digital Object Identifier: doi:10.1007/BF02183741
[3] Alexander, K. S. (2008). The effect of disorder on polymer depinning transitions. Comm. Math. Phys. 279 117–146.
Mathematical Reviews (MathSciNet):
MR2377630
Zentralblatt MATH:
05306089
Digital Object Identifier: doi:10.1007/s00220-008-0425-5
[4] Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Applications of Mathematics (New York). Stochastic Modelling and Applied Probability 51. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1978607
[5] Bertoin, J. (1999). Intersection of independent regenerative sets. Probab. Theory Related Fields 114 97–121.
Mathematical Reviews (MathSciNet):
MR1697141
Zentralblatt MATH:
0937.60043
Digital Object Identifier: doi:10.1007/s004400050223
[6] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.
Mathematical Reviews (MathSciNet):
MR898871
[7] Bolthausen, E. and Sznitman, A.-S. (2002). Ten Lectures on Random Media. DMV Seminar 32. Birkhäuser, Basel.
Mathematical Reviews (MathSciNet):
MR1890289
Zentralblatt MATH:
1075.60128
[8] Bovier, A. (2006). Statistical Mechanics of Disordered Systems. A Mathematical Perspective. Cambridge Univ. Press, Cambridge.
Mathematical Reviews (MathSciNet):
MR2252929
Zentralblatt MATH:
1108.82002
[9] Bovier, A. and Külske, C. (1996). There are no nice interfaces in (2+1)-dimensional SOS models in random media. J. Statist. Phys. 83 751–759.
[10] Caravenna, F., Giacomin, G. and Zambotti, L. (2007). A renewal theory approach to periodic copolymers with adsorption. Ann. Appl. Probab. 17 1362–1398.
Mathematical Reviews (MathSciNet):
MR2344310
Zentralblatt MATH:
1136.82391
Digital Object Identifier: doi:10.1214/105051607000000159
Project Euclid: euclid.aoap/1186755243
[11] Chover, J., Ney, P. and Wainger, S. (1973). Functions of probability measures. J. Analyse Math. 26 255–302.
Mathematical Reviews (MathSciNet):
MR348393
Zentralblatt MATH:
0276.60018
Digital Object Identifier: doi:10.1007/BF02790433
[12] Derrida, B., Hakim, V. and Vannimenus, J. (1992). Effect of disorder on two-dimensional wetting. J. Statist. Phys. 66 1189–1213.
Mathematical Reviews (MathSciNet):
MR1156401
Zentralblatt MATH:
0900.82051
Digital Object Identifier: doi:10.1007/BF01054419
[13] Doney, R. A. (1997). One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Related Fields 107 451–465.
Mathematical Reviews (MathSciNet):
MR1440141
Zentralblatt MATH:
0883.60022
Digital Object Identifier: doi:10.1007/s004400050093
[14] Fisher, D. S. (1995). Critical behavior of random transverse-field Ising spin chains. Phys. Rev. B 51 6411–6461.
[15] Fisher, M. E. (1984). Walks, walls, wetting, and melting. J. Statist. Phys. 34 667–729.
Mathematical Reviews (MathSciNet):
MR751710
Zentralblatt MATH:
0589.60098
Digital Object Identifier: doi:10.1007/BF01009436
[16] Forgacs, G., Luck, J. M., Nieuwenhuizen, Th. M. and Orland, H. (1986). Wetting of a disordered substrate: Exact critical behavior in two dimensions. Phys. Rev. Lett. 57 2184–2187.
[17] Garsia, A. and Lamperti, J. (1962/1963). A discrete renewal theorem with infinite mean. Comment. Math. Helv. 37 221–234.
Mathematical Reviews (MathSciNet):
MR148121
Zentralblatt MATH:
0114.08803
Digital Object Identifier: doi:10.1007/BF02566974
[18] Giacomin, G. (2007). Random Polymer Models. Imperial College Press, London.
Mathematical Reviews (MathSciNet):
MR2380992
Zentralblatt MATH:
1125.82001
[19] Giacomin, G. (2008). Renewal convergence rates and correlation decay for homogeneous pinning models. Electron. J. Probab. 13 513–529.
Mathematical Reviews (MathSciNet):
MR2386741
[20] Giacomin, G., Lacoin, H. and Toninelli, F. L. (2008). Marginal relevance of disorder for pinning models. Commun. Pure Appl. Math. To appear.
[21] Giacomin, G. and Toninelli, F. L. (2006). The localized phase of disordered copolymers with adsorption. ALEA Lat. Am. J. Probab. Math. Stat. 1 149–180 (electronic).
Mathematical Reviews (MathSciNet):
MR2249653
Zentralblatt MATH:
1134.82006
[22] Giacomin, G. and Toninelli, F. L. (2006). Smoothing effect of quenched disorder on polymer depinning transitions. Comm. Math. Phys. 266 1–16.
Mathematical Reviews (MathSciNet):
MR2231963
Zentralblatt MATH:
1113.82032
Digital Object Identifier: doi:10.1007/s00220-006-0008-2
[23] Guerra, F. and Toninelli, F. L. (2002). Quadratic replica coupling in the Sherrington–Kirkpatrick mean field spin glass model. J. Math. Phys. 43 3704–3716.
[24] Imry, Y. and Ma, S.-K. (1975). Random-field instability of the ordered state of continuous symmetry. Phys. Rev. Lett. 35 1399–1401.
[25] Mézard, M. and Montanari, A. Information, Physics and Computation. Oxford Univ. Press. To appear.
Mathematical Reviews (MathSciNet):
MR2518205
[26] Monthus, C. and Garel, T. (2007). On the multifractal statistics of the local order parameter at random critical points: Application to wetting transitions with disorder. Phys. Rev. E 76 021114.
[27] Ney, P. (1981). A refinement of the coupling method in renewal theory. Stochastic Process. Appl. 11 11–26.
Mathematical Reviews (MathSciNet):
MR608004
Zentralblatt MATH:
0451.60082
Digital Object Identifier: doi:10.1016/0304-4149(81)90018-1
[28] Talagrand, M. (2003). Spin Glasses: A Challenge for Mathematicians: Cavity and Mean Field Models. Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 46. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR1993891
[29] Talagrand, M. (2007). Large deviations, Guerra’s and A.S.S. schemes, and the Parisi hypothesis. J. Stat. Phys. 126 837–894.
Mathematical Reviews (MathSciNet):
MR2311888
Zentralblatt MATH:
1133.82330
Digital Object Identifier: doi:10.1007/s10955-006-9108-9
[30] Toninelli, F. L. (2008). A replica-coupling approach to disordered pinning models. Comm. Math. Phys. 280 389–401.
Mathematical Reviews (MathSciNet):
MR2395475
Zentralblatt MATH:
05306107
Digital Object Identifier: doi:10.1007/s00220-008-0469-6
[31] Toninelli, F. L. (2007). Critical properties and finite-size estimates for the depinning transition of directed random polymers. J. Stat. Phys. 126 1025–1044.
Mathematical Reviews (MathSciNet):
MR2311896
Zentralblatt MATH:
1122.82053
Digital Object Identifier: doi:10.1007/s10955-006-9123-x
[32] Toninelli, F. L. (2007). Correlation lengths for random polymer models and for some renewal sequences. Electron. J. Probab. 12 613–636 (electronic).
Mathematical Reviews (MathSciNet):
MR2318404
Zentralblatt MATH:
1136.82013
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