×

zbMATH — the first resource for mathematics

Marginal relevance of disorder for pinning models. (English) Zbl 1189.60173
The effect of disorder on pinning and wetting models is discussed. Let \(\alpha\) be the return probability exponent. The authors prove disorder relevance both for the general \(\alpha={1\over 2}\) pinning model and for the hierarchical pinning model proposed by Derrida, Hakim, and Nannimenus, in the sense that a shift of the quenched critical point with respect to the annealed one. In both cases the Gaussian disorder used and shown that the shift is at least of order \(\exp(-{1\over \beta^4})\) for \(\beta\) small, if \(\beta^2\) is the disorder variance.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F17 Functional limit theorems; invariance principles
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Abraham, Phase transitions and critical phenomena pp 1– (1986)
[2] Aizenman, Localization at large disorder and at extreme energies: an elementary derivation, Comm Math Phys 157 (2) pp 245– (1993) · Zbl 0782.60044
[3] Alexander, The effect of disorder on polymer depinning transitions, Comm Math Phys 279 (1) pp 117– (2008) · Zbl 1175.82034
[4] Alexander, K. S.; Zygouras, N. Equality of critical points for polymer depinning transitions with loop exponent one. Preprint. arXiv: 0811.1902, 2008. · Zbl 1187.82054
[5] Alexander, K. S.; Zygouras, N. Quenched and annealed critical points in polymer pinning models. Preprint. arXiv: 0805.1708, 2008. · Zbl 1188.82154
[6] Bhattacharjee, Directed polymers with random interaction: marginal relevance and novel criticality, Phys Rev Lett 70 pp 49– (1993)
[7] Buffet, Directed polymers on trees: a martingale approach, J Phys A 26 (8) pp 1823– (1993) · Zbl 0773.60035
[8] Chung, Probability limit theorems assuming only the first moment. I, Mem Am Math Soc 1951 (6) pp 1– (1951)
[9] Derrida, Renormalisation group study of a disordered model, J Phys A 17 (16) pp 3223– (1984)
[10] Derrida, Fractional moment bounds and disorder relevance for pinning models, Comm Math Phys 287 (3) pp 867– (2009) · Zbl 1226.82028
[11] Derrida, Effect of disorder on two-dimensional wetting, J Statist Phys 66 (5) pp 1189– (1992) · Zbl 0900.82051
[12] Doney, One-sided local large deviation and renewal theorems in the case of infinite mean, Probab Theory Related Fields 107 (4) pp 451– (1997) · Zbl 0883.60022
[13] Feller, An introduction to probability theory and its applications (1971) · Zbl 0219.60003
[14] Fisher, Walks, walls, wetting, and melting, J Statist Phys 34 (5) pp 667– (1984) · Zbl 0589.60098
[15] Forg??cs, Phase transitions and critical phenomena pp 135– (1991)
[16] Forg??cs, Wetting of a disordered substrate: exact critical behavior in two dimensions, Phys Rev Lett 57 pp 2184– (1986)
[17] Gangardt, Wetting transition on a one-dimensional disorder, J Stat Phys 130 (3) pp 483– (2008) · Zbl 1139.82311
[18] Giacomin, Random polymer models (2007) · Zbl 1125.82001 · doi:10.1142/p504
[19] Giacomin, Hierarchical pinning models, quadratic maps and quenched disorder, Probab Theory Related Fields · Zbl 1190.60093
[20] Giacomin, G.; Lacoin, H.; Toninelli, F. L. Disorder relevance at marginality and critical point shift. Preprint. arXiv: 0906.1942 [math-ph], 2009. · Zbl 1210.82036
[21] Giacomin, Smoothing effect of quenched disorder on polymer depinning transitions, Comm Math Phys 266 (1) pp 1– (2006) · Zbl 1113.82032
[22] Giacomin, Smoothing of depinning transitions for directed polymers with quenched disorder, Phys Rev Lett 96 (7) (2006) · Zbl 1113.82032
[23] Grosberg, An investigation of the configurational statistics of a polymer chain in an external field by the dynamical renormalization group method, Soviet Phys JETP 64 (3) pp 493– (1986)
[24] Grosberg, Theory of phase transitions of the coil-globule type in a heteropolymer chain with disordered sequence of links, Soviet Phys JETP 64 (6) pp 1284– (1986)
[25] Harris, Effect of random defects on the critical behaviour of Ising models, J Phys C 7 pp 1671– (1974)
[26] Harris, The theory of branching processes (1963) · Zbl 0117.13002 · doi:10.1007/978-3-642-51866-9
[27] Lacoin, Hierarchical pinning model with site disorder: disorder is marginally relevant, Probab Theory Related Fields · Zbl 1201.60095
[28] Lacoin, Proceedings of the Summer School ???Spin glasses??? (Paris, June 2007)
[29] Available online at: http://people.math.jussieu.fr/???lacoin/spinglass.pdf
[30] Stepanow, The Green’s function approach to adsorption of a random heteropolymer onto surfaces, J Phys A 35 pp 4229– (2002) · Zbl 1053.82016
[31] Tang, Rare-event induced binding transition of heteropolymers, Phys Rev Lett 86 pp 830– (2001)
[32] Toninelli, Disordered pinning models and copolymers: beyond annealed bounds, Ann Appl Probab 18 (4) pp 1569– (2008) · Zbl 1157.60090
[33] Toninelli, A replica-coupling approach to disordered pinning models, Comm Math Phys 280 (2) pp 389– (2008) · Zbl 1207.82026
[34] Toninelli, Coarse graining, fractional moments and the critical slope of random copolymers, Electron J Probab 14 (20) pp 531– (2009) · Zbl 1189.60186 · doi:10.1214/EJP.v14-612
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.