×

zbMATH — the first resource for mathematics

Disorder relevance at marginality and critical point shift. (English. French summary) Zbl 1210.82036
Summary: Recently the renormalization group predictions on the effect of disorder on pinning models have been put on mathematical grounds. The picture is particularly complete if the disorder is relevant or irrelevant in the Harris criterion sense: the question addressed is whether quenched disorder leads to a critical behavior which is different from the one observed in the pure, i.e. annealed, system. The Harris criterion prediction is based on the sign of the specific heat exponent of the pure system, but it yields no prediction in the case of vanishing exponent. This case is called marginal, and the physical literature is divided on what one should observe for marginal disorder, notably there is no agreement on whether a small amount of disorder leads or not to a difference between the critical point of the quenched system and the one for the pure system. In [Commun. Pure Appl. Math. 63, No. 2, 233–265 (2010; Zbl 1189.60173)] we have proven that the two critical points differ at marginality of at least \(\exp(-c/\beta^4\)), where \(c>0\) and \(\beta^2\) is the disorder variance, for \(\beta \in (0, 1)\) and Gaussian IID disorder. The purpose of this paper is to improve such a result: we establish in particular that the \(\exp(-c/\beta^4\)) lower bound on the shift can be replaced by \(\exp(-c(b)/\beta^b)\), \(c(b)>0\) for \(b>2\) \((b=2\) is the known upper bound and it is the result claimed in [B. Derrida, V. Hakim and J. Vannimenus, J. Stat. Phys. 66, No. 5–6, 1189–1213 (1992; Zbl 0900.82051)]), and we deal with very general distribution of the IID disorder variables. The proof relies on coarse graining estimates and on a fractional moment change of measure argument based on multi-body potential modifications of the law of the disorder.

MSC:
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B27 Critical phenomena in equilibrium statistical mechanics
60K37 Processes in random environments
PDF BibTeX XML Cite
Full Text: DOI EuDML arXiv
References:
[1] K. S. Alexander. The effect of disorder on polymer depinning transitions. Comm. Math. Phys. 279 (2008) 117-146. · Zbl 1175.82034 · doi:10.1007/s00220-008-0425-5
[2] K. S. Alexander and N. Zygouras. Quenched and annealed critical points in polymer pinning models. Comm. Math. Phys. 291 (2009) 659-689. · Zbl 1188.82154 · doi:10.1007/s00220-009-0882-5
[3] K. S. Alexander and N. Zygouras. Equality of critical points for polymer depinning transitions with loop exponent one. Ann. Appl. Probab. 20 (2010) 356-366. · Zbl 1187.82054 · doi:10.1214/09-AAP621
[4] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation . Cambridge Univ. Press, Cambridge, 1987. · Zbl 0617.26001
[5] K. L. Chung and P. Erdös. Probability limit theorems assuming only the first moment I. Mem. Amer. Math. Soc. 6 (1951) 1-19. · Zbl 0042.37601
[6] F. Comets and N. Yoshida. Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 (2006) 1746-1770. · Zbl 1104.60061 · doi:10.1214/009117905000000828
[7] B. Derrida, G. Giacomin, H. Lacoin and F. L. Toninelli. Fractional moment bounds and disorder relevance for pinning models. Comm. Math. Phys. 287 (2009) 867-887. · Zbl 1226.82028 · doi:10.1007/s00220-009-0737-0
[8] B. Derrida, V. Hakim and J. Vannimenus. Effect of disorder on two-dimensional wetting. J. Stat. Phys. 66 (1992) 1189-1213. · Zbl 0900.82051 · doi:10.1007/BF01054419
[9] R. A. Doney. One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Related Fields 107 (1997) 451-465. · Zbl 0883.60022 · doi:10.1007/s004400050093
[10] M. E. Fisher. Walks, walls, wetting, and melting. J. Stat. Phys. 34 (1984) 667-729. · Zbl 0589.60098 · doi:10.1007/BF01009436
[11] G. Forgacs, J. M. Luck, T. M. Nieuwenhuizen and H. Orland. Wetting of a disordered substrate: Exact critical behavior in two dimensions. Phys. Rev. Lett. 57 (1986) 2184-2187.
[12] G. Giacomin. Random Polymer Models . Imperial College Press, London, 2007. · Zbl 1125.82001
[13] G. Giacomin. Renewal sequences, disordered potentials, and pinning phenomena. In Spin Glasses: Statics and Dynamics 235-270. Progress in Probability 62 . Birkhäuser, Basel, 2009. · Zbl 1194.82042
[14] G. Giacomin, H. Lacoin and F. L. Toninelli. Marginal relevance of disorder for pinning models. Comm. Pure Appl. Math. 63 (2010) 233-265. · Zbl 1189.60173 · doi:10.1002/cpa.20301
[15] G. Giacomin and F. L. Toninelli. Smoothing effect of quenched disorder on polymer depinning transitions. Comm. Math. Phys. 266 (2006) 1-16; Smoothing of depinning transitions for directed polymers with quenched disorder. Phys. Rev. Lett. 96 (2006) 070602. · Zbl 1113.82032 · doi:10.1007/s00220-006-0008-2
[16] G. Giacomin and F. L. Toninelli. On the irrelevant disorder regime of pinning models. Ann. Probab. 37 (2009) 1841-1873. · Zbl 1181.60148 · doi:10.1214/09-AOP454
[17] A. B. Harris. Effect of random defects on the critical behaviour of ising models. J. Phys. C 7 (1974) 1671-1692.
[18] H. Lacoin. Hierarchical pinning model with site disorder: Disorder is marginally relevant. Probab. Theory Related Fields . To appear. Available at . · Zbl 1201.60095 · doi:10.1007/s00440-009-0226-6 · arxiv.org
[19] F. L. Toninelli. A replica-coupling approach to disordered pinning models. Comm. Math. Phys. 280 (2008) 389-401. · Zbl 1207.82026 · doi:10.1007/s00220-008-0469-6
[20] F. L. Toninelli. Coarse graining, fractional moments and the critical slope of random copolymers. Electron. J. Probab. 14 (2009) 531-547. · Zbl 1189.60186 · emis:journals/EJP-ECP/_ejpecp/viewarticle0a4c.html · eudml:227069
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.