Abstract
This chapter is devoted to showing that, when α ≥ 1 ∕ 2, quenched and annealed critical points are different for every β > 0, with explicit estimates on the difference. Such a result follows from upper bounds on the free energy that are obtained by estimating fractional moments (of order less than one) of the partition function. Estimates for every β > 0, notably for arbitrarily small values of β, are obtained by using a change of measure argument on the law of the disorder and by coarse graining techniques. Proving such estimates becomes harder and harder as α approaches 1 ∕ 2, i.e. the marginal disorder case in the Harris’ sense: for α = 1 ∕ 2 the Harris criterion yields no prediction and whether quenched and annealed critical points differed or not has been a debated issue in the physical literature.
Keywords
- Partition Function
- Correlation Length
- Coarse Graining
- Random Environment
- Large Deviation Principle
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
K.S. Alexander, The effect of disorder on polymer depinning transitions. Commun. Math. Phys. 279, 117–146 (2008)
K.S. Alexander, N. Zygouras, Quenched and annealed critical points in polymer pinning models. Commun. Math. Phys. 291, 659–689 (2009)
M. Aizenman, S. Molchanov, Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245–278 (1993)
Q. Berger, H. Lacoin, The effect of disorder on the free-energy for the random walk pinning model: smoothing of the phase transition and low temperature asymptotics. J. Stat. Phys. 42, 322–341 (2011)
Q. Berger, F.L. Toninelli, On the critical point of the random walk pinning model in dimension d = 3. Electron. J. Probab. 15, 654–683 (2010)
S.M. Bhattacharjee, S. Mukherji, Directed polymers with random interaction: marginal relevance and novel criticality. Phys. Rev. Lett. 70, 49–52 (1993)
N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation (Cambridge University Press, Cambridge, 1987)
M. Birkner, R. Sun, Annealed vs quenched critical points for a random walk pinning model. Ann. Inst. H. Poincaré (B) Probab. Stat. 46, 414–441 (2010)
M. Birkner, R. Sun, Disorder relevance for the random walk pinning model in dimension 3. Ann. Inst. H. Poincaré (B) Probab. Stat. 47, 259–293 (2011)
M. Birkner, A. Greven, F. den Hollander, Quenched large deviation principle for words in a letter sequence. Probab. Theory Relat. Fields 148, 403–456 (2010)
T. Bodineau, G. Giacomin, H. Lacoin, F.L. Toninelli, Copolymers at selective interfaces: new bounds on the phase diagram. J. Stat. Phys. 132, 603–626 (2008)
F. Caravenna, G. Giacomin, On constrained annealed bounds for pinning and wetting models. Electron. Commun. Probab. 10, 179–189 (2005)
D. Cheliotis, F. den Hollander, Variational characterization of the critical curve for pinning of random polymers. arXiv:1005.3661
B. Derrida, G. Giacomin, H. Lacoin, F.L. Toninelli, Fractional moment bounds and disorder relevance for pinning models. Commun. Math. Phys. 287, 867–887 (2009)
B. Derrida, V. Hakim, J. Vannimenus, Effect of disorder on two-dimensional wetting. J. Stat. Phys. 66, 1189–1213 (1992)
G. Forgacs, J.M. Luck, Th. M. Nieuwenhuizen, H. Orland, Wetting of a disordered substrate: exact critical behavior in two dimensions. Phys. Rev. Lett. 57, 2184–2187 (1986)
D.M. Gangardt, S.K. Nechaev, Wetting transition on a one-dimensional disorder. J. Stat. Phys. 130, 483–502 (2008)
G. Giacomin, H. Lacoin, F.L. Toninelli, Hierarchical pinning models, quadratic maps and quenched disorder. Probab. Theory Relat. Fields 147, 185–216 (2010)
G. Giacomin, H. Lacoin, F.L. Toninelli, Marginal relevance of disorder for pinning models. Commun. Pure Appl. Math. 63, 233–265 (2010)
G. Giacomin, H. Lacoin, F.L. Toninelli, Disorder relevance at marginality and critical point shift. Ann. Inst. H. Poincaré (B) Probab. Stat. 47, 148–175 (2011)
A.Y. Grosberg, E.I. Shakhnovich, An investigation of the configurational statistics of a polymer chain in an external field by the dynamical renormalization group method. Sov. Phys. JETP 64, 493–501 (1986)
A.Y. Grosberg, E.I. Shakhnovich, Theory of phase transitions of the coil-globule type in a heteropolymer chain with disordered sequence of links. Sov. Phys. JETP 64, 1284–1290 (1986)
A.B. Harris, Effect of random defects on the critical behaviour of Ising models. J. Phys. C 7, 1671–1692 (1974)
H. Lacoin, Hierarchical pinning model with site disorder: disorder is marginally relevant. Probab. Theory Relat. Fields 148, 159–175 (2010)
H. Lacoin, New bounds for the free energy of directed polymer in dimension 1+1 and 1+2. Commun. Math. Phys. 294, 471–503 (2010)
H. Lacoin, Influence of spatial correlation for directed polymers. Ann. Probab. 39, 139–175 (2011)
T. Morita, Statistical mechanics of quenched solid solutions with application to magnetically dilute alloys. J. Math. Phys. 5, 1401–1405 (1966)
S. Stepanow, A.L. Chudnovskiy, The Green’s function approach to adsorption of a random heteropolymer onto surfaces. J. Phys. A Math. Gen. 35, 4229–4238 (2002)
L.-H. Tang, H. Chaté, Rare-event induced binding transition of heteropolymers. Phys. Rev. Lett. 86, 830–833 (2001)
F.L. Toninelli, A replica-coupling approach to disordered pinning models. Commun. Math. Phys. 280, 389–401 (2008)
F.L. Toninelli, Disordered pinning models and copolymers: beyond annealed bounds. Ann. Appl. Probab. 18, 1569–1587 (2008)
F.L. Toninelli, Coarse graining, fractional moments and the critical slope of random copolymers. Electron. J. Probab. 14, 531–547 (2009)
A. Yilmaz, O. Zeitouni, Differing averaged and quenched large deviations for random walks in random environments in dimensions two and three. Commun. Math. Phys. (to appear). arXiv:0910.1169
N. Zygouras, Strong disorder in semidirected random polymers. arXiv:1009.2693
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Giacomin, G. (2011). Critical Point Shift: The Fractional Moment Method. In: Disorder and Critical Phenomena Through Basic Probability Models. Lecture Notes in Mathematics(), vol 2025. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21156-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-21156-0_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21155-3
Online ISBN: 978-3-642-21156-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
