Electronic Journal of Probability

Disorder relevance without Harris Criterion: the case of pinning model with $\gamma $-stable environment

Hubert Lacoin and Julien Sohier

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Abstract

We investigate disorder relevance for the pinning of a renewal whose inter-arrival law has tail exponent $\alpha >0$ when the law of the random environment is in the domain of attraction of a stable law with parameter $\gamma \in (1,2)$. We prove that in this case, the effect of disorder is not decided by the sign of the specific heat exponent as predicted by Harris criterion but that a new criterion emerges to decide disorder relevance. More precisely we show that when $\alpha >1-\gamma ^{-1}$ there is a shift of the critical point at every temperature whereas when $\alpha < 1-\gamma ^{-1}$, at high temperature the quenched and annealed critical points coincide, and the critical exponents are identical.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 50, 26 pp.

Dates
Received: 21 October 2016
Accepted: 4 May 2017
First available in Project Euclid: 13 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1497319467

Digital Object Identifier
doi:10.1214/17-EJP66

Mathematical Reviews number (MathSciNet)
MR3666013

Zentralblatt MATH identifier
1368.60104

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 82B27: Critical phenomena 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Pinning model disorder relevance stable laws Harris criterion

Rights
Creative Commons Attribution 4.0 International License.

Citation

Lacoin, Hubert; Sohier, Julien. Disorder relevance without Harris Criterion: the case of pinning model with $\gamma $-stable environment. Electron. J. Probab. 22 (2017), paper no. 50, 26 pp. doi:10.1214/17-EJP66. https://projecteuclid.org/euclid.ejp/1497319467


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References

  • [1] M. Aizenman and J. Wehr, Rounding effects of quenched randomness on first-order phase transitions, Comm. Math. Phys. 130 (1990) 489–528.
  • [2] T. Alberts, K. Khanin and J. Quastel, The intermediate disorder regime for directed polymers in dimension 1+1, Ann. Probab. 42 (2014) 1212–1256.
  • [3] S. Albeverio and X. Y. Zhou, Free energy and some sample path properties of a random walk with random potential, J. Statist. Phys. 83 (1996) 573–622.
  • [4] K. S. Alexander, The effect of disorder on polymer depinning transitions, Commun. Math. Phys. 279 (2008), 117–146.
  • [5] K.S. Alexander and V. Sidoravicius, Pinning of polymers and interfaces by random potentials, Ann. Appl. Probab. 16 (2006) 636–669.
  • [6] K.S. Alexander and N. Zygouras, Quenched and annealed critical points in polymer pinning models, Comm. Math. Phys. 291 (2009) 659–689.
  • [7] Q. Berger, F. Caravenna, J. Poisat, R. Sun and N. Zygouras, The Critical Curves of the Random Pinning and Copolymer Models at Weak Coupling, Comm. Math. Phys. 326 (2014) 507–530.
  • [8] Q. Berger and H. Lacoin, Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift, J. Inst. Math. Jussieu, Firstview (2016) 1–42.
  • [9] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variations, Cambridge University Press (1987).
  • [10] Birkner M., A Condition for Weak Disorder for Directed Polymers in Random Environment, Elec. Comm. Probab. 9 (2004) 22–25.
  • [11] F. Caravenna, R. Sun and N. Zygouras, The continuum disordered pinning model Prob. Theor. Relat. Fields. 164 (2016) 17–59.
  • [12] F. Caravenna, R. Sun and N. Zygouras, Polynomial chaos and scaling limits of disordered systems J. Eur. Math. Soc. 19 (2017) 1–65.
  • [13] F. Caravenna, F. Toninelli, N. Torri, Universality for the pinning model in the weak coupling regime, to appear on Ann. Probab.
  • [14] F. Caravena and F. Den Hollander, A general smoothing inequality for disordered polymers, Elec. Comm. Probab. 18 (2013) Article 76.
  • [15] 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.
  • [16] B. Derrida, V. Hakim and J. Vannimenus, Effect of disorder on two-dimensional wetting, J. Statist. Phys. 66 (1992) 1189–1213.
  • [17] R. A. Doney, One-sided local large deviation and renewal theorems in the case of infinite mean, Probab. Theory Relat. Fields 107 (1997) 451–465.
  • [18] W. Feller, An introduction to probability theory and its applications. Vol. II, New York–London–Sydney, John Wiley and Sons, 1966.
  • [19] G. Giacomin, Random polymer models, Imperial College Press, World Scientific (2007).
  • [20] G. Giacomin, Disorder and critical phenomena through basic probability models, École d’Été de Probabilités de Saint-Flour XL 2010, Springer Lecture Notes in Mathematics 2025 (2011).
  • [21] G. Giacomin, H. Lacoin and F. L. Toninelli, Hierarchical pinning models, quadratic maps and quenched disorder, Probab. Theor. Rel. Fields 147 (2010) 185–216.
  • [22] G. Giacomin, H. Lacoin and F. L. Toninelli, Marginal relevance of disorder for pinning models, Commun. Pure Appl. Math. 63 (2010) 233–265.
  • [23] G. Giacomin and F. L. Toninelli, Smoothing effect of quenched disorder on polymer depinning transitions, Commun. Math. Phys. 266 (2006) 1–16.
  • [24] G. Giacomin and F. L. Toninelli, The localized phase of disordered copolymers with adsorption, ALEA 1 (2006) 149–180.
  • [25] A. B. Harris, Effect of Random Defects on the Critical Behaviour of Ising Models, J. Phys. C 7 (1974) 1671–1692.
  • [26] F. den Hollander, Random Polymers, École d’ Été de Probabilités de Saint-Flour XXXVII, 2007 Springer Lecture Notes in Mathematics 1974 (2009).
  • [27] Y. Imri and S-k Ma, Random-Field Instability of the Ordered State of Continuous Symmetry, Phys. Rev. Lett. 35 (1975) 1399.
  • [28] H. Lacoin, The martingale approach to disorder irrelevance for pinning models, Elec. Comm. Probab. 15 (2010) 418–427.
  • [29] H. Lacoin, Marginal relevance for the $\gamma $-stable pinning (preprint) arXiv:1612.02389.
  • [30] R. Rhodes, J. Sohier, and V. Vargas, Levy multiplicative chaos and star scale invariant random measures, Ann. Probab. 42 (2014) 689–724.
  • [31] A. Shamov, On Gaussian multiplicative chaos, J. Func. Anal. 270 (2016) 3224–3261.
  • [32] F. L. Toninelli, A replica-coupling approach to disordered pinning models, Commun. Math. Phys. 280 (2008) 389–401.
  • [33] F.L. Toninelli, Coarse graining, fractional moments and the critical slope of random copolymers, Electron. Journal Probab. 14 (2009), 531–547.
  • [34] Z. Shi, Branching Random Walks, École d’Été de Probabilités de Saint-Flour XLII – 2012, Lecture Notes in Mathematics 2151 Springer 2015.