Cheliotis, Dimitris; Chino, Yuki; Poisat, Julien
The random pinning model with correlated disorder given by a renewal set
Annales Henri Lebesgue, Volume 2 (2019), p. 281-329

KeywordsPinning model, localization transition, free energy, correlated disorder, renewal, disorder relevance, Harris criterion, smoothing inequality, second moment.

Abstract

We investigate the effect of correlated disorder on the localization transition undergone by a renewal sequence with loop exponent $\alpha >0$, when the correlated sequence is given by another independent renewal set with loop exponent $\stackrel{^}{\alpha }>0$. Using the renewal structure of the disorder sequence, we compute the annealed critical point and exponent. Then, using a smoothing inequality for the quenched free energy and second moment estimates for the quenched partition function, combined with decoupling inequalities, we prove that in the case $\stackrel{^}{\alpha }>2$ (summable correlations), disorder is irrelevant if $\alpha <1/2$ and relevant if $\alpha >1/2$, which extends the Harris criterion for independent disorder. The case $\stackrel{^}{\alpha }\in \left(1,2\right)$ (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for $\alpha >1/\stackrel{^}{\alpha }$, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case $\stackrel{^}{\alpha }\in \left(0,1\right)$ is somewhat special but treated for completeness: in this case, disorder has no effect on the quenched free energy, but the annealed model exhibits a phase transition.

References

[AB18] Alexander, Kenneth S.; Berger, Quentin Pinning of a renewal on a quenched renewal, Electron. J. Probab., Volume 23 (2018), 6, 48 pages | MR 3771743 | Zbl 1390.60341

[AS06] Alexander, Kenneth S.; Sidoravicius, Vladas Pinning of polymers and interfaces by random potentials, Ann. Appl. Probab., Volume 16 (2006) no. 2, pp. 636-669 | Article | MR 2244428 | Zbl 1145.82010

[Asm03] Asmussen, Søren Applied probability and queues, Springer, Applications of Mathematics, Volume 51 (2003), xii+438 pages | MR 1978607 | Zbl 1029.60001

[AZ09] Alexander, Kenneth S.; Zygouras, Nikos Quenched and annealed critical points in polymer pinning models, Commun. Math. Phys., Volume 291 (2009) no. 3, pp. 659-689 | MR 2534789 | Zbl 1188.82154

[BCP + 14] Berger, Quentin; Caravenna, Francesco; Poisat, Julien; Sun, Rongfeng; Zygouras, Nikos The critical curve of the random pinning and copolymer models at weak coupling, Commun. Math. Phys., Volume 326 (2014) no. 2, pp. 507-530 | Article | MR 3165465 | Zbl 1320.82074

[Ber13] Berger, Quentin Comments on the influence of disorder for pinning model in correlated Gaussian environment, ALEA, Lat. Am. J. Probab. Math. Stat., Volume 10 (2013) no. 2, pp. 953-977 | MR 3151746 | Zbl 1282.82030

[Ber14] Berger, Quentin Pinning model in random correlated environment: appearance of an infinite disorder regime, J. Stat. Phys., Volume 155 (2014) no. 3, pp. 544-570 | Article | MR 3192173 | Zbl 1297.82043

[BGT89] Bingham, Nicholas H.; Goldie, Charles M.; Teugels, Jozef L. Regular variation, Cambridge University Press, Encyclopedia of Mathematics and Its Applications, Volume 27 (1989), 512 pages | MR 1015093 | Zbl 0667.26003

[BL12] Berger, Quentin; Lacoin, Hubert Sharp critical behavior for pinning models in a random correlated environment, Stochastic Processes Appl., Volume 122 (2012) no. 4, pp. 1397-1436 | MR 2914757 | Zbl 1266.82080

[BL16] Berger, Quentin; Lacoin, Hubert Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift, J. Inst. Math. Jussieu, Volume 17 (2016) no. 2, pp. 205-346 | MR 3773271 | Zbl 1405.60139

[BP15] Berger, Quentin; Poisat, Julien On the critical curve of the pinning and copolymer models in correlated Gaussian environment, Electron. J. Probab., Volume 20 (2015), 71, 35 pages | Article | MR 3361259 | Zbl 1323.82022

[CdH13a] Caravenna, Francesco; den Hollander, Frank A general smoothing inequality for disordered polymers, Electron. Commun. Probab., Volume 18 (2013), 76, 15 pages | MR 3109631 | Zbl 1329.60329

[CdH13b] Cheliotis, Dimitris; den Hollander, Frank Variational characterization of the critical curve for pinning of random polymers, Ann. Probab., Volume 41 (2013) no. 3B, pp. 1767-1805 | Article | MR 3098058 | Zbl 1281.60083

[CSZ16] Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos The continuum disordered pinning model, Probab. Theory Relat. Fields, Volume 164 (2016) no. 1-2, pp. 17-59 | Article | MR 3449385 | Zbl 1341.82040

[CSZ17] Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos Polynomial chaos and scaling limits of disordered systems, J. Eur. Math. Soc., Volume 19 (2017) no. 1, pp. 1-65 | Article | MR 3584558 | Zbl 1364.82026

[CTT17] Caravenna, Francesco; Toninelli, Fabio Lucio; Torri, Niccoló Universality for the pinning model in the weak coupling regime, Ann. Probab., Volume 45 (2017) no. 4, pp. 2154-2209 | Article | MR 3693960 | Zbl 1376.82036

[DGLT09] Derrida, Bernard; Giacomin, Giambattista; Lacoin, Hubert; Toninelli, Fabio Lucio Fractional moment bounds and disorder relevance for pinning models, Commun. Math. Phys., Volume 287 (2009) no. 3, pp. 867-887 | MR 2486665 | Zbl 1226.82028

[dH09] den Hollander, Frank Random Polymer models: École d’Été de Probabilités de Saint-Flour XXXVII-2007, Springer, Lecture Notes in Mathematics, Volume 1974 (2009), xiii+258 pages | Zbl 1173.82002

[Dur10] Durrett, Rick Probability: Theory and examples, Cambridge University Press, Cambridge Series in Statistical and Probabilistic Mathematics, Volume 31 (2010) | MR 2722836 | Zbl 1202.60001

[Fel68] Feller, William An introduction to probability and its applications. Vol. 1, John Wiley & Sons (1968), xviii+509 pages | Zbl 0155.23101

[Fel71] Feller, William An introduction to probability and its applications. Vol. 2, John Wiley & Sons (1971), xxiv+669 pages | Zbl 0219.60003

[Fre82] Frenk, Johannes B. G. The behaviour of the renewal sequence in case the tail of the waiting-time distribution is regularly varying with index $-1$, Adv. Appl. Probab., Volume 14 (1982) no. 4, pp. 870-874 | Article | Zbl 0508.60070

[GHM99] Georgii, Hans-Otto; Häggström, Olle; Maes, Christian The random geometry of equilibrium phases, Academic Press Inc. (Phase transitions and critical phenomena) Volume 18 (1999), pp. 1-142 | Article

[Gia07] Giacomin, Giambattista Random polymer models, World Scientific (2007), xvi+242 pages | MR 2380992 | Zbl 1125.82001

[Gia11] Giacomin, Giambattista Disorder and Critical Phenomena Through Basic Probability Models. École d’Été de Probabilités de Saint-Flour XL-2010, Springer, Lecture Notes in Mathematics, Volume 2025 (2011), xi+130 pages | MR 2816225 | Zbl 1230.82004

[GL62] Garsia, Adriano; Lamperti, John A discrete renewal theorem with infinite mean, Comment. Math. Helv., Volume 37 (1962), pp. 221-234 | Article | MR 148121 | Zbl 0114.08803

[GT06] Giacomin, Giambattista; Toninelli, Fabio Lucio Smoothing effect of quenched disorder on polymer depinning transitions, Commun. Math. Phys., Volume 266 (2006) no. 1, pp. 1-16 | Article | MR 2231963 | Zbl 1113.82032

[GTL10] Giacomin, Giambattista; Toninelli, Fabio Lucio; Lacoin, Hubert Marginal relevance of disorder for pinning models, Commun. Pure Appl. Math., Volume 63 (2010) no. 2, pp. 233-265 | Article | MR 2588461 | Zbl 1189.60173

[Har74] Harris, A. B. Effect of random defects on the critical behaviour of Ising model, J. Phys. C, Solid State Phys., Volume 7 (1974) no. 9, pp. 1671-1692 | Article

[JPS06] Jeon, Jae-Hyung; Park, Pyeong Jun; Sung, Wokyung The effect of sequence correlation on bubble statistics in double-stranded DNA, J. Chem. Phys., Volume 125 (2006), 164901 | Article

[Lac10] Lacoin, Hubert The martingale approach to disorder irrelevance for pinning models, Electron. Commun. Probab., Volume 15 (2010), pp. 418-427 | Article | MR 2726088 | Zbl 1221.82058

[LS17] Lacoin, Hubert; Sohier, Julien Disorder relevance without Harris Criterion: the case of pinning model with $\gamma$-stable environment, Electron. J. Probab., Volume 22 (2017), 50, 26 pages | MR 3666013 | Zbl 1368.60104

[Nag12] Nagaev, Sergey V. The renewal theorem in the absence of power moments, Theory Probab. Appl., Volume 56 (2012) no. 1, pp. 166-175 | Zbl 1238.60094

[Poi13a] Poisat, Julien On quenched and annealed critical curves of random pinning model with finite range correlations, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 49 (2013), pp. 456-482 | Article | MR 3088377 | Zbl 1276.82024

[Poi13b] Poisat, Julien Ruelle-Perron-Frobenius operator approach to the annealed pinning model with Gaussian long-range correlated disorder, Markov Process. Relat. Fields, Volume 19 (2013) no. 3, pp. 577-606 | MR 3156968 | Zbl 1321.82021

[Spi13] Spitzer, Frank Principles of random walk, Springer, Graduate Texts in Mathematics, Volume 34 (2013) | Zbl 0979.60002

[ST94] Samorodnitsky, Gennady; Taqqu, Murad S. Stable non-Gaussian random processes, Chapman & Hall, Stochastic Modeling (1994), xviii+632 pages | Zbl 0925.60027

[Ton08] Toninelli, Fabio Lucio A replica-coupling approach to disordered pinning models, Commun. Math. Phys., Volume 280 (2008) no. 2, pp. 389-401 | MR 2395475 | Zbl 1207.82026

[WH83] Weinrib, Abel; Halperin, Bertrand I. Critical phenomena in systems with long-range-correlated disorder, Phys. Rev. B, Volume 27 (1983), pp. 413-427 | Article