### Metadata

### 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 $\widehat{\alpha}>0$. 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 $\widehat{\alpha}>2$ (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 $\widehat{\alpha}\in (1,2)$ (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for $\alpha >1/\widehat{\alpha}$, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case $\widehat{\alpha}\in (0,1)$ 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] Pinning of a renewal on a quenched renewal, Electron. J. Probab., Volume 23 (2018), 6, 48 pages | Zbl 1390.60341

[AS06] Pinning of polymers and interfaces by random potentials, Ann. Appl. Probab., Volume 16 (2006) no. 2, pp. 636-669 | Zbl 1145.82010

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

[AZ09] Quenched and annealed critical points in polymer pinning models, Commun. Math. Phys., Volume 291 (2009) no. 3, pp. 659-689 | Zbl 1188.82154

[BCP + 14] 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

[Ber13] 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 | Zbl 1282.82030

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

[BGT89] Regular variation, Cambridge University Press, Encyclopedia of Mathematics and Its Applications, Volume 27 (1989), 512 pages | Zbl 0667.26003

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

[BL16] 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 | Zbl 1405.60139

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

[CdH13a] A general smoothing inequality for disordered polymers, Electron. Commun. Probab., Volume 18 (2013), 76, 15 pages | Zbl 1329.60329

[CdH13b] Variational characterization of the critical curve for pinning of random polymers, Ann. Probab., Volume 41 (2013) no. 3B, pp. 1767-1805 | Zbl 1281.60083

[CSZ16] The continuum disordered pinning model, Probab. Theory Relat. Fields, Volume 164 (2016) no. 1-2, pp. 17-59 | Zbl 1341.82040

[CSZ17] Polynomial chaos and scaling limits of disordered systems, J. Eur. Math. Soc., Volume 19 (2017) no. 1, pp. 1-65 | Zbl 1364.82026

[CTT17] Universality for the pinning model in the weak coupling regime, Ann. Probab., Volume 45 (2017) no. 4, pp. 2154-2209 | Zbl 1376.82036

[DGLT09] Fractional moment bounds and disorder relevance for pinning models, Commun. Math. Phys., Volume 287 (2009) no. 3, pp. 867-887 | Zbl 1226.82028

[dH09] 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] Probability: Theory and examples, Cambridge University Press, Cambridge Series in Statistical and Probabilistic Mathematics, Volume 31 (2010) | Zbl 1202.60001

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

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

[Fre82] 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

[GHM99] The random geometry of equilibrium phases, Academic Press Inc. (Phase transitions and critical phenomena) Volume 18 (1999), pp. 1-142 | Article

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

[Gia11] 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 | Zbl 1230.82004

[GL62] A discrete renewal theorem with infinite mean, Comment. Math. Helv., Volume 37 (1962), pp. 221-234 | Zbl 0114.08803

[GT06] Smoothing effect of quenched disorder on polymer depinning transitions, Commun. Math. Phys., Volume 266 (2006) no. 1, pp. 1-16 | Article | Zbl 1113.82032

[GTL10] Marginal relevance of disorder for pinning models, Commun. Pure Appl. Math., Volume 63 (2010) no. 2, pp. 233-265 | Article | Zbl 1189.60173

[Har74] 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

[JPS06] The effect of sequence correlation on bubble statistics in double-stranded DNA, J. Chem. Phys., Volume 125 (2006), 164901 | Article

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

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

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

[Poi13a] 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 | Zbl 1276.82024

[Poi13b] 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 | Zbl 1321.82021

[Spi13] Principles of random walk, Springer, Graduate Texts in Mathematics, Volume 34 (2013)

[ST94] Stable non-Gaussian random processes, Chapman & Hall, Stochastic Modeling (1994), xviii+632 pages | Zbl 0925.60027

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

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