Path properties of the disordered pinning model in the delocalized regime

Alexander, Kenneth S.; Zygouras, Nikos

doi:10.1214/13-aap930

Cited by 7 publications

(7 citation statements)

References 14 publications

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“…The analogous question even in the one dimensional set-up is not trivial even if by now rather sharp results are available [4]. Precise path description in the localized phase raises a number of issues too, in particular there are all the issues that have been treated, not always with complete success, in the one dimensional set-up (see [28,Ch.…”

Section: 4mentioning

confidence: 99%

Pinning and disorder relevance for the lattice Gaussian free field

Giacomin

Lacoin

2017

J. Eur. Math. Soc.

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Abstract. This paper provides a rigorous study of the localization transition for a Gaussian free field on Z d interacting with a quenched disordered substrate that acts on the interface when its height is close to zero. The substrate has the tendency to localize or repel the interface at different sites and one can show that a localization-delocalization transition takes place when varying the average pinning potential h: the free energy density is zero in the delocalized regime, that is for h smaller than a threshold hc, and it is positive for h > hc. For d ≥ 3 we compute hc and we show that the transition happens at the same value as for the annealed model. However we can show that the critical behavior of the quenched model differs from the one of the annealed one. While the phase transition of the annealed model is of first order, we show that the quenched free energy is bounded above by ((h−hc)+) 2 times a positive constant and that, for Gaussian disorder, the quadratic behavior is sharp. Therefore this provides an example in which a relevant disorder critical exponent can be made explicit: in theoretical physics disorder is said to be relevant when the disorder changes the critical behavior of a system and, while there are cases in which it is known that disorder is relevant, the exact critical behavior is typically unknown. For d = 2 we are not able to decide whether the quenched and annealed critical points coincide, but we provide an upper bound for the difference between them.

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Section: 4mentioning

confidence: 99%

Pinning and disorder relevance for the lattice Gaussian free field

Giacomin

Lacoin

2017

J. Eur. Math. Soc.

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“…In the last twenty years this question has been addressed, first by theoretical physicists (see e.g. [26] and references therein) and then by mathematicians [4,5,3,8,27,35,36,37,42,46] (see also [32,33] for reviews), for a simple model of a 1dimensional interface interacting with a substrate: for this model the interface is given by the graph of a random walk which takes random energy rewards when it touches a defect line. In this case, the pure system has the remarkable quality of being what physicists call exactly solvable, meaning that there exists an explicit expression for the free energy [29].…”

Section: Introductionmentioning

confidence: 99%

Pinning and disorder relevance for the lattice Gaussian Free Field II: the two dimensional case

Lacoin

2015

Preprint

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This paper continues a study initiated in [34], on the localization transition of a lattice free field on Z d interacting with a quenched disordered substrate that acts on the interface when its height is close to zero. The substrate has the tendency to localize or repel the interface at different sites. A transition takes place when the average pinning potential h goes past a threshold hc: from a delocalized phase h < hc, where the field is macroscopically repelled by the substrate to a localized one h > hc where the field sticks to the substrate. Our goal is to investigate the effect of the presence of disorder on this phase transition. We focus on the two dimensional case (d = 2) for which we had obtained so far only limited results. We prove that the value of hc(β) is the same as for the annealed model, for all values of β and that in a neighborhood of hc. Moreover we prove that, in contrast with the case d ≥ 3 where the free energy has a quadratic behavior near the critical point, the phase transition is of infinite order lim u→0+ log f(β, hc(β) + u)

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“…(2) from the proof we see that C β can be chosen independent of β if we assume (1.6) (see Remark 1.8); (3) the finite order big-jump transition at h b has disappeared, but the C ∞ regularity estimate on the free energy leaves open the possibility of an infinite order transition; (4) nevertheless, (1.26) tells us that the loops in the localized regime do not have macroscopic size, so the large loop phenomenon is washed out by the disorder; (5) we have decided to leave aside the delicate analysis of the path behavior in the delocalized phase: we certainly expect that results like in [4,34] can be adapted, but only under stronger conditions on Ψ(m, N ) (and the problem is already present for β = 0, see Remark 1.6).…”

Section: Recall Remark 14 and Set Hmentioning

confidence: 99%

Localization, big-jump regime and the effect disorder for a class of generalized pinning models

Giacomin,

Havret

2020

Preprint

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One dimensional pinning models have been widely studied in the physical and mathematical literature, also in presence of disorder. Roughly speaking, they undergo a transition between a delocalized phase and a localized one. In mathematical terms these models are obtained by modifying the distribution of a discrete renewal process via a Boltzmann factor with an energy that contains only one body potentials. For some more complex models, notably pinning models based on higher dimensional renewals, it has been shown that other phases may be present.We study a generalization of the one dimensional pinning model in which the energy may depend in a nonlinear way on the contact fraction: this class of models contains the circular DNA case considered for example in [7]. We give a full solution of this generalized pinning model in absence of disorder and show that another transition appears. In fact the systems may display up to three different regimes: delocalization, partial localization and full localization. What happens in the partially localized regime can be explained in terms of the "big-jump" phenomenon for sums of heavy tail random variables under conditioning.We then show that disorder completely smears this second transition and we are back to the delocalization versus localization scenario. In fact we show that the disorder, even if arbitrarily weak, is incompatible with the presence of a big-jump.

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Path properties of the disordered pinning model in the delocalized regime

Cited by 7 publications

References 14 publications

Pinning and disorder relevance for the lattice Gaussian free field

Pinning and disorder relevance for the lattice Gaussian free field

Pinning and disorder relevance for the lattice Gaussian Free Field II: the two dimensional case

Localization, big-jump regime and the effect disorder for a class of generalized pinning models

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