The Derrida--Retaux conjecture on recursive models

Chen, Xinxing; Dagard, Victor; Derrida, Bernard; Hu, Yueyun; Lifshits, Mikhail; Shi, Zhan

doi:10.48550/arxiv.1907.01601

Cited by 8 publications

(22 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…What the disordered critical behavior really is remains mathematically a fully open issue. Substantial progress on this problem has been recently achieved, but not for the pinning models itself: the critical behavior of a relevant disorder case for one class of copolymer pinning models and of a simplified version of the hierarchical pinning model have been identified respectively in [1] and in [9].…”

mentioning

confidence: 99%

The disordered lattice free field pinning model approaching criticality

Giacomin¹,

Lacoin²

2019

Preprint

View full text Add to dashboard Cite

We continue the study, initiated in [21], of the localization transition of a lattice free field φ = (φ(x)) x∈Z d , d ≥ 3, in presence of a quenched disordered substrate. The presence of the substrate affects the interface at the spatial sites in which the interface height is close to zero. This corresponds to the Hamiltonian Moreover, we give a precise description of the trajectories of the field in the same regime: the absolute value of the field is 2σ 2 d | log(h − hc(β))| to leading order when h hc(β) except on a vanishing fraction of sites (σ 2 d is the single site variance of the free field).

show abstract

mentioning

confidence: 99%

The disordered lattice free field pinning model approaching criticality

Giacomin¹,

Lacoin²

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…One of the main reason is that the model is expected to be in the strong or infinite disorder universality class [17,26,43], see also the more recent contribution [22]. In this line there have been also some mathematical progress [10,16], but they do not impact directly the pinning model.…”

Section: Exploiting the Legendre Transform And The Key Role Of The St...mentioning

confidence: 99%

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

Giacomin,

Havret

2020

Preprint

View full text Add to dashboard Cite

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.

show abstract

“…Since then, a body of work with an increasing level of generality has emerged [67,28,37,9] ultimately considering critical conditioned Bienaymé-Galton-Watson tree (with finite variance) for the underlying tree and independent car arrivals whose laws may depend on the degree of the vertices [32]. See [10,29] for the case of supercritical trees. In this broad context, it was shown that a sharp phase transition appears for the parking process: there is a critical "density" of cars (depending on the combinatorial details of the model) such that below this density, almost all cars manage to park, whereas above this density, a positive proportion of cars do not find a parking spot.…”

Section: Introductionmentioning

confidence: 99%

Parking on Cayley trees & Frozen Erdös-Rényi

Contat,

Curien

2021

Preprint

View full text Add to dashboard Cite

Consider a uniform rooted Cayley tree T n with n vertices and let m cars arrive sequentially, independently, and uniformly on its vertices. Each car tries to park on its arrival node, and if the spot is already occupied, it drives towards the root of the tree and parks as soon as possible. Lackner & Panholzer [56] established a phase transition for this process when m ≈ n 2 . In this work, we couple this model with a variant of the classical Erdős-Rényi random graph process. This enables us to describe the phase transition for the size of the components of parked cars using a modification of the multiplicative coalescent which we name the frozen multiplicative coalescent. The geometry of critical parked clusters is also studied. Those trees are very different from Bienaymé-Galton-Watson trees and should converge towards the growth-fragmentation trees canonically associated to the 3/2-stable process that already appeared in the study of random planar maps.

show abstract

The Derrida--Retaux conjecture on recursive models

Cited by 8 publications

References 21 publications

The disordered lattice free field pinning model approaching criticality

The disordered lattice free field pinning model approaching criticality

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

Parking on Cayley trees & Frozen Erdös-Rényi

Contact Info

Product

Resources

About