Renewal Sequences, Disordered Potentials, and Pinning Phenomena

Giacomin, Giambattista

doi:10.1007/978-3-7643-9891-0_11

Cited by 6 publications

(5 citation statements)

References 59 publications

(114 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…and, when L(·) is trivial, ∂ a f(0, h c (0) + a) behaves like (a constant times) a (1−α)/α for α ∈ (0, 1) (this is detailed for example in [13]) and like a constant for α ≥ 1. This suggests that the expansion (1.10) cannot work for α > 1/2, because the second-order term, for a ց 0, becomes larger than the first order term (a max(1/α,1) ).…”

Section: Introductionmentioning

confidence: 98%

Disorder relevance at marginality and critical point shift

Giacomin¹,

Lacoin²,

Toninelli³

2011

Ann. Inst. H. Poincaré Probab. Statist.

Self Cite

View full text Add to dashboard Cite

Recently the renormalization group predictions on the effect of disorder on pinning models have been put on mathematical grounds. The picture is particularly complete if the disorder is relevant or irrelevant in the Harris criterion sense: the question addressed is whether quenched disorder leads to a critical behavior which is different from the one observed in the pure, i.e. annealed, system. The Harris criterion prediction is based on the sign of the specific heat exponent of the pure system, but it yields no prediction in the case of vanishing exponent. This case is called marginal, and the physical literature is divided on what one should observe for marginal disorder, notably there is no agreement on whether a small amount of disorder leads or not to a difference between the critical point of the quenched system and the one for the pure system. In [14] we have proven that the two critical points differ at marginality of at least exp(−c/β 4 ), where c > 0 and β 2 is the disorder variance, for β ∈ (0, 1) and Gaussian IID disorder. The purpose of this paper is to improve such a result: we establish in particular that the exp(−c/β 4 ) lower bound on the shift can be replaced by exp(−c(b)/β b ), c(b) > 0 for b > 2 (b = 2 is the known upper bound and it is the result claimed in [8]), and we deal with very general distribution of the IID disorder variables. The proof relies on coarse graining estimates and on a fractional moment-change of measure argument based on multi-body potential modifications of the law of the disorder.

show abstract

Section: Introductionmentioning

confidence: 98%

Disorder relevance at marginality and critical point shift

Giacomin¹,

Lacoin²,

Toninelli³

2011

Ann. Inst. H. Poincaré Probab. Statist.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Although most mathematically rigorous work is relatively recent, there is an extensive physics literature on polymer pinning models; see the recent book [8] and the surveys [9,16] and references therein. In [1] (see also [15] for a slightly weaker statement with simpler proof), it was proven that for 1 < c < 3/2, and for c = 3/2 with ∞ n=1 1/nϕ(n) 2 < ∞, for sufficiently small β, one has u q c (β) = u a c (β) and the specific heat exponents are the same.…”

mentioning

confidence: 99%

Equality of critical points for polymer depinning transitions with loop exponent one

Alexander¹,

Zygouras²

2010

Ann. Appl. Probab.

View full text Add to dashboard Cite

We consider a polymer with configuration modeled by the trajectory\ud of a Markov chain, interacting with a potential of form u + Vn when it visits a\ud particular state 0 at time n, with {Vn} representing i.i.d. quenched disorder. There\ud is a critical value of u above which the polymer is pinned by the potential. A\ud particular case not covered in a number of previous studies is that of loop exponent\ud one, in which the probability of an excursion of length n takes the form φ(n)/n for\ud some slowly varying φ; this includes simple random walk in two dimensions. We\ud show that in this case, at all temperatures, the critical values of u in the quenched\ud and annealed models are equal, in contrast to all other loop exponents, for which\ud these critical values are known to differ at least at low temperatures

show abstract

“…A number of predictions have been made in the physics literature, sometimes contradictory, [15,13,27,32,33,28,36]. It is however now well established under which condition the nature of the transition is the same for the pure system and in the case of a weak disorder [17,20,23,21,22] as well as how the transition point is shifted [1,2,3,12]. Concerning the nature of the transition, the main result [18,19] is that in presence of disorder, the transition is always smooth, implying the impossibility of first order transitions or of diverging specific heats.…”

Section: Appendixmentioning

confidence: 99%