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Comments on the Influence of Disorder for Pinning Model in Correlated Gaussian Environment
We study the random pinning model, in the case of a Gaussian environment
presenting power-law decaying correlations, of exponent decay a>0. We comment
on the annealed (i.e. averaged over disorder) model, which is far from being
trivial, and we discuss the influence of disorder on the critical properties of
the system. We show that the annealed critical exponent \nu^{ann} is the same
as the homogeneous one \nu^{pur}, provided that correlations are decaying fast
enough (a>2). If correlations are summable (a>1), we also show that the
disordered phase transition is at least of order 2, showing disorder relevance
if \nu^{pur}<2. If correlations are not summable (a<1), we show that the phase
transition disappears.Comment: 23 pages, 1 figure Modifications in v2 (outside minor typos):
Assumption 1 on correlations has been simplified for more clarity; Theorem 4
has been improved to a more general underlying renewal distribution; Remark
2.1 added, on the assumption on the correlations in the summable cas
Pinning model in random correlated environment: appearance of an infinite disorder regime
We study the influence of a correlated disorder on the localization phase
transition in the pinning model. When correlations are strong enough, a strong
disorder regime arises: large and frequent attractive regions appear in the
environment. We present here a pinning model in random binary ({-1,1}-valued)
environment. Defining strong disorder via the requirement that the probability
of the occurrence of a large attractive region is sub-exponential in its size,
we prove that it coincides with the fact that the critical point is equal to
its minimal possible value. We also stress that in the strong disorder regime,
the phase transition is smoother than in the homogeneous case, whatever the
critical exponent of the homogeneous model is: disorder is therefore always
relevant. We illustrate these results with the example of an environment based
on the sign of a Gaussian correlated sequence, in which we show that the phase
transition is of infinite order in presence of strong disorder. Our results
contrast with results known in the literature, in particular in the case of an
IID disorder, where the question of the influence of disorder on the critical
properties is answered via the so-called Harris criterion, and where a
conventional relevance/irrelevance picture holds.Comment: 27 pages, some corrections made in v
Sharp critical behavior for pinning model in random correlated environment
This article investigates the effect for random pinning models of long range
power-law decaying correlations in the environment. For a particular type of
environment based on a renewal construction, we are able to sharply describe
the phase transition from the delocalized phase to the localized one, giving
the critical exponent for the (quenched) free-energy, and proving that at the
critical point the trajectories are fully delocalized. These results contrast
with what happens both for the pure model (i.e. without disorder) and for the
widely studied case of i.i.d. disorder, where the relevance or irrelevance of
disorder on the critical properties is decided via the so-called Harris
Criterion.Comment: 37 pages, 2 figure
On the critical curves of the Pinning and Copolymer models in Correlated Gaussian environment
We investigate the disordered copolymer and pinning models, in the case of a
correlated Gaussian environment with summable correlations, and when the return
distribution of the underlying renewal process has a polynomial tail. As far as
the copolymer model is concerned, we prove disorder relevance both in terms of
critical points and critical exponents, in the case of non-negative
correlations. When some of the correlations are negative, even the annealed
model becomes non-trivial. Moreover, when the return distribution has a finite
mean, we are able to compute the weak coupling limit of the critical curves for
both models, with no restriction on the correlations other than summability.
This generalizes the result of Berger, Caravenna, Poisat, Sun and Zygouras
\cite{cf:BCPSZ} to the correlated case. Interestingly, in the copolymer model,
the weak coupling limit of the critical curve turns out to be the maximum of
two quantities: one generalizing the limit found in the IID case
\cite{cf:BCPSZ}, the other one generalizing the so-called Monthus bound.Comment: 35 page
Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift
The effect of disorder for pinning models is a subject which has attracted
much attention in theoretical physics and rigorous mathematical physics. A
peculiar point of interest is the question of coincidence of the quenched and
annealed critical point for a small amount of disorder. The question has been
mathematically settled in most cases in the last few years, giving in
particular a rigorous validation of the Harris Criterion on disorder relevance.
However, the marginal case, where the return probability exponent is equal to
, i.e. where the inter-arrival law of the renewal process is given by
where is a slowly varying function, has been left
partially open. In this paper, we give a complete answer to the question by
proving a simple necessary and sufficient criterion on the return probability
for disorder relevance, which confirms earlier predictions from the literature.
Moreover, we also provide sharp asymptotics on the critical point shift: in the
case of the pinning (or wetting) of a one dimensional simple random walk, the
shift of the critical point satisfies the following high temperature
asymptotics This
gives a rigorous proof to a claim of B. Derrida, V. Hakim and J. Vannimenus
(Journal of Statistical Physics, 1992).Comment: 34 Page
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