102 research outputs found
Renewal sequences, disordered potentials, and pinning phenomena
We give an overview of the state of the art of the analysis of disordered
models of pinning on a defect line. This class of models includes a number of
well known and much studied systems (like polymer pinning on a defect line,
wetting of interfaces on a disordered substrate and the Poland-Scheraga model
of DNA denaturation). A remarkable aspect is that, in absence of disorder, all
the models in this class are exactly solvable and they display a
localization-delocalization transition that one understands in full detail.
Moreover the behavior of such systems near criticality is controlled by a
parameter and one observes, by tuning the parameter, the full spectrum of
critical behaviors, ranging from first order to infinite order transitions.
This is therefore an ideal set-up in which to address the question of the
effect of disorder on the phase transition,notably on critical properties. We
will review recent results that show that the physical prediction that goes
under the name of Harris criterion is indeed fully correct for pinning models.
Beyond summarizing the results, we will sketch most of the arguments of proof.Comment: 32 pages. Notes from lectures given at the summer school on Spin
Glasses (June 25- July 6, 2007). Birkhauser, eds. A. Boutet de Monvel and A.
Bovier (to appear
Renewal convergence rates and correlation decay for homogeneous pinning models
A class of discrete renewal processes with super-exponentially decaying
inter-arrival distributions coincides with the infinite volume limit of general
homogeneous pinning models in their localized phase. Pinning models are
statistical mechanics systems to which a lot of attention has been devoted both
for their relevance for applications and because they are solvable models
exhibiting a non-trivial phase transition. The spatial decay of correlations in
these systems is directly mapped to the speed of convergence to equilibrium for
the associated renewal processes. We show that close to criticality, under
general assumptions, the correlation decay rate, or the renewal convergence
rate, coincides with the inter-arrival decay rate. We also show that, in
general, this is false away from criticality. Under a stronger assumption on
the inter-arrival distribution we establish a local limit theorem, capturing
thus the sharp asymptotic behavior of correlations.Comment: 13 page
On constrained annealed bounds for pinning and wetting models
The free energy of quenched disordered systems is bounded above by the free
energy of the corresponding annealed system. This bound may be improved by
applying the annealing procedure, which is just Jensen inequality, after having
modified the Hamiltonian in a way that the quenched expressions are left
unchanged. This procedure is often viewed as a partial annealing or as a
constrained annealing, in the sense that the term that is added may be
interpreted as a Lagrange multiplier on the disorder variables.
In this note we point out that, for a family of models, some of which have
attracted much attention, the multipliers of the form of empirical averages of
local functions cannot improve on the basic annealed bound from the viewpoint
of characterizing the phase diagram. This class of multipliers is the one that
is suitable for computations and it is often believed that in this class one
can approximate arbitrarily well the quenched free energy.Comment: 10 page
Periodic copolymers at selective interfaces: A Large Deviations approach
We analyze a (1+1)-dimension directed random walk model of a polymer dipped
in a medium constituted by two immiscible solvents separated by a flat
interface. The polymer chain is heterogeneous in the sense that a single
monomer may energetically favor one or the other solvent. We focus on the case
in which the polymer types are periodically distributed along the chain or, in
other words, the polymer is constituted of identical stretches of fixed length.
The phenomenon that one wants to analyze is the localization at the interface:
energetically favored configurations place most of the monomers in the
preferred solvent and this can be done only if the polymer sticks close to the
interface. We investigate, by means of large deviations, the energy-entropy
competition that may lead, according to the value of the parameters (the
strength of the coupling between monomers and solvents and an asymmetry
parameter), to localization. We express the free energy of the system in terms
of a variational formula that we can solve. We then use the result to analyze
the phase diagram.Comment: Published at http://dx.doi.org/10.1214/105051604000000800 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
On the irrelevant disorder regime of pinning models
Recent results have lead to substantial progress in understanding the role of
disorder in the (de)localization transition of polymer pinning models. Notably,
there is an understanding of the crucial issue of disorder relevance and
irrelevance that is now rigorous. In this work, we exploit interpolation and
replica coupling methods to obtain sharper results on the irrelevant disorder
regime of pinning models. In particular, in this regime, we compute the first
order term in the expansion of the free energy close to criticality and this
term coincides with the first order of the formal expansion obtained by field
theory methods. We also show that the quenched and quenched averaged
correlation length exponents coincide, while, in general, they are expected to
be different. Interpolation and replica coupling methods in this class of
models naturally lead to studying the behavior of the intersection of certain
renewal sequences and one of the main tools in this work is precisely renewal
theory and the study of these intersection renewals.Comment: Published in at http://dx.doi.org/10.1214/09-AOP454 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Sharp asymptotic behavior for wetting models in (1+1)-dimension
We consider continuous and discrete (1+1)-dimensional wetting models which
undergo a localization/delocalization phase transition. Using a simple approach
based on Renewal Theory we determine the precise asymptotic behavior of the
partition function, from which we obtain the scaling limits of the models and
an explicit construction of the infinite volume measure (thermodynamic limit)
in all regimes, including the critical one.Comment: 14 pages, 1 figur
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