arXiv subjects

Francesco Caravenna

Publications and source records attributed to Francesco Caravenna.

51 records · Page 3Linked to original sources

arXiv2010

Localization for (1+1)-dimensional pinning models with $(\nabla + Δ)$-interaction

We study the localization/delocalization phase transition in a class of directed models for a homogeneous linear chain attracted to a defect line. The self-interaction of the chain is of mixed gradient and Laplacian kind, whereas the attraction to the defect line is of $δ$-pinning type, with strength $ε\geq 0$. It is known that, when the self-interaction is purely Laplacian, such models undergo a non-trivial phase transition: to localize the chain at the defect line, the reward $ε$ must be greater than a strictly positive critical threshold $ε_c > 0$. On the other hand, when the self-interaction is purely gradient, it is known that the transition is trivial: an arbitrarily small reward $ε> 0$ is sufficient to localize the chain at the defect line ($ε_c = 0$). In this note we show that in the mixed gradient and Laplacian case, under minimal assumptions on the interaction potentials, the transition is always trivial, that is $ε_c = 0$.

math.PR↗

arXiv2009

A polymer in a multi-interface medium

We consider a model for a polymer chain interacting with a sequence of equispaced flat interfaces through a pinning potential. The intensity $δ\in \mathbb {R}$ of the pinning interaction is constant, while the interface spacing $T=T_N$ is allowed to vary with the size $N$ of the polymer. Our main result is the explicit determination of the scaling behavior of the model in the large $N$ limit, as a function of $(T_N)_N$ and for fixed $δ>0$. In particular, we show that a transition occurs at $T_N=O(\log N)$. Our approach is based on renewal theory.

math.PR↗

arXiv2009

Scaling limits of $(1+1)$-dimensional pinning models with Laplacian interaction

We consider a random field $φ:\{1,...,N\}\to \mathbb{R}$ with Laplacian interaction of the form $\sum_iV(Δφ_i)$, where $Δ$ is the discrete Laplacian and the potential $V(\cdot)$ is symmetric and uniformly strictly convex. The pinning model is defined by giving the field a reward $\varepsilon\ge0$ each time it touches the x-axis, that plays the role of a defect line. It is known that this model exhibits a phase transition between a delocalized regime $(\varepsilon<\varepsilon_c)$ and a localized one $(\varepsilon>\varepsilon_c)$, where $0<\varepsilon_c<\infty$. In this paper we give a precise pathwise description of the transition, extracting the full scaling limits of the model. We show, in particular, that in the delocalized regime the field wanders away from the defect line at a typical distance $N^{3/2}$, while in the localized regime the distance is just $O((\log N)^2)$. A subtle scenario shows up in the critical regime $(\varepsilon=\varepsilon_c)$, where the field, suitably rescaled, converges in distribution toward the derivative of a symmetric stable Lévy process of index 2/5. Our approach is based on Markov renewal theory.

math.PR↗

arXiv2009

Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction

We consider a random field $φ:\{1,...,N\}\to\mathbb{R}$ as a model for a linear chain attracted to the defect line $φ=0$, that is, the x-axis. The free law of the field is specified by the density $\exp(-\sum_iV(Δφ_i))$ with respect to the Lebesgue measure on $\mathbb{R}^N$, where $Δ$ is the discrete Laplacian and we allow for a very large class of potentials $V(\cdot)$. The interaction with the defect line is introduced by giving the field a reward $\varepsilon\ge0$ each time it touches the x-axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative. We show that both models undergo a phase transition as the intensity $\varepsilon$ of the pinning reward varies: both in the pinning ($a=\mathrm{p}$) and in the wetting ($a=\mathrm{w}$) case, there exists a critical value $\varepsilon_c^a$ such that when $\varepsilon>\varepsilon_c^a$ the field touches the defect line a positive fraction of times (localization), while this does not happen for $\varepsilon<\varepsilon_c^a$ (delocalization). The two critical values are nontrivial and distinct: $0<\varepsilon_c^{\mat hrm{p}}<\varepsilon_c^{\mathrm{w}}<\infty$, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at $\varepsilon=\varepsilon_c^{\mathrm{p}}$ is delocalized. On the other hand, the transition in the wetting model is of first order and for $\varepsilon=\varepsilon_c^{\mathrm{w}}$ the field is localized. The core of our approach is a Markov renewal theory description of the field.

math.PR↗

arXiv2009

Depinning of a polymer in a multi-interface medium

In this paper we consider a model which describes a polymer chain interacting with an infinity of equi-spaced linear interfaces. The distance between two consecutive interfaces is denoted by T = T_N and is allowed to grow with the size N of the polymer. When the polymer receives a positive reward for touching the interfaces, its asymptotic behavior has been derived in a previous paper, showing that a transition occurs when T_N \approx log(N). In the present paper, we deal with the so-called depinning case, i.e., the polymer is repelled rather than attracted by the interfaces. Using techniques from renewal theory, we determine the scaling behavior of the model for large N as a function of T_N, showing that two transitions occur, when T_N \approx N^{1/3} and when T_N \approx N^{1/2} respectively.

math.PR↗

arXiv2008

Large scale behavior of semiflexible heteropolymers

We consider a general discrete model for heterogeneous semiflexible polymer chains. Both the thermal noise and the inhomogeneous character of the chain (the disorder) are modeled in terms of random rotations. We focus on the quenched regime, i.e., the analysis is performed for a given realization of the disorder. Semiflexible models differ substantially from random walks on short scales, but on large scales a Brownian behavior emerges. By exploiting techniques from tensor analysis and non-commutative Fourier analysis, we establish the Brownian character of the model on large scale and we obtain an expression for the diffusion constant. We moreover give conditions yielding quantitative mixing properties.

math.PR↗

arXiv2008

The quenched critical point of a diluted disordered polymer model

We consider a model for a polymer interacting with an attractive wall through a random sequence of charges. We focus on the so-called diluted limit, when the charges are very rare but have strong intensity. In this regime, we determine the quenched critical point of the model, showing that it is different from the annealed one. The proof is based on a rigorous renormalization procedure. Applications of our results to the problem of a copolymer near a selective interface are discussed.

math.PR↗

arXiv2008

Invariance principles for random walks conditioned to stay positive

Let $\{S_n\}$ be a random walk in the domain of attraction of a stable law $\mathcal{Y}$, i.e. there exists a sequence of positive real numbers $(a_n)$ such that $S_n/a_n$ converges in law to $\mathcal{Y}$. Our main result is that the rescaled process $(S_{\lfloor nt\rfloor}/a_n, t\ge 0)$, when conditioned to stay positive, converges in law (in the functional sense) towards the corresponding stable Lévy process conditioned to stay positive. Under some additional assumptions, we also prove a related invariance principle for the random walk killed at its first entrance in the negative half-line and conditioned to die at zero.

math.PR↗

arXiv2007

A renewal theory approach to periodic copolymers with adsorption

We consider a general model of a heterogeneous polymer chain fluctuating in the proximity of an interface between two selective solvents. The heterogeneous character of the model comes from the fact that the monomer units interact with the solvents and with the interface according to some charges that they carry. The charges repeat themselves along the chain in a periodic fashion. The main question concerning this model is whether the polymer remains tightly close to the interface, a phenomenon called localization, or whether there is a marked preference for one of the two solvents, thus yielding a delocalization phenomenon. In this paper, we present an approach that yields sharp estimates for the partition function of the model in all regimes (localized, delocalized and critical). This, in turn, makes possible a precise pathwise description of the polymer measure, obtaining the full scaling limits of the model. A key point is the closeness of the polymer measure to suitable Markov renewal processes, Markov renewal theory being one of the central mathematical tools of our analysis.

math.PR↗

arXiv2007

Tightness conditions for polymer measures

We give sufficient conditions for tightness in the space C([0,1]) for sequences of probability measures which enjoy a suitable decoupling between zero level set and excursions. Applications of our results are given in the context of (homogeneous, periodic and disordered) random walk models for polymers and interfaces.

math.PR↗

arXiv2006

Infinite volume limits of polymer chains with periodic charges

The aim of this paper is twofold: - To give an elementary and self-contained proof of an explicit formula for the free energy for a general class of polymer chains interacting with an environment through periodic potentials. This generalizes a result in [Bolthausen and Giacomin, AAP 2005] in which the formula is derived by using the Donsker-Varadhan Large Deviations theory for Markov chains. We exploit instead tools from renewal theory. - To identify the infinite volume limits of the system. In particular, in the different regimes we encounter transient, null recurrent and positive recurrent processes (which correspond to delocalized, critical and localized behaviors of the trajectories). This is done by exploiting the sharp estimates on the partition function of the system obtained by the renewal theory approach. The precise characterization of the infinite volume limits of the system exposes a non-uniqueness problem. We will however explain in detail how this (at first) surprising phenomenon is instead due to the presence of a first-order phase transition.

math.PR↗

arXiv2005

Random walk models and probabilistic techniques for inhomogeneous polymer chains

Modeling of polymer chains has received a lot of attention in mathematics. In fact, probabilistic models that naturally arise in statistical mechanics have been widely studied by mathematicians for the very challenging and novel problems that they pose. The physical situation that we consider in this thesis is that of a polymer in the proximity of an interface between two selective solvents, in the case when the interaction of the monomers with the solvents and the interface may vary from monomer to monomer (inhomogeneous polymer). In interesting cases thee is a phase transition between a state in which the polymer sticks very close to the interface (localized regime) and a state in which it wanders away from it (delocalized regime). The mechanism underlying such a transition is an energy/entropy competition. Our task has been to study random walk models of polymer chains with the purpose of understanding this competition in a deep and quantitative way. Despite the fact that the definition of these models is extremely elementary, their analysis is not simple at all, and several interesting questions are still open. In this Ph.D. thesis we present new results that answer some of these questions. The analysis performed has required the application of a wide range of techniques, including large deviations, concentration inequalities, renewal theory, fluctuation theory for random walks. A numerical and statistical study has been performed too. Finally we prove a local limit theorem for random walks conditioned to stay positive.

math.PR↗

arXiv2005

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.

math.PR↗

arXiv2005

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.

math.PR↗

arXiv2005

A Local Limit Theorem for random walks conditioned to stay positive

We consider a real random walk S_n = X_1 + ... + X_n attracted (without centering) to the normal law: this means that for a suitable norming sequence a_n we have the weak convergence S_n / a_n --> f(x) dx, where f(x) is the standard normal density (this happens in particular by the CLT when X_1 has zero mean and finite variance σ^2, with a_n = σ\sqrt{n}). A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit Theorems, in the lattice and nonlattice case respectively. Now let C_n denote the event (S_1 > 0, ..., S_n > 0) and let S_n^+ denote the random variable S_n conditioned on C_n: it is known that S_n^+ / a_n --> f^+(x) \dd x, where f^+(x) := x \exp(-x^2/2) \ind_{x > 0}. What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence. We also consider the particular case when X_1 has an absolutely continuous law: in this case the uniform convergence of the density of S_n^+ / a_n towards f^+(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so--called Fluctuation Theory for random walks.

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