arXiv subjects

Hubert Lacoin

Publications and source records attributed to Hubert Lacoin.

70 records · Page 4Linked to original sources

arXiv2011

Superdiffusivity for Brownian motion in a Poissonian potential with long range correlation II: upper bound on the volume exponent

This paper continues a study on trajectories of Brownian Motion in a field of soft trap whose radius distribution is unbounded. We show here for both point-to-point and point-to-plane model the volume exponent (the exponent associated to transversal fluctuation of the trajectories) is strictly less than one and give an explicit upper bound that depends on the parameters of the problem. In some specific cases, this upper bound matches the lower bound proved in the first part of this work and we get the exact value of the volume exponent.

math.PR↗

arXiv2011

Superdiffusivity for Brownian Motion in a Poissonian Potential with Long Range Correlation I: Lower Bound on the Volume Exponent

We study trajectories of d-dimensional Brownian Motion in Poissonian potential up to the hitting time of a distant hyper-plane. Our Poissonian potential V can be associated to a field of traps whose centers location is given by a Poisson Point process and whose radii are IID distributed with a common distribution that has unbounded support; it has the particularity of having long-range correlation. We focus on the case where the law of the trap radii has power-law decay and prove that superdiffusivity hold under certain condition, and get a lower bound on the volume exponent. Results differ quite much with the one that have been obtained for the model with traps of bounded radii by Wühtrich: the superdiffusivity phenomenon is enhanced by the presence of correlation.

math.PR↗

arXiv2011

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.

math.PR↗

arXiv2011

The martingale approach to disorder irrelevance for pinning models

This paper presents a very simple and self-contained proof of disorder irrelevance for inhomogeneous pinning models with return exponent alpha in the Interval (0,1/2). We also give a new upper bound for the contact fraction of the disordered model at criticality.

math.PR↗

arXiv2011

A two cities theorem for the parabolic Anderson model

The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_tu(t,z)=Δu(t,z)+ξ(z)u(t,z)$ on $(0,\infty)\times {\mathbb{Z}}^d$ with random potential $(ξ(z):z\in{\mathbb{Z}}^d)$. We consider independent and identically distributed potentials, such that the distribution function of $ξ(z)$ converges polynomially at infinity. If $u$ is initially localized in the origin, that is, if $u(0,{z})={\mathbh1}_0({z})$, we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem.

math.PR↗

arXiv2010

Influence of spatial correlation for directed polymers

In this paper, we study a model of a Brownian polymer in $\mathbb {R}_+\times \mathbb {R}^d$, introduced by Rovira and Tindel [J. Funct. Anal. 222 (2005) 178--201]. Our investigation focuses mainly on the effect of strong spatial correlation in the environment in that model in terms of free energy, fluctuation exponent and volume exponent. In particular, we prove that under some assumptions, very strong disorder and superdiffusivity hold at all temperatures when $d\ge3$ and provide a novel approach to Petermann's superdiffusivity result in dimension one [Superdiffusivity of directed polymers in random environment (2000) Ph.D. thesis]. We also derive results for a Brownian model of pinning in a nonrandom potential with power-law decay at infinity.

math.PR↗

arXiv2010

A scaling limit theorem for the parabolic Anderson model with exponential potential

The parabolic Anderson problem is the Cauchy problem for the heat equation with random potential and localized initial condition. In this paper we consider potentials which are constant in time and independent exponentially distributed in space. We study the growth rate of the total mass of the solution in terms of weak and almost sure limit theorems, and the spatial spread of the mass in terms of a scaling limit theorem. The latter result shows that in this case, just like in the case of heavy tailed potentials, the mass gets trapped in a single relevant island with high probability.

math.PR↗

arXiv2010

The effect of disorder on the free-energy for the Random Walk Pinning Model: smoothing of the phase transition and low temperature asymptotics

We consider the continuous time version of the Random Walk Pinning Model (RWPM), studied in [5,6,7]. Given a fixed realization of a random walk Y$ on Z^d with jump rate rho (that plays the role of the random medium), we modify the law of a random walk X on Z^d with jump rate 1 by reweighting the paths, giving an energy reward proportional to the intersection time L_t(X,Y)=\int_0^t \ind_{X_s=Y_s}\dd s: the weight of the path under the new measure is exp(beta L_t(X,Y)), beta in R. As beta increases, the system exhibits a delocalization/localization transition: there is a critical value beta_c, such that if beta>beta_c the two walks stick together for almost-all Y realizations. A natural question is that of disorder relevance, that is whether the quenched and annealed systems have the same behavior. In this paper we investigate how the disorder modifies the shape of the free energy curve: (1) We prove that, in dimension d larger or equal to three 3, the presence of disorder makes the phase transition at least of second order. This, in dimension larger or equal to 4, contrasts with the fact that the phase transition of the annealed system is of first order. (2) In any dimension, we prove that disorder modifies the low temperature asymptotic of the free energy.

math-ph↗

arXiv2010

Polymer dynamics in the depinned phase: metastability with logarithmic barriers

We consider the stochastic evolution of a (1 + 1)-dimensional polymer in the depinned regime. At equilibrium the system exhibits a double well structure: the polymer lies(essentially) either above or below the repulsive line. As a consequence one expects a metastable behavior with rare jumps between the two phases combined with a fast thermalization inside each phase. However the energy barrier between these two phases is only logarithmic in the system size L and therefore the two relevant time scales are only polynomial in L with no clear-cut separation between them. The whole evolution is governed by a subtle competition between the diffusive behavior inside one phase and the jumps across the energy barriers. In particular the usual scenario in which the tunneling time coincides with the exponential of the energy barrier breaks down. Our main results are: (i) a proof that the mixing time of the system lies between L^{5/2} and L^{5/2+2}; (ii) the identification of two regions associated with the positive and negative phase of the polymer together with the proof of the asymptotic exponentiality of the tunneling time between them with rate equal to a half of the spectral gap.

math.PR↗

arXiv2009

Desordre et phenomenes de localisation

Cette these est consacree a l' etude de differents modeles aleatoires de polymeres. On s'interesse en particulier a l'influence du desordre sur la localisation des trajectoires pour les modeles d'accrochage et pour les polymeres diriges en milieu aleatoire. En plus des modeles classiques dans Zd, nous abordons l' etude de modeles dit hierarchiques, construits sur une suite de reseaux auto-similaires, tres present dans la litterature physique. Les resultats que nous avons obtenus concernent principalement l' energie libre et le phenomene de surdiffusivite. En particulier, nous prouvons: (1) la pertinence du desordre a toute temperature dans pour le modele d' accrochage desordonne en dimension 1 + 1, (2) l' occurence d' un desordre tres fort a toute temperature en dimension 1 + 2 pour les polymeres diriges en milieu aleatoire. This thesis studies models of random directed polymers. We focus on the influence of disorder on localization of the trajectories for pinning model and directed polymers in random environment. In addition to the classical Zd models, we pay a particular attention to so-called hierachical models, buildt on a sequence of self-similar lattices, that are frequently studied in the physics literature. The results we obtain concern mainly free energy and superdiffusivity properties. In particular we present the proof that: (1) disorder is relevant at arbitrary high temperature for pinning models in dimension 1 + 1, (2) very strong disorder holds at all temperature in dimension 1 + 2 for directed polymers in random environment.

math.PR↗

arXiv2009

New bounds for the free energy of directed polymers in dimension 1+1 and 1+2

We study the free energy of the directed polymer in random environment in dimension 1+1 and 1+2. For dimension 1, we improve the statement of Comets and Vargas concerning very strong disorder by giving sharp estimates on the free energy at high temperature. In dimension 2, we prove that very strong disorder holds at all temperatures, thus solving a long standing conjecture in the field.

math-ph↗

arXiv2009

Marginal relevance of disorder for pinning models

The effect of disorder on pinning and wetting models has attracted much attention in theoretical physics. In particular, it has been predicted on the basis of the Harris criterion that disorder is relevant (annealed and quenched model have different critical points and critical exponents) if the return probability exponent alpha, a positive number that characterizes the model, is larger than 1/2. Weak disorder has been predicted to be irrelevant (i.e. coinciding critical points and exponents) if alpha < 1/2. Recent mathematical work has put these predictions on firm grounds. In renormalization group terms, the case alpha = 1/2 is a 'marginal case' and there is no agreement in the literature as to whether one should expect disorder relevance or irrelevance at marginality. The question is particularly intriguing also because the case alpha = 1/2 includes the classical models of two-dimensional wetting of a rough substrate, of pinning of directed polymers on a defect line in dimension (3+1) or (1+1) and of pinning of an heteropolymer by a point potential in three-dimensional space. Here we prove disorder relevance both for the general alpha = 1/2 pinning model and for the hierarchical version of the model proposed by B. Derrida, V. Hakim and J. Vannimenus (JSP, 1992), in the sense that we prove a shift of the quenched critical point with respect to the annealed one. In both cases we work with Gaussian disorder and we show that the shift is at least of order exp(-1/β^4) for beta small, if beta is the standard deviation of the disorder.

math-ph↗

arXiv2009

Disorder relevance at marginality and critical point shift

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 a previous work (arXiv:0811.0723) we have proven that the two critical points differ at marginality of at least exp(-c/beta^4), where c>0 and beta^2 is the disorder variance, for beta in (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/beta^4) lower bound on the shift can be replaced by exp(-c(b)/beta^b), c(b)>0 for b>2 (b=2 is the known upper bound and it is the result claimed in [Derrida, Hakim, Vannimenus, JSP 1992]), 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.

math-ph↗

arXiv2009

Directed Polymers on Hierarchical Lattices with site disorder

We study a polymer model on hierarchical lattices very close to the one introduced and studied in \cite{DGr, CD}. For this model, we prove the existence of free energy and derive the necessary and sufficient condition for which very strong disorder holds for all $\gb$, and give some accurate results on the behavior of the free energy at high-temperature. We obtain these results by using a combination of fractional moment method and change of measure over the environment to obtain an upper bound, and second moment method to get a lower bound. We also get lower bounds on the fluctuation exponent of $\log Z_n$, and study the infinite polymer measure in the weak disorder phase.

math.PR↗

arXiv2009

Hierarchical pinning model with site disorder: Disorder is marginally relevant

We study a hierarchical disordered pinning model with site disorder for which, like in the bond disordered case [6, 9], there exists a value of a parameter b (enters in the definition of the hierarchical lattice) that separates an irrelevant disorder regime and a relevant disorder regime. We show that for such a value of b the critical point of the disordered system is different from the critical point of the annealed version of the model. The proof goes beyond the technique used in [9] and it takes explicitly advantage of the inhomogeneous character of the Green function of the model.

math.PR↗

arXiv2009

Hierarchical pinning models, quadratic maps and quenched disorder

We consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by B. Derrida, V. Hakim and J. Vannimenius in 1992, which can be re-interpreted as an infinite dimensional dynamical system with random initial condition (the disorder). It is defined through a recurrence relation for the law of a random variable {R_n}_{n=1,2,...}, which in absence of disorder (i.e., when the initial condition is degenerate) reduces to a particular case of the well-known Logistic Map. The large-n limit of the sequence of random variables 2^{-n} log R_n, a non-random quantity which is naturally interpreted as a free energy, plays a central role in our analysis. The model depends on a parameter alpha>0, related to the geometry of the hierarchical lattice, and has a phase transition in the sense that the free energy is positive if the expectation of R_0 is larger than a certain threshold value, and it is zero otherwise. It was conjectured by Derrida et al. (1992) that disorder is relevant (respectively, irrelevant or marginally relevant) if 1/2 1/2 we find the correct scaling form (for weak disorder) of the critical point shift.

math.PR↗