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23
Pinning and wetting transition for (1+1)dimensional fields with Laplacian interaction
 Ann. Probab
"... Abstract. We consider a random field ϕ: {1,..., N} → R as a model for a linear chain attracted to the defect line ϕ = 0, i.e. the x–axis. The free law of the field is specified by the density exp ` − P i V (∆ϕi) ´ with respect to the Lebesgue measure on R N, where ∆ is the discrete Laplacian and w ..."
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Cited by 25 (4 self)
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Abstract. We consider a random field ϕ: {1,..., N} → R as a model for a linear chain attracted to the defect line ϕ = 0, i.e. the x–axis. The free law of the field is specified by the density exp ` − P i V (∆ϕi) ´ with respect to the Lebesgue measure on R N, where ∆ is the discrete Laplacian and we allow for a very large class of potentials V (·). The interaction with the defect line is introduced by giving the field a reward ε ≥ 0 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 ε of the pinning reward varies: both in the pinning (a = p) and in the wetting (a = w) case, there exists a critical value ε a c such that when ε> ε a c the field touches the defect line a positive fraction of times (localization), while this does not happen for ε < ε a c (delocalization). The two critical values are nontrivial and distinct: 0 < ε p c < ε w c < ∞, 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 ε = ε p c is delocalized. On the other hand, the transition in the wetting model is of first order and for ε = ε w c the field is localized. The core of our approach is a Markov renewal theory description of the field. 1. Introduction and
On the approach to equilibrium for a polymer with adsorption and repulsion, Electronic Journal of Probability 13
, 2008
"... Abstract. We consider paths of a one–dimensional simple random walk conditioned to come back to the origin after L steps, L ∈ 2N. In the pinning model each path η has a weight λ N(η) , where λ> 0 and N(η) is the number of zeros in η. When the paths are constrained to be non–negative, the polymer ..."
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Cited by 14 (5 self)
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Abstract. We consider paths of a one–dimensional simple random walk conditioned to come back to the origin after L steps, L ∈ 2N. In the pinning model each path η has a weight λ N(η) , where λ> 0 and N(η) is the number of zeros in η. When the paths are constrained to be non–negative, the polymer is said to satisfy a hard– wall constraint. Such models are well known to undergo a localization/delocalization transition as the pinning strength λ is varied. In this paper we study a natural “spin flip ” dynamics for these models and derive several estimates on its spectral gap and mixing time. In particular, for the system with the wall we prove that relaxation to equilibrium is always at least as fast as in the free case (i.e. λ = 1 without the wall), where the gap and the mixing time are known to scale as L −2 and L 2 log L, respectively. This improves considerably over previously known results. For the system without the wall we show that the equilibrium phase transition has a clear dynamical manifestation: for λ � 1 relaxation is again at least as fast as the diffusive free case, but in the strictly delocalized phase (λ < 1) the gap is shown to be O(L −5/2), up to logarithmic corrections. As an application of our bounds, we prove stretched exponential relaxation of local functions in the localized regime.
Invariance principles for random walks conditioned to stay positive
, 2006
"... Let {Sn} be a random walk in the domain of attraction of a stable law Y, i.e. there exists a sequence of positive real numbers (an) such that Sn/an converges in law to Y. Our main result is that the rescaled process (S ⌊nt⌋/an, t ≥ 0), when conditioned to stay positive for all the time, converges i ..."
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Cited by 13 (4 self)
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Let {Sn} be a random walk in the domain of attraction of a stable law Y, i.e. there exists a sequence of positive real numbers (an) such that Sn/an converges in law to Y. Our main result is that the rescaled process (S ⌊nt⌋/an, t ≥ 0), when conditioned to stay positive for all the time, converges in law (in the functional sense) towards the corresponding stable Lévy process conditioned to stay positive in the same sense. Under some additional assumptions, we also prove a related invariance principle for the random walk killed at its first entrance in the negative halfline and conditioned to die at zero.
Nonexistence of random gradient Gibbs measures in continuous interface models in d
, 2008
"... Abstract We consider statistical mechanics models of continuous spins in a disordered environment. These models have a natural interpretation as effective interface models. It is wellknown that without disorder there are no interface Gibbs measures in infinite volume in dimension d = 2, while ther ..."
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Cited by 9 (1 self)
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Abstract We consider statistical mechanics models of continuous spins in a disordered environment. These models have a natural interpretation as effective interface models. It is wellknown that without disorder there are no interface Gibbs measures in infinite volume in dimension d = 2, while there are "gradient Gibbs measures" describing an infinitevolume distribution for the increments of the field, as was shown by Funaki and Spohn. In the present paper we show that adding a disorder term prohibits the existence of such gradient Gibbs measures for general interaction potentials in d = 2. This nonexistence result generalizes the simple case of Gaussian fields where it follows from an explicit computation. In d = 3 where random gradient Gibbs measures are expected to exist, our method provides a lower bound of the order of the inverse of the distance on the decay of correlations of Gibbs expectations w.r.t. the distribution of the random environment.
Scaling limit for a class of gradient fields with nonconvex potentials
, 2007
"... We consider gradient fields (φx: x ∈ Z d) whose law takes the GibbsBoltzmann form Z −1 exp{ − P 〈x,y〉 V (φy − φx)} where the sum runs over nearest neighbors. We assume that V admits the representation Z V (η) = − log (dκ) exp ˆ − 1 2κη2 ˜ where is a positive measure with compact support in (0, ..."
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Cited by 9 (2 self)
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We consider gradient fields (φx: x ∈ Z d) whose law takes the GibbsBoltzmann form Z −1 exp{ − P 〈x,y〉 V (φy − φx)} where the sum runs over nearest neighbors. We assume that V admits the representation Z V (η) = − log (dκ) exp ˆ − 1 2κη2 ˜ where is a positive measure with compact support in (0, ∞). Hence V is symmetric and nonconvex in general. While for strictly convex V ’s the translationinvariant, ergodic gradient Gibbs measures are completely characterized by their tilt, a nonconvex potential as above may lead to several ergodic gradient Gibbs measures with zero tilt. Still, every ergodic, zerotilt gradient Gibbs measure for the potential V from above scales to a Gaussian free field.
Decay of covariances, uniqueness of ergodic component and scaling limit for a class of ∇φ systems with nonconvex potential
, 2009
"... We consider a gradient interface model on the lattice with interaction potential which is a nonconvex perturbation of a convex potential. Using a technique which decouples the neighboring vertices sites into even and odd vertices, we show for a class of nonconvex potentials: the uniqueness of ergod ..."
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Cited by 5 (2 self)
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We consider a gradient interface model on the lattice with interaction potential which is a nonconvex perturbation of a convex potential. Using a technique which decouples the neighboring vertices sites into even and odd vertices, we show for a class of nonconvex potentials: the uniqueness of ergodic component for ∇φ Gibbs measures, the decay of covariances, the scaling limit and the strict convexity of the surface tension.
Scaling limit and cuberoot fluctuations in sos surfaces above a wall
, 2013
"... Consider the classical (2 + 1)dimensional SolidOnSolid model above a hard wall on an L × L box of Z2. The model describes a crystal surface by assigning a nonnegative integer height ηx to each site x in the box and 0 heights to its boundary. The probability of a surface configuration η is prop ..."
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Cited by 5 (4 self)
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Consider the classical (2 + 1)dimensional SolidOnSolid model above a hard wall on an L × L box of Z2. The model describes a crystal surface by assigning a nonnegative integer height ηx to each site x in the box and 0 heights to its boundary. The probability of a surface configuration η is proportional to exp(−βH(η)), where β is the inversetemperature and H(η) sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures. First we show that with high probability the height of almost all sites is concentrated on two levels, H(L) = ⌊(1/4β) logL ⌋ and H(L) − 1. Moreover, for most values of L the height is concentrated on the single value H(L). Next, we study the ensemble of level lines corresponding to the heights (H(L),H(L)−1,...). We prove that w.h.p. there is a unique macroscopic level line for each height. Furthermore, when taking a diverging sequence of system sizes Lk, the rescaled macroscopic level line at height H(Lk) − n has a limiting shape if the fractional parts of (1/4β) logLk converge to a noncritical value. The scaling limit is an explicit convex subset of the unit square Q and its boundary has a flat component on the boundary of Q. Finally, the highest macroscopic level line has L 1/3+o(1) k fluctuations along the flat part of the boundary of its limiting shape.
Random walk versus random line
, 904
"... Abstract: We consider random walks Xn in Z+, obeying a detailed balance condition, with a weak drift towards the origin when Xn ր ∞. We reconsider the equivalence in law between a random walk bridge and a 1+1 dimensional SolidOnSolid bridge with a corresponding Hamiltonian. Phase diagrams are disc ..."
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Cited by 2 (1 self)
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Abstract: We consider random walks Xn in Z+, obeying a detailed balance condition, with a weak drift towards the origin when Xn ր ∞. We reconsider the equivalence in law between a random walk bridge and a 1+1 dimensional SolidOnSolid bridge with a corresponding Hamiltonian. Phase diagrams are discussed in terms of recurrence versus wetting. A drift −δX −1 n + O(X −2 n) of the random walk yields a SolidOnSolid potential with an attractive well at the origin and a repulsive tail δ(2+δ) 8 X−2 n + O(X−3 n) at infinity, showing complete wetting for δ ≤ 1 and critical partial wetting for δ> 1.