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## Localization and delocalization of random interfaces (2006)

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342 | The two-dimensional ising model
- McCoy, Wu
- 1973
(Show Context)
Citation Context ...nals. This method was then successively improved, first in [67] to prove exponential decay of the 2-point function for this class of models, and then 2 Actually, this was done earlier by McCoy and Wu =-=[77]-=-, but they failed to interpret properly what they had computed. 26in [19] to establish precise estimates on the critical behavior in any dimensions for possibly long-range Gaussian interactions. Conc... |

169 |
On the extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the dffusion equation
- Brascamp, Lieb
(Show Context)
Citation Context ...or some c = c(d). P(ϕ0 ≥ T) ≍ e −cT2 , Remark 9. Of course, the tails have also Gaussian decay in the case of uniformly strictly convex interaction V , as follows, e.g., from Brascamp-Lieb inequality =-=[23]-=-. 1.2.5 Thermodynamical criteria of localization The above quantities are those one would very much like to compute in every situations. Unfortunately, this often turns out to be too difficult, and we... |

109 |
2007 Gaussian free fields for mathematicians
- Sheffield
(Show Context)
Citation Context ...ntinuous Gaussian free field with suitable covariance also holds for any interaction V which is uniformly strictly convex [78]. A more detailed introduction to the Gaussian free field can be found in =-=[84]-=-. 1.3 Discrete effective interface models: basic properties Let us now turn to the case of discrete effective interface models. It turns out that they have very different behavior in different situati... |

68 | G.: Entropic repulsion and the maximum of the twodimensional harmonic
- Bolthausen, Deuschel, et al.
- 2001
(Show Context)
Citation Context ...orresponding quantity for the Brownian bridge. In higher dimension the results are more interesting, since the extrema of the field turn out to be much larger than the typical values. It is proved in =-=[15]-=- that the maximum of the 2-dimensional finite-range Gaussian field in the box ΛN satisfies, for any δ > 0, (∣ ∣∣∣ sup ϕi − 2 √ ) g2 log N ∣ ≥ δ log N = 0 . (11) lim N→∞ PΛN i∈ΛN 10where gd was introd... |

53 |
The Kosterlitz-Thouless transition in two-dimensional abelian spin systems and the Coulomb gas
- Fröhlich, Spencer
- 1981
(Show Context)
Citation Context ...is still poorly understood. There are no result concerning the Gaussian asymptotics. The stronger results known to date are those given in the celebrated (and difficult!) work of Fröhlich and Spencer =-=[53]-=-, who proved that, at small enough β, for any i, j and Λ large enough. var β P (ϕi − ϕj) ≍ log |j − i| , Λ 14Open Problem 3. Prove the existence of a roughening transition directly in terms of the in... |

49 | Rigorous probabilistic analysis on equilibrium crystal shapes,
- Bodineau, Ioffe, et al.
- 2000
(Show Context)
Citation Context ...t still excellent reviews of Fisher [50] and Bricmont et al [25]. Finally, concerning macroscopic variational formula, as well as various related issues for lattice gases, I refer to the review paper =-=[11]-=-. 1.1 The free model The interface is described by a function ϕ : Zd → R (continuous effective interface models) or ϕ : Zd → Z (discrete effective interface models); the quantity ϕi is interpreted as ... |

46 | On homogenization and scaling limit of some gradient perturbations of a massless free field
- Naddaf, Spencer
- 1997
(Show Context)
Citation Context ...ion of the Brownian motion killed as it exits V . Actually convergence to a continuous Gaussian free field with suitable covariance also holds for any interaction V which is uniformly strictly convex =-=[78]-=-. A more detailed introduction to the Gaussian free field can be found in [84]. 1.3 Discrete effective interface models: basic properties Let us now turn to the case of discrete effective interface mo... |

45 |
D.: The Random Walk Representation of Classical Spin Systems and Correlation Inequalities. It The Skeleton Inequalities
- Brydges, Frohlich, et al.
- 1983
(Show Context)
Citation Context ...localized phase for weak enough self-potential, while localization always occur in the latter case. After these early works limited to the one-dimensional model, the first rigorous work I am aware is =-=[26]-=- in which the authors introduced a new random walk representation (different from the one described in the introduction) and applied it in particular to prove exponential decay of the 2-point function... |

45 | Smoothing effect of quenched disorder on polymer depinning transitions
- Giacomin, Toninelli
(Show Context)
Citation Context ...hen the interface still almost surely gets localized (in the sense that there is a density of pinned sites) even when w is slightly negative (that is, in average the reward is actually a penalty). In =-=[59]-=-, it is proved for the same type of models that 34U(x) c 2q 2 x 2 Figure 8: The pinning potential considered in [46] and its quadratic approximation yielding the effective mass. the presence of disor... |

32 | Pinning of polymers and interfaces by random potentials
- Alexander, Sidoravicius
(Show Context)
Citation Context ...C.1.] for a proof. 5.5 Additional results Random potential There have also been several works on the study of localization by a random pinning potential, in particular in dimension 1. For example, in =-=[6, 79]-=-, it is proved (in a rather general 1-dimensional setup) that if the pinning potential at site i is given by w + Wi, with w ∈ R a constant, and Wi a family of i.i.d. real-valued random variables with ... |

32 | Height fluctuations in the honeycomb dimer model,
- Kenyon
- 2008
(Show Context)
Citation Context ...nterfaces is identical to that of their continuous counterpart. In particular they should have Gaussian asymptotics. This turns out to be quite delicate, and the only rigorous works I am aware of are =-=[70, 71, 56]-=-: they establish weak convergence 13Figure 3: The oriented level lines of a discrete effective interface; the orientation specify the type of the contour, i.e. whether it is increasing or decreasing ... |

30 | Entropic repulsion of the lattice free field,
- Bolthausen, Deuschel, et al.
- 1995
(Show Context)
Citation Context ...inite-range Gaussian field in the box ΛN satisfies, for any δ > 0, lim N→∞ PΛN (∣ ∣∣∣ sup ϕi − 2 √ g2 log N ∣ i∈ΛN ) ≥ δ log N = 0 . (11) 10where gd was introduced in (8). Similarly, it is proved in =-=[16, 17]-=- that the maximum of the d-dimensional, d ≥ 3, finite-range Gaussian field in the box ΛN satisfies, for any δ > 0, ∣ ) PΛN lim N→∞ PΛN (∣ ∣∣∣ sup ϕi − √ √ 2dgd log N ∣ ≥ δ√log N i∈ΛN = 0 . (12) It is ... |

30 | Dominos and the Gaussian free field
- Kenyon
- 2001
(Show Context)
Citation Context ...nterfaces is identical to that of their continuous counterpart. In particular they should have Gaussian asymptotics. This turns out to be quite delicate, and the only rigorous works I am aware of are =-=[70, 71, 56]-=-: they establish weak convergence 13Figure 3: The oriented level lines of a discrete effective interface; the orientation specify the type of the contour, i.e. whether it is increasing or decreasing ... |

28 |
Weakly pinned random walk on the wall: pathwise descriptions of the phase transition.
- Isozaki, Yoshida
- 2001
(Show Context)
Citation Context ... continuous. Open Problem 10. Determine the nature of the phase transition in the twodimensional model. In dimension d = 1, however, the understanding is pretty much complete. After an initial result =-=[68]-=-, restricted to a particular choice of underlying random walk, the following result, valid for essentially arbitrary interaction/underlying random walk was proved in [42] . Suppose that the interactio... |

27 |
Random surfaces in statistical mechanics: roughning, rounding, wetting,
- Bricmont, Mellouki, et al.
- 1986
(Show Context)
Citation Context ...an interface with a neutral hard wall, i.e. the phenomenon of entropic repulsion; since excellent reviews about this (and related) topic can be found in [13, 58] (for continuous effective models) and =-=[25]-=- (for discrete ones), our discussion will stay rather superficial. The presence of the hard wall at the sites of ΛM is modeled by the positivity def constraint ΩM,+ = {ϕi ≥ 0, ∀i ∈ ΛM}. The measure de... |

27 | Extremes of the discrete two-dimensional Gaussian free
- Daviaud
- 2006
(Show Context)
Citation Context ...ate the geometry and distribution of these spikes. The most interesting picture emerges when d = 2. A detailed study of the spikes of the 2-dimensional nearest-neighbor Gaussian model can be found in =-=[34]-=-. The main results can be stated as follows. • The spikes are rather fat. Let 0 < λ < 1, 0 < ε < 1, and DN(λ) def { = sup a ∈ N : ∃i ∈ ΛεN, min |j−i|∞≤a ϕj > 2λ √ } g2 log N Then log DN(λ) 1 λ lim = −... |

27 |
On the symmetry of the Gibbs states in two-dimensional lattice systems.
- Pfister
- 1981
(Show Context)
Citation Context ... Hamiltonian enjoys a continuous symmetry: H(ϕ) = H(ϕ + c), for any c ∈ R, since the formal Hamiltonian is actually only a function of the gradient field ∇ϕ. By standard Mermin-Wagner– type arguments =-=[45, 80, 24, 66]-=-, it then follows that this continuous symmetry has to be present also at the level of the infinite-volume Gibbs measures, when d = 1 or 2 and the interaction does not decay too slowly (a condition au... |

25 | A rigorous renormalization group method for interfaces in random media.
- Bovier, Kulske
- 1994
(Show Context)
Citation Context ...hat the disorder is fixed, not sampled from some given distribution). Finally, pinning of an SOS interface by spatial disorder not restricted to a plane but present everywhere in space was studied in =-=[21, 22]-=-. The main results are that: 1) In dimensions d ≥ 3, the interface is rigid, provided that β be large enough and the disorder sufficiently weakly coupled to the field. 2) In dimensions d ≤ 2, the inte... |

25 |
Large deviations and concentration properties for ∇φ interface models, Probab. Theory Relat. Fields 117
- Deuschel, Giacomin, et al.
- 2000
(Show Context)
Citation Context ...ess, it turns out that it is possible to derive a generalization of this representation valid for the whole class of uniformly strictly convex interactions. Such a generalization has been proposed in =-=[41]-=- and is a probabilistic reformulation of an earlier result, in the PDE context, by Helffer and Sjöstrand [64]. It works as follows: One constructs a stochastic process (Φ(t), X(t)) where • Φ( · ) is a... |

24 |
On estimating the derivatives of symmetric diffusions in stationary random environments, with applications to the ∇φ interface model
- Delmotte, Deuschel
(Show Context)
Citation Context ...Λ(ϕi, ϕj) = EΛ ( E Λ i,· ∫ τΛ 0 ) 1 {X(s)=j}ds . (6) Thanks to the ellipticity of the random walk X(t) under the assumption of strict convexity, it is possible to obtain some Aronson type bounds, see =-=[61, 37]-=-, showing that this RWRE has the same qualitative behavior as the random walk in the Gaussian case. This explains why most of the results that have been obtained for the Gaussian model also hold in th... |

24 |
Scaling limits of equilibrium wetting models
- Deuschel, Giacomin, et al.
- 2005
(Show Context)
Citation Context ... 0. This is the case, e.g., for the Gaussian model in dimension d ≥ 3. This shows that the wetting transition is a continuous transition in that case. It is also known to be continuous in dimension 1 =-=[42]-=-, but nothing at all in known in the two-dimensional case, although it is clearly expected that the transition is also continuous. Open Problem 10. Determine the nature of the phase transition in the ... |

22 | 2D models of statistical physics with continuous symmetry: the case of singular interactions
- Ioffe, Shlosman, et al.
(Show Context)
Citation Context ... Hamiltonian enjoys a continuous symmetry: H(ϕ) = H(ϕ + c), for any c ∈ R, since the formal Hamiltonian is actually only a function of the gradient field ∇ϕ. By standard Mermin-Wagner– type arguments =-=[45, 80, 24, 66]-=-, it then follows that this continuous symmetry has to be present also at the level of the infinite-volume Gibbs measures, when d = 1 or 2 and the interaction does not decay too slowly (a condition au... |

21 |
The Statistical Mechanics of anharmonic lattices,
- Brascamp, Lebowitz, et al.
- 1975
(Show Context)
Citation Context ...ecise information that what is given here can be obtained. Remark 7. Apart from the random walk representation, there is only one general tool to prove localization: the Brascamp-Lieb inequality, see =-=[24]-=-. Unfortunately, the class to which this approach applies, if already quite large, is still much too limited. Namely, it is required that V satisfies one of the following conditions: 1. V (x) = ax 2 +... |

21 |
On the decay of correlations in SO(n)- symmetric ferromagnets
- McBryan, Spencer
- 1977
(Show Context)
Citation Context ...horizontal slab of height 2ℓ. 4.2 Main results Continuous effective interface models Of course, once constrained inside a slab, the interface is localized in any dimension. It has also been proved in =-=[76]-=- that it becomes massive in any dimension, but with an estimate for the mass that is only correct in dimensions 3 and above. This result has later been improved in order to get the qualitatively corre... |

19 | On the layering transition of an SOS surface interacting with a wall
- Cesi, Martinelli
- 1996
(Show Context)
Citation Context ..., at which the thickness of the film increases by one microscopic unit. This phenomenon has undergone a detailed rigorous study, at sufficiently large β, for the discrete SOS model with W(x) = |x| in =-=[44, 30, 74]-=-. Let us now turn to rough interfaces. For systems above their roughening temperatures (e.g. Ising model in dimension d = 3 between Tr and Tc, the 2dimensional Ising model at any subcritical temperatu... |

18 |
Solvable model with a roughening transition for a planar Ising ferromagnet Phys.
- Abraham
- 1980
(Show Context)
Citation Context ...one-dimensional effective interface models was initiated by a desire to better understand the wetting phenomenon discovered through exact computations by D. Abraham in the two-dimensional Ising model =-=[1]-=- 2 ; more on that in Section 6. It was thus considered useful to prove a similar result in the simpler context of one-dimensional effective interface models. Of particular interest to these earlier wo... |

16 | Non-Gaussian surface pinned by a weak potential, Probab
- Deuschel, Velenik
(Show Context)
Citation Context ...e variance, as well as exponential decay of the 2-point function by Bolthausen and Brydges [14]. All these results were limited to Gaussian interactions. A more general approach was then developed in =-=[43]-=- in order to treat the case of non-Gaussian (but uniformly strictly convex) interactions and, as a side-product, provided stronger results such as the correct tail for the one-site marginals. This met... |

16 |
Aspects of statistical mechanics of random surfaces
- Giacomin
- 2001
(Show Context)
Citation Context ...e very good reviews and lecture notes covering in depth these issues. For additional informations on effective interface models I recommend in particular the lecture notes by Funaki [55] and Giacomin =-=[58]-=-. Bolthausen has also written several nice review papers on various topics covered or not in these notes; among them, I think his review on entropic repulsion [12] is really quite enlightening. Concer... |

15 |
Entropic repulsion for the free field: pathwise characterization in d
- Deuschel, Giacomin
- 1999
(Show Context)
Citation Context ...ctly convex interactions [40]. In the Gaussian case, even more is known about the repelled field: Once the new average is subtracted, it is weakly converging to the unconstrained infinitevolume field =-=[39]-=-, which means that both fields look locally the same: There exists a sequence aM, with limM→∞ aM/ √ 4gd log M = 1, such that P +(M) ( · + aM) M→∞ =⇒ P . I emphasize however that the two fields have th... |

15 |
Correlation inequalities in the thermodynamic limit for classical and quantum systems
- J, Park
- 1990
(Show Context)
Citation Context ...hat the fluctuations of the discrete models should be dominated by those of the continuous ones. There are indeed a few comparison inequalities of this type, but they are restricted to V (x) = 1 2 x2 =-=[52]-=- (the subscripts “D” and “C” serve to distinguish between the discrete and continuous models): var β P (ϕ0) ≤ var β Λ,D P (ϕ0), Λ,C (13) E β Λ,D (ecϕ0) ≤ E β Λ,C (ecϕ0), (14) for any c ≥ 0. Actually t... |

15 |
On the correlation for Kac-like models in the convex
- Helffer, Sjöstrand
- 1994
(Show Context)
Citation Context ...lass of uniformly strictly convex interactions. Such a generalization has been proposed in [41] and is a probabilistic reformulation of an earlier result, in the PDE context, by Helffer and Sjöstrand =-=[64]-=-. It works as follows: One constructs a stochastic process (Φ(t), X(t)) where • Φ( · ) is a diffusion on R Zd with invariant measure PΛ; • given a trajectory ϕ( · ) of the process Φ, X(t) is an, in ge... |

14 | Critical behavior of the massless free field at the depinning transition
- Bolthausen, Velenik
(Show Context)
Citation Context ...xponential decay of the 2-point function for this class of models, and then 2 Actually, this was done earlier by McCoy and Wu [77], but they failed to interpret properly what they had computed. 26in =-=[19]-=- to establish precise estimates on the critical behavior in any dimensions for possibly long-range Gaussian interactions. Concerning discrete effective interface models, the situation is as follows: i... |

14 |
There are no nice interfaces
- Bovier, Kulske
- 1996
(Show Context)
Citation Context ...hat the disorder is fixed, not sampled from some given distribution). Finally, pinning of an SOS interface by spatial disorder not restricted to a plane but present everywhere in space was studied in =-=[21, 22]-=-. The main results are that: 1) In dimensions d ≥ 3, the interface is rigid, provided that β be large enough and the disorder sufficiently weakly coupled to the field. 2) In dimensions d ≤ 2, the inte... |

13 |
Localisation-delocalisation transition in a solid-on-solid model with a pinning potential
- Burkhardt
- 1981
(Show Context)
Citation Context ... more on that in Section 6. It was thus considered useful to prove a similar result in the simpler context of one-dimensional effective interface models. Of particular interest to these earlier works =-=[31, 85, 27, 2]-=- was the dramatic difference in behavior between cases where the localizing self-potential was coupled or not with a positivity constraint: in the former case there is a delocalized phase for weak eno... |

13 |
Entropic repulsion of the lattice free field. II. The 0-boundary case
- Deuschel
- 1996
(Show Context)
Citation Context ...cally. For example, the minimum of the field in the box ΛN is much smaller than that of the centered repelled field, compare (19) and (11). The case N = M, i.e. the measure P +(N) has been treated in =-=[38]-=- and [40]; ΛN it turns out that (19) still holds, with the same constant, provided the sup is restricted to i ∈ ΛǫN, 0 < ǫ < 1). Moreover, estimates for the growth of the interface near to the boundar... |

13 |
Layering transition in SOS model with external magnetic
- Dinaburg, Mazel
- 1996
(Show Context)
Citation Context ..., at which the thickness of the film increases by one microscopic unit. This phenomenon has undergone a detailed rigorous study, at sufficiently large β, for the discrete SOS model with W(x) = |x| in =-=[44, 30, 74]-=-. Let us now turn to rough interfaces. For systems above their roughening temperatures (e.g. Ising model in dimension d = 3 between Tr and Tc, the 2dimensional Ising model at any subcritical temperatu... |

12 |
Random walk representations and entropic repulsion for gradient models
- Bolthausen
- 2001
(Show Context)
Citation Context ...re notes by Funaki [55] and Giacomin [58]. Bolthausen has also written several nice review papers on various topics covered or not in these notes; among them, I think his review on entropic repulsion =-=[12]-=- is really quite enlightening. Concerning discrete heights models, I would refer to the older, but still excellent reviews of Fisher [50] and Bricmont et al [25]. Finally, concerning macroscopic varia... |

12 | Phase coexistence of gradient Gibbs states. - Biskup, Kotecky - 2007 |

11 |
A note on wetting transition for gradient fields, Stochastic Process
- Caputo, Velenik
(Show Context)
Citation Context ...imensions. Unfortunately, the above result in dimension 2 has only been established for the Gaussian model; an extension to uniformly strictly convex V would also allow an extension of the results of =-=[29]-=-, discussed in Section 6, to this class of interactions. Open Problem 6. Extend (21) (when d = 2) to the case of uniformly strictly convex interactions V . Disordered wall In the above, the wall was c... |

11 |
Equilibrium fluctuations for ∇ϕ interface model.
- Giacomin, Olla, et al.
- 2001
(Show Context)
Citation Context ...Λ(ϕi, ϕj) = EΛ ( E Λ i,· ∫ τΛ 0 ) 1 {X(s)=j}ds . (6) Thanks to the ellipticity of the random walk X(t) under the assumption of strict convexity, it is possible to obtain some Aronson type bounds, see =-=[61, 37]-=-, showing that this RWRE has the same qualitative behavior as the random walk in the Gaussian case. This explains why most of the results that have been obtained for the Gaussian model also hold in th... |

10 | Enhanced interface repulsion from quenched hard-wall randomness
- Bertacchi, Giacomin
- 2002
(Show Context)
Citation Context ...y convex interactions V . Disordered wall In the above, the wall was considered to be perfectly planar. I briefly mention here some studies of this phenomenon in the presence of a rough substrate. In =-=[7]-=-, the wall is modeled by a family of i.i.d. random variables σi, i ∈ Z d , d ≥ 3 independent of the interface ϕ. The constraint becomes of course ϕi ≥ σi, for all i ∈ Z d . It is proved that the behav... |

10 |
Pinning of an interface by a weak potential
- Dunlop, Magnen, et al.
- 1992
(Show Context)
Citation Context ...ove exponential decay of the 2-point function under the measure P a,b in dimensions d ≥ 3 for the Gaussian model. A study of the much more delicate two-dimensional case was then done by Dunlop et al. =-=[48]-=-, who proved that the field is localized for any strictly positive values of a and b, in the sense that E a,b (|ϕ0|) < ∞. This result was improved to show finiteness of the variance, as well as expone... |

10 |
On an invariance principle for phase separation lines
- Greenberg, Ioffe
(Show Context)
Citation Context ...where this can be sometimes proved is d = 1. In that case, it is for example possible to prove that interfaces in the 2D Ising model have the same Brownian asymptotics as their effective counterparts =-=[28, 63]-=-. 1.2 Continuous effective interface models: basic properties I start by discussing basic properties of continuous effective interface models, as these are in general better understood than their disc... |

10 | Universality of critical behaviour in a class of recurrent random walks
- Hryniv, Velenik
(Show Context)
Citation Context ...ch piece the desired event has strictly positive probability, uniformly in ℓ, which gives an additional factor exp(−O(N/ℓ 2 )). An analogous reasoning yields the corresponding upper bound; see, e.g., =-=[65]-=-. In the case of the SOS model, it is only proved that when d = 2 and β ≫ 1, the above quantity is of order exp(−O(ℓ)). This is one example of the fact that in phenomena depending on the behavior of s... |

10 |
Entropic repulsion for a Gaussian lattice field with certain finite range interaction
- Sakagawa
(Show Context)
Citation Context ...rom the mathematical point of view, due to the lack of nearly all the main tools used in the study of effective interfaces: no nice random walk representation, no FKG inequalities, etc. I refer 15to =-=[82, 72]-=- for rigorous results (concerning the entropic repulsion phenomenon) on such models in dimensions 5 and more. 2 Massive model 2.1 Description of the model A very crude way to localize the interface is... |

9 |
An exactly solved model with a wetting transition
- Abraham, Smith
- 1986
(Show Context)
Citation Context ...he sense that otherwise the critical behavior would be different). Remark 23. The only case previously rigorously studied in the literature is the one-dimensional continuous SOS model with W(x) = |x| =-=[5]-=-. This case turns out to be exactly solvable; in particular, the authors also obtain some explicit constants, which are out of reach in the general case. See also the heuristic discussion in [50]. Rem... |

9 | Entropic repulsion of an interface in an external field, Probab. Theory Related Fields 129
- Velenik
(Show Context)
Citation Context ...f the asymptotic behavior of the free energy of the constrained interface. Remark 13. A completely different proof of (24), valid for arbitrary uniformly strictly convex interactions, can be found in =-=[86]-=-; it makes use of the results of the preceding section on entropic repulsion, together with correlation inequalities. However, it does not permit to recover the estimates (23) and (22) for the varianc... |

8 |
The pinning of an interface by a planar defect
- Chalker
- 1982
(Show Context)
Citation Context ...there was a series of papers [31, 27, 85, 33] (see also [50]) analyzing the same question in the simpler settings of one-dimensional effective interface models (still through exact computations), and =-=[32]-=- establishing the existence of the wetting transition in the SOS model in any dimension. Only recently has the rigorous analysis of this problem been reconsidered, both providing stronger and more gen... |

8 |
A remark on the low temperature behavior of an SOS interface
- Lebowitz, Mazel
- 1996
(Show Context)
Citation Context ..., at which the thickness of the film increases by one microscopic unit. This phenomenon has undergone a detailed rigorous study, at sufficiently large β, for the discrete SOS model with W(x) = |x| in =-=[44, 30, 74]-=-. Let us now turn to rough interfaces. For systems above their roughening temperatures (e.g. Ising model in dimension d = 3 between Tr and Tc, the 2dimensional Ising model at any subcritical temperatu... |

7 |
Surfaces and Peierls contours: 3-d wetting and 2-d Ising percolation
- Abraham, Newman
- 1989
(Show Context)
Citation Context ...g these terms yields the claimed result. Remark 11. To see that this repulsion effect is really due to the downward spikes, it is interesting to compare with what happens in the wedding cake model of =-=[3, 4]-=-; see Fig. 4. The latter is a discrete model of random surface which has the following properties: 1. The difference between neighboring heights is 0, 1 or −1; 2. a connected region of constant height... |

7 | Wall repulsion and mutual interface repulsion: a harmonic crystal model in high dimensions. Stochastic Process
- Bertacchi, Giacomin
- 2004
(Show Context)
Citation Context ...ply directly. I learned from G. Giacomin, however, that the case of two fields with non-commuting covariance matrices, which he has treated [57], is more subtle, and actually requires a new proof. In =-=[9]-=-, the authors consider two d-dimensional, d ≥ 3, Gaussian interfaces ϕ1 and ϕ2 , with the constraint that ϕ2 i ≥ ϕ1i ≥ 0 for all i. The main result is that the height of ϕ1 is still given (at leading ... |

7 | On entropic reduction of fluctuations
- Bodineau, Giacomin, et al.
- 2001
(Show Context)
Citation Context ...itively increase fluctuations, but there is a delicate problem of exchange of limits, and there are examples in which fluctuations at zero-temperature are much larger than at finite temperatures, see =-=[10]-=-). The behavior when ⃗n is vertical, i.e. the case ψ ≡ 0, is more interesting. It is expected that there is a phase transition, the roughening transition, at an inverse temperature 0 < βr < ∞ such tha... |

7 | Absence of a wetting transition for a pinned harmonic crystal in dimensions three and
- Bolthausen, Deuschel, et al.
- 2000
(Show Context)
Citation Context ...n the dimension and on the tail of the gradient interaction. 1. V (x) = x 2 : ηc > 0 if and only if d ≤ 2. 2. V Lipschitz: ηc > 0 for all d ≥ 1. The “only if” part of the first statement is proved in =-=[18]-=-, while the “if” part and the second statement are proved in [29]; heuristic for these results are provided in Subsection 6.3. Remark 21. The usual derivation of Gaussian effective interface models fr... |

7 |
Pinning of a rough interface by an external potential
- Leeuwen, Hilhorst
- 1981
(Show Context)
Citation Context ... more on that in Section 6. It was thus considered useful to prove a similar result in the simpler context of one-dimensional effective interface models. Of particular interest to these earlier works =-=[31, 85, 27, 2]-=- was the dramatic difference in behavior between cases where the localizing self-potential was coupled or not with a positivity constraint: in the former case there is a delocalized phase for weak eno... |

6 |
The wetting transition in a random surface model
- Abraham, Newman
- 1991
(Show Context)
Citation Context ...g these terms yields the claimed result. Remark 11. To see that this repulsion effect is really due to the downward spikes, it is interesting to compare with what happens in the wedding cake model of =-=[3, 4]-=-; see Fig. 4. The latter is a discrete model of random surface which has the following properties: 1. The difference between neighboring heights is 0, 1 or −1; 2. a connected region of constant height... |

6 |
A note on the decay of correlations under δ-pinning, Probab
- Ioffe, Velenik
(Show Context)
Citation Context ...but uniformly strictly convex) interactions and, as a side-product, provided stronger results such as the correct tail for the one-site marginals. This method was then successively improved, first in =-=[67]-=- to prove exponential decay of the 2-point function for this class of models, and then 2 Actually, this was done earlier by McCoy and Wu [77], but they failed to interpret properly what they had compu... |

6 |
A note on Green’s function for random walk in four dimensions
- LAWLER
- 1994
(Show Context)
Citation Context ...so depending on the transition kernel p( · ), such that covP(ϕ0, ϕi) = (Rd + o(1)) |i| 2−d . Notice that this result requires that the transition kernel has slightly more than moments of order d (see =-=[73]-=- for a more on that), a condition satisfied when (3) holds. We see that the corresponding limiting field has very strong correlations (not summable). In particular, the mass satisfies mP(x) ≡ 0. This ... |

5 |
Localization and decay of correlations for a pinned lattice free field in dimension two
- Bolthausen, Brydges
(Show Context)
Citation Context ...tive values of a and b, in the sense that E a,b (|ϕ0|) < ∞. This result was improved to show finiteness of the variance, as well as exponential decay of the 2-point function by Bolthausen and Brydges =-=[14]-=-. All these results were limited to Gaussian interactions. A more general approach was then developed in [43] in order to treat the case of non-Gaussian (but uniformly strictly convex) interactions an... |

5 |
Entropic repulsion for massless fields, Stochastic Process
- Deuschel, Giacomin
(Show Context)
Citation Context ...M, and therefore of the same order as the repulsion height. The above estimate remains qualitatively true (that is, without matching upper and lower bounds) for uniformly strictly convex interactions =-=[40]-=-. In the Gaussian case, even more is known about the repelled field: Once the new average is subtracted, it is weakly converging to the unconstrained infinitevolume field [39], which means that both f... |

5 |
Cassie’s law and concavity of wall tension with respect to disorder
- Dunlop, Topolski
(Show Context)
Citation Context ... the wetting transition in the presence of a longrange wall/interface interaction. Disordered wall The model in the presence of a disordered substrate has also been studied in several works, see e.g. =-=[35, 36, 49, 20]-=-. Two types of disorder have been considered: random pinning potentials (similar to what is discussed in Subsection 5.5), and rough walls (similar to what is discussed in Subsection 3.3). The main con... |

5 |
Proof of confinement of static quarks in 3-dimensional U(1) lattice gauge theory for all values of the coupling constant
- Göpfert, Mack
- 1982
(Show Context)
Citation Context ... as β → 0. 1.3.3 Dimension 3 and higher The behavior in dimensions 3 and higher is expected to be radically different: For any β > 0, the interface should be localized and massive. This was proved in =-=[62]-=- for the DG model in dimension 3 (note that localization is not very surprising since the same is also true for continuous effective interface models; it is the exponential decay of correlations that ... |

5 | Polymer pinning at an interface.
- Pétrélis
- 2006
(Show Context)
Citation Context ...C.1.] for a proof. 5.5 Additional results Random potential There have also been several works on the study of localization by a random pinning potential, in particular in dimension 1. For example, in =-=[6, 79]-=-, it is proved (in a rather general 1-dimensional setup) that if the pinning potential at site i is given by w + Wi, with w ∈ R a constant, and Wi a family of i.i.d. real-valued random variables with ... |

5 |
Ornstein-Zernike theory for finiterange Ising models above tc. Probab. Theory Relat
- Campanino, Ioffe, et al.
(Show Context)
Citation Context ...where this can be sometimes proved is d = 1. In that case, it is for example possible to prove that interfaces in the 2D Ising model have the same Brownian asymptotics as their effective counterparts =-=[28, 63]-=-. 1.2 Continuous effective interface models: basic properties I start by discussing basic properties of continuous effective interface models, as these are in general better understood than their disc... |

5 | Large deviations and interacting random walks - Bolthausen - 2002 |

4 |
On the repulsion of an interface above a correlated substrate
- Bertacchi, Giacomin
- 2003
(Show Context)
Citation Context ...flat wall. • Almost-Gaussian tails: Assume that there exists Q > 0 such that lim r→∞ r−2P(σ0 ≥ r) = −1/2Q . Then the repulsion height is almost-surely given (at leading order) by √ 4(gd + Q)log N. In =-=[8]-=-, the wall was itself sampled according to the law of a Gaussian effective interface model. It is then proved that, at leading order, this strongly correlated substrate gives almost-surely rise to pre... |

4 |
Large-field versus small-field expansions and Sobolev inequalities
- Lemberger
- 1995
(Show Context)
Citation Context ...upled to the field. 2) In dimensions d ≤ 2, the interface is never rigid. Mean-field regime The critical behavior of the covariance has also been obtained in a mean-field regime in [46, 47], see also =-=[75]-=-. I briefly describe the setting and the result in order to show the difference with the regime discussed in this section. The measure considered in [46] is the following perturbation of the Gaussian ... |

4 |
Binding of a domain wall in the planar Ising ferromagnet
- Abraham
- 1981
(Show Context)
Citation Context ... more on that in Section 6. It was thus considered useful to prove a similar result in the simpler context of one-dimensional effective interface models. Of particular interest to these earlier works =-=[31, 85, 27, 2]-=- was the dramatic difference in behavior between cases where the localizing self-potential was coupled or not with a positivity constraint: in the former case there is a delocalized phase for weak eno... |

3 |
The pinning of a domain wall by weakened bonds in two dimensions
- Chalker
- 1981
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Nonexistence of one- and two-dimensional Gibbs fields with noncompact group of continuous symmetries
- Dobrushin, Shlosman
(Show Context)
Citation Context ... Hamiltonian enjoys a continuous symmetry: H(ϕ) = H(ϕ + c), for any c ∈ R, since the formal Hamiltonian is actually only a function of the gradient field ∇ϕ. By standard Mermin-Wagner– type arguments =-=[45, 80, 24, 66]-=-, it then follows that this continuous symmetry has to be present also at the level of the infinite-volume Gibbs measures, when d = 1 or 2 and the interaction does not decay too slowly (a condition au... |

3 |
The phase transition in the discrete Gaussian chain with 1/r 2 interaction energy
- Fröhlich, Zegarliński
- 1991
(Show Context)
Citation Context ...ved that the one-dimensional DG model with p(i) ∼ |i| −r describes a rigid interface at any temperatures if 1 < r < 2, while it is always rough when r > 2. The marginal case r = 2 has been studied in =-=[54]-=-, where it is proved that there is a roughening transition from a rigid to a rough phase as the temperature increases. Moreover in the rough phase, var P β(ϕi − ϕj) ≥ c(β)log |j − i|. Membranes Beside... |

3 |
On stochastic domination in the Brascamp{Lieb framework, preprint (2001), accepted for publication on
- Giacomin
(Show Context)
Citation Context ...nnected, to simplify the argument. 4 The only estimate where the Gaussian assumption was used in the proof in [19], see Footnote 2 therein, can be extended using Brascamp-Lieb inequality, as shown in =-=[60]-=-. 290 def Let us write B = B, and define Bk , k ≥ 1, as being the set of cells obtained by adding to Bk−1 all its neighboring cells. Let also ¯ k be the largest value of k such that Bk def ⊂ Λ, and s... |

3 |
Pinning by a sparse potential, Stochastic Process
- Janvresse, Rue, et al.
(Show Context)
Citation Context ...rresponding deterministic case (Wi ≡ 0). In a different spirit, the case of a diluted pinning potential (that is, a pinning potential taking value ζ > 0 or 0 at each site of the box) is considered in =-=[69]-=-. It is proved that, for d = 1 or 2, the interface is localized (again, in the sense that there is a density of pinned sites) if and only if the sites at which the pinning potential is non-zero have p... |

3 |
Entropic repulsion for two dimensional multi-layered harmonic crystals
- Sakagawa
(Show Context)
Citation Context ...og N and is therefore unaffected by the presence of ϕ2 (there is no “pressure” on the lower interface from the upper one). The height of ϕ2 itself is given by ( √ 4g1 d + √ 4(g1 d + g2 d ))√log N. In =-=[83]-=-, the corresponding question was considered for K ≥ 2 two-dimensional Gaussian interfaces above a hard wall. The results obtained are completely similar. 4 Confinement between two walls 4.1 Descriptio... |

3 | Smoothening effect of quenched disorder on polymer depinning transition
- Giacomin, Toninelli
(Show Context)
Citation Context ...hen the interface still almost surely gets localized (in the sense that there is a density of pinned sites) even when w is slightly negative (that is, in average the reward is actually a penalty). In =-=[59]-=-, it is proved for the same type of models that the presence of disorder induces a smoothening of the phase transition in the sense that it becomes higher order than in the corresponding deterministic... |

2 | An equilibrium lattice model of wetting on rough substrates
- Borgs, Coninck, et al.
- 1999
(Show Context)
Citation Context ... the wetting transition in the presence of a longrange wall/interface interaction. Disordered wall The model in the presence of a disordered substrate has also been studied in several works, see e.g. =-=[35, 36, 49, 20]-=-. Two types of disorder have been considered: random pinning potentials (similar to what is discussed in Subsection 5.5), and rough walls (similar to what is discussed in Subsection 3.3). The main con... |

2 |
Pinning and roughening of one-dimensional models of interfaces and steps
- Chui, Weeks
- 1981
(Show Context)
Citation Context ... that the width diverges (sub-linearly) with the system size. 36was able to prove its existence for the two-dimensional Ising model [1]. Immediately following this work, there was a series of papers =-=[31, 27, 85, 33]-=- (see also [50]) analyzing the same question in the simpler settings of one-dimensional effective interface models (still through exact computations), and [32] establishing the existence of the wettin... |

2 |
Mass generation for an interface in the mean field regime
- Dunlop, Magnen, et al.
- 1992
(Show Context)
Citation Context ... w is slightly negative (that is, in average the reward is actually a penalty). In [59], it is proved for the same type of models that 34U(x) c 2q 2 x 2 Figure 8: The pinning potential considered in =-=[46]-=- and its quadratic approximation yielding the effective mass. the presence of disorder induces a smoothening of the phase transition in the sense that it becomes higher order than in the corresponding... |

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Is short-range “critical” wetting a first-order transition
- Fisher, Jin
- 1992
(Show Context)
Citation Context ...understood, even rigorously, in the two-dimensional Ising model, the situation in the three-dimensional Ising model is still controversial, even among physicists. In particular, it has been suggested =-=[51]-=- that another kind of effective models should be considered if one wants to correctly predict the behavior observed in this model. From a mathematical point of view, these new effective models are too... |

2 |
Lectures on probability theory and statistics, volume 1869
- Dembo, Funaki
(Show Context)
Citation Context ...ntionned. There are very good reviews and lecture notes covering in depth these issues. For additional informations on effective interface models I recommend in particular the lecture notes by Funaki =-=[55]-=- and Giacomin [58]. Bolthausen has also written several nice review papers on various topics covered or not in these notes; among them, I think his review on entropic repulsion [12] is really quite en... |

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private communication
- Giacomin
(Show Context)
Citation Context ...an and independent, so the results of the present section apply directly. I learned from G. Giacomin, however, that the case of two fields with non-commuting covariance matrices, which he has treated =-=[57]-=-, is more subtle, and actually requires a new proof. In [9], the authors consider two d-dimensional, d ≥ 3, Gaussian interfaces ϕ1 and ϕ2 , with the constraint that ϕ2 i ≥ ϕ1i ≥ 0 for all i. The main ... |

1 |
Wetting of heterogeneous surfaces at the mesoscopic scale
- Coninck, Dobrovolny, et al.
(Show Context)
Citation Context ... the wetting transition in the presence of a longrange wall/interface interaction. Disordered wall The model in the presence of a disordered substrate has also been studied in several works, see e.g. =-=[35, 36, 49, 20]-=-. Two types of disorder have been considered: random pinning potentials (similar to what is discussed in Subsection 5.5), and rough walls (similar to what is discussed in Subsection 3.3). The main con... |

1 |
Rigorous generalization of Young’s law for heterogeneous and rough substrates
- Coninck, Miracle-Solé, et al.
- 2003
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Private communication
- Sakagawa
(Show Context)
Citation Context ...ℓ log PΛN(|ϕi| ≤ ℓ, ∀i ∈ ΛN) = ⎪⎩ −2 ) (d = 1), exp(−O(ℓ)) (d = 2), exp(−O(ℓ2 (24) )) (d ≥ 3), for all N > N0(ℓ). 23The last result has very recently been given a sharper form in dimensions d ≥ 3 in =-=[81]-=-; it turns out, unsurprisingly, that this probability has the same leadingorder behavior as the corresponding i.i.d. Gaussian field, |ΛN | −1 log PΛN(|ϕi| ≤ ℓ, ∀i ∈ ΛN) = exp(− ℓ2 (1 + o(1)). 2gd The ... |

1 |
Height fluctuations in the honeycomb dimer model
- Geoffroy
- 2005
(Show Context)
Citation Context ...nterfaces is identical to that of their continuous counterpart. In particular they should have Gaussian asymptotics. This turns out to be quite delicate, and the only rigorous works I am aware of are =-=[70, 71, 56]-=-: they establish weak convergence of a suitably rescaled version of the SOS interface at β = ∞ to the continuous 13Figure 3: The oriented level lines of a discrete effective interface; the orientatio... |

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Entropic repulsion for a class of interface models in high dimensions
- Kurt
- 2005
(Show Context)
Citation Context ...e from the mathematical point of view, due to the lack of nearly all the main tools used in the study of effective interfaces: no nice random walk representation, no FKG inequalities, etc. I refer to =-=[82, 72]-=- for rigorous results (concerning the entropic repulsion phenomenon) on such models in dimensions 5 and more. 152 Massive model 2.1 Description of the model A very crude way to localize the interface... |