The scaling limit of the membrane model
On the integer lattice we consider the discrete membrane model, a random interface in which the field has Laplacian interaction. We prove that, under appropriate rescaling, the discrete membrane model converges to the continuum membrane model in . Namely, it is shown that the scaling limit in is a Hölder continuous random field, while in the membrane model converges to a random distribution. As a by-product of the proof in , we obtain the scaling limit of the maximum. This work complements the analogous results of CarJDScaling in .
Key words and phrases:Membrane model, scaling limit, random interface, continuum membrane model, Green’s function
2000 Mathematics Subject Classification:31B30, 60J45, 60G15, 82C20
The main object of study in this article is the membrane model (MM), also known as discrete bilaplacian model. The membrane model is a special instance of a more general class of interface models in which the interaction of the system is governed by the exponential of an Hamiltonian function . More specifically, random interfaces are fields , whose distribution is determined by the probability measure on , , with density
where is a finite subset, is the 1-dimensional Lebesgue measure on , is the Dirac measure at and is a normalising constant. We are imposing zero boundary conditions i.e. almost surely for all , but the definition holds for more general boundary conditions. A relevant example is where the Hamiltonian is driven by a convex function of the gradient, that is, , convex, and the sum being over nearest neighbours. The most well-known among these interfaces is the discrete Gaussian free field (DGFF) when . The quadratic potential allows one to have various tools at one’s disposal, like the random walk representation of covariances and inequalities like FKG. These tools can be generalised to (strictly) convex potentials in the form of the Brascamp–Lieb inequality and the Helffer–Sjöstrand random walk representation. We refer to NaddafSpencer, giacomin2001, Funaki, velenikloc for an overview. Outside the convex regime, the non-convex regime was recently studied for example in cotar2009strict, biskup:spohn.
A very natural probabilistic question one can ask oneself is: “What happens to a random interface when one rescales it suitably?”. In in the example of the DGFF the scaling limit is the Brownian bridge. In the limit, the continuum Gaussian free field, is not a random variable and can only be interpreted in the language of distribution theory (see for example Sheff, biskup:survey). The importance of the continuum Gaussian free field in relies on its universality property due to conformal invariance, and links it to other stochastic processes like SLE, CLE, and Liouville quantum gravity. The recent developments concerning the extreme value theory of DGFF (and, more generally, log-correlated fields) have shown impressive connections also to number theory, branching processes and random matrices.
In comparison to the DGFF, the membrane model has received slightly less attention, mainly due to the technical challenges intrinsic of the model. It is the Gaussian interface for which
and is the discrete Laplacian defined by
In case , we will denote the measure with Hamiltonian (1.1) by . Introduced by Sakagawa in the probabilistic literature, the MM looks for certain aspects very similar to the DGFF: it is log-correlated in , has a supercritical regime in and is subcritical in . In particular in there is no thermodynamic limit of the measures as . The MM displays however certain crucial difficulties, in that for example it exhibits no random walk representation, and several correlation inequalities are lacking. Nonetheless it is possible, via analytic and numerical methods, to obtain sharp results on its behaviour. Examples are the study of the entropic repulsion and pinning effects (CaravennaDeuschel_pin, Kurt_d4, Kurt_d5, BCK17, AKW16), extreme value theory (CCHInterfaces), and connections to other statistical mechanics models (CHR). In this framework we present our work which aims at determining the scaling limit of the bilaplacian model. The answer in was given by CarJDScaling, who also look at the situation in which a pinning force is added to the model. We complement their work by determining the scaling limit in all . We also mention HrVe, who consider general semiflexible membranes as well with a different scaling approach. Their results are derived using an integrated random walk representation which is difficult to adapt in higher dimensions.
The main contributions of this article are as follows:
in we consider the discrete membrane model on a box of side-length and interpolate it in a continuous way. We show that the process converges to a real-valued process with continuous trajectories and the convergence takes place in the space of continuous functions (see Theorem 2.1). The utility of this type of convergence is that it yields the scaling limit of the discrete maximum exploiting the continuous mapping theorem (Corollary 2.2). While the limiting maximum of the discrete membrane model in was derived by CCHInterfaces, in the problem remains open as far as the authors know (tightness can be derived from ding:roy:ofer). The limit field also turns to be Hölder continuous with exponent less than in and less than in .
The proof of the above facts is based on two basic steps: tightness and finite dimensional convergence. Tightness depends on the gradient estimates of the discrete Green’s functions which were very recently derived in Mueller:Sch:2017; finite dimensional convergence follows from the convergence of the Green’s function.
In the limiting process on a sufficiently nice domain will be a fractional Gaussian field with Hurst parameter on . The theory of fractional Gaussian fields was surveyed recently in LSSW. The authors there construct the continuum membrane model using characteristic functionals. We take here a bit different route and give a representation using the eigenvalues of the biharmonic operator in the continuum. We remark however that these eigenvalues differ from the square of the Laplacian eigenvalues due to boundary conditions. The GFF theory which is based on (the first order Sobolev space) needs to be replaced by (second order Sobolev space).
Our main result is given in Theorem 3.11. Its proof is again split into two steps: finite dimensional convergence and tightness. Both steps crucially require an approximation result of PDEs given by thomee: there he gives quantitative estimates on the approximation of solutions of PDEs involving “nice” elliptic operators by their discrete counterparts. We believe that the techniques used in that article might have implications in the development of the theory of the membrane model, in particular the idea of tackling boundary values by rescaling the standard discrete Sobolev norm around the boundary. Especially in this allows one to overcome the difficulty of extending estimates from the bulk up to the boundary, which is generally one stumbling block in the study of the MM.
In we also consider the infinite volume membrane model on . We show in Lemma 4.3 that the limit is the fractional Gaussian field of Hurst parameter on (see LSSW) and we prove in Theorem 4.4 the convergence with the help of characteristic functionals. We utilise the classical result of Fernique:1968 (recently extended in the tempered distribution setting by BOY2017) stating that convergence of tempered distributions is equivalent to that of their characteristic functions. Technical tools useful for this scope are the explicit Fourier transform of the infinite volume Green’s function and the Poisson summation formula.
We stress that, regardless of the dimension, the field is always rescaled as for . Heuristically, the factor corresponds to the order of growth of the variance of the model in a box, which we recall here for completeness.
In if denotes the distance to the boundary of one has for some constant (Mueller:Sch:2017, Theorem 1.1)
In let us denote the bulk of by for . Then from Cip13 we have: there exists a constant such that
Asymptotics up to the boundary are not known to the best of the authors’ knowledge. The approach of thomee allows to circumvent this lack of estimates.
In the infinite volume covariance satisfies (Sakagawa, Lemma 5.1)
Interestingly this reflects the behavior of the characteristic singular solution (fundamental solution) of the biharmonic equation, which is
The reader can consult MayborodaReg, MitreaMitrea and references therein for sharp pointwise estimates of the Green’s function of the bilaplacian in general domains and for regularity properties of the biharmonic Green’s function.
We would like to conclude the Introduction with a few open questions:
Is the maximum of the discrete membrane model at criticality scaling to a randomly shifted Gumbel, as predicted by ding:roy:ofer?
What will the scaling limit be for interfaces with mixed Hamiltonian of the form , convex functions (in particular, )? Results on these models were shown in Caravenna/Borecki:2010 in .
Structure of the paper In Section 2 we handle the case , while in Section 3 we treat the finite-volume case in . In Section 4 we analyse the case of the infinite-volume model in . To keep the article self-contained in Appendix A we discuss the results from thomee and also deduce a quantitative version of the approximation result proved there.
Acknowledgements The first author is supported by the grant 613.009.102 of the Netherlands Organisation for Scientific Research (NWO) and was supported by the EPSRC grant EP/N004566/1 and the Dutch stochastics cluster STAR (Stochastics – Theoretical and Applied Research) while affiliated with the University of Bath. The first and third author acknowledge the MFO grant RiP 1706s. The third author also thanks the NETWORKS grant in the Netherlands and the University of Leiden where a part of the work was carried out. All authors are very grateful to Vidar Thomée who kindly provided the paper thomee. Thanks also to Francesco Caravenna and Noemi Kurt for helpful discussions, and to Stefan Müller and Florian Schweiger for sharing their article Mueller:Sch:2017. F. Schweiger also observed that Theorem 2.5 yields global Hölder continuity, and also how to improve the Hölder exponent in (Lemma 2.6). Finally it is our pleasure to thank an anonymous referee for his/her insightful comments and careful reading which improved the quality and readability of the paper.
We fix a constant throughout the whole paper. In the following always denotes a universal constant whose value however may change in each occurence. We will use to denote convergence in distribution. We denote, for any , , the “integer part” of as and similarly is the “fractional part” of .
2. Convergence in
2.1. Description of the limiting field
Let and , where . Let be the MM on and let be the covariance function for this model. It is known (Kurt_thesis, Section 1) to satisfy the following discrete boundary value problem for all :
First we want to define a continuous interpolation of the discrete field to have convergence in the space of continuous functions. There are many ways to define the field . We take one of the simplest geometric ways which is akin to the interpolation of simple random walk trajectories in Donsker’s invariance principle. Mind that we take the domain as a square since the recent gradient estimates and convergence of the Green’s function of Mueller:Sch:2017 can be applied easily.
Interpolation in . Let . Then lies in the square box with vertices , where are the standard basis vectors of . Suppose is a point in the triangle . Then we can write with And in this case we define
Similarly, if then we define
Thus the interpolated field is defined by
where , .
Interpolation in . In the interpolated field can be defined in the same way as above. We use tetrahedrons to define the interpolated field as
where and are pairwise different.
Note that in both we have
From the above construction it follows that, for each , is a continuous function on . This shows that can be considered as a random variable taking values in where is the space of continuous functions on and is its Borel -algebra. Also recall the definition of Green’s function: the Green’s function for the biharmonic operator is such that for every fixed , it solves the equation
in the space , the completion of with respect to the norm
In the above equations , the continuum bilaplacian, acts on the component, and is the Hessian. The detailed properties of such spaces are needed in so we defer the discussions on them to Section 3. We denote the continuum Green’s function by to indicate the dependence on the domain .
We are now ready to state our main result for the case . It shows that the convergence of the above described process occurs in the space of continuous functions.
Theorem 2.1 (Scaling limit in ).
Consider the interpolated membrane model in and as above. Then there exists a centered continuous Gaussian process with covariance on such that converges in distribution to in the space of all continuous functions on . Furthermore the process is almost surely Hölder continuous with exponent , for every resp. in resp. .
An immediate consequence of the continuous mapping theorem is that, as ,
It is easy to see that for any square or a cube in the lattice,
Hence . So combining these observations we obtain the scaling limit of the maximum of the discrete membrane model in lower dimensions.
Let and let . Then as
2.2. Proof of the scaling limit (Theorem 2.1)
The proof follows the general methodology of a functional CLT, namely, we first show the tightness of the interpolated field and secondly we show that the finite dimensional distributions converge. As a by-product of the proof, the limiting Gaussian process will be well-defined, that is, its covariance function will be positive definite. The finite dimensional convergence follows easily from the very recent work of Mueller:Sch:2017 where the convergence of the discrete Green’s function to the continuum one is shown. Tightness also requires the crucial bounds on gradients which were derived in the same article. Since we have interpolated the field continuously and not piece-wise in boxes or cubes one of the main efforts is to deduce moment bounds from integer lattice points.
Tightness and Hölder continuity
To derive the tightness we need the following ingredients. The first one consists in the following bounds for the discrete Green’s function and its gradients which follow from Mueller:Sch:2017. We define the directional derivative of a function as
and the discrete gradient as
For functions of several variables we use a subscript to indicate the variable with respect to which a derivative is taken, for example in we take the discrete derivative in the direction in the variable and in in the variable , and means we are taking the gradient in the variable. We now state some bounds on the covariance function and its gradient from Mueller:Sch:2017, where they appear in a more general version.
Lemma 2.3 (Mueller:Sch:2017).
Now from the estimate 3 and the fact that
one can observe the following Fact.
Next we want to show that the sequence is tight in . We use the following theorem, whose proof follows from that of Theorem 14.9 of kallenberg:foundations.
Let be continuous processes on with values in a complete separable metric space . Assume that is tight in and that for constants
uniformly in . Then is tight in and for every the limiting processes are almost surely Hölder continuous with exponent .
Observe that the process is Gaussian, and since from Lemma 2.3 it follows that , it is easy to see that is tight. Again, using the properties of Gaussian laws, to show (2.1) it is enough to show the following the lemma.
Let in and in . Then there exists a constant (which depends on in ) such that
for all , uniformly in .
This Lemma will immediately give (2.1) and hence the Hölder continuity of the limiting field.
The field is almost surely Hölder continuous with exponent , where in and in .
Now we show the proof of the Lemma.
Proof of Lemma 2.6.
First we consider . We fix a and let . We split the proof into a few cases.
Suppose belong to the same smallest square box in the lattice . First assume , that is, the points are in the interior and not touching the top and right boundaries. In this case if we have and . Then by definition of the interpolation we have
So from the above expression we have
Again if and , or if and then we consider the point on the line segment joining and such that is the point of intersection of the line segment joining and the diagonal joining Then we have using the above computations
Now the other case, that is, when follows from above by continuity.
Suppose do not belong to the same smallest square box in the lattice . In this case if then one can obtain (2.2) by the above case and a suitable point in between. So we assume . Depending on whether and belong to the discrete lattice we split the proof in two broad cases. We will use bounds on mixed discrete derivatives for a better control of finite differences of the Green’s function.
Suppose at least one between does not belong to . Then
Note that for the last inequality we have used our assumption .
Now we consider . Let . We split the proof into cases similar to those of . We give a brief description. For Case 1, suppose belong to the same smallest cube in the lattice . First assume . In this case if and then it follows from the definition of interpolation
Now from Fact 2.4 and the fact that we have (2.2). Note that this is a particular case of lying in the same tetrahedral portion of the cube. Hence if lie in the same tetrahedral portion of the cube then by similar arguments (2.2) holds. If do not lie in the same tetrahedral part then we consider points (at most 3) on the line segment joining them such that two consecutive between , the selected points and lie in the same tetrahedral part. Then applying the previous argument we can obtain (2.2). Now the case when follows by continuity. For Case 2, we describe Sub-case 2(a) which turns out to be simpler in . The rest of the argument is similar to that in . Suppose with . Then
Without loss of generality assume . Then
Hence (2.2) follows. ∎
Finite dimensional convergence
The main content of this Subsubsection is to show
With the notation of Theorem 2.1, for all ,
To show the finite dimensional convergence we use Corollary 1.4 of Mueller:Sch:2017 (in their setting the domain was but the result works for as well). We observe that for , one has where satisfies for with the following boundary value problem ( is defined in Appendix A):
Let be the Gaussian process on such that for all , where is the Green’s function for the biharmonic equation with homogeneous Dirichlet boundary conditions (it will be a by-product of this proof that such a process exists). First we consider . For we have
Then using Fact 2.4 we have and hence converges to zero in probability as tends to infinity.
Again if then
and converges to by Corollary 1.4 of Mueller:Sch:2017. Also if then . Hence .