Regularity of the surface tension for thestochastic interface models.

Joint work with Scott Armstrong (Courant)

Wei Wu

University of Warwick

The Discrete Gaussian free field

Consider φ : QL := [−L,L]d ∩ Zd → R, sampled from themeasure

ZL,ξ :=

RQL

exp

(−∑

e

12

(∇φ(e))2

)1φ|∂QL

=ξ dφ.

This is a finite dimensional Gaussian-Hilbert space where theprobability density is proportional to exp

(−1

2(φ, φ)∇).

I φ(x) is multivariate GaussianI 〈φ(x)〉 = harmonic extension of ξ at xI Cov(φ(x), φ(y)) = ∆−1

L (x , y)

The ∇φ model

Consider φ : QL := [−L,L]d ∩ Zd → R, sampled from themeasure

ZL,ξ :=

RQL

exp

(−∑

e

V (∇φ(e))

)1φ|∂QL

=ξ dφ

=

RQL

exp

(−∑

e

V (∇φ(e)− ξ)

)1φ|∂QL

=0 dφ

WhereI V is even.I Uniformly convex: λ ≤ V ′′(t) ≤ Λ.

I Regularity: V ∈ C2,γ(R) for some γ > 0.

Infinite volume limit exists and the infinite ∇φ measure isclassified by the slope ξ = p · x . (Funaki-Spohn)

The ∇φ model

Figure: A realization of the random surface (by C. Gu).

History

I First introduced by Brascamp, Lebowitz, Lieb (1975) as“anharmonic crystals”. Conjectured large scale behavior isGaussian like.

I Renormalization group approach V (x) = x2/2 + small(80s-early 90: Brydges-Yau, Dimock-Hurd, ...)

I CLT for the infinite volume Gibbs state (Naddaf-Spencer97), via Helffer-Sjöstrand.

I CLT in finite volume. Level line to SLE4 (Miller 11)I Dynamical CLT (Giacomin, Olla, Spohn 01)I Surface tensions. (Funaki-Spohn 97,

Deuschel-Giacomin-Ioffe 01, Sheffield 03, Dario 18)I Extreme values beyond the Gaussian case (Belius-W.; W.-

Zeitouni).

The Naddaf-Spencer CLT

TheoremLet g ∈ L2 (Rd). Define

ΦR (g) := R−d/2∑x∈Zd

∇iφ(x)g( x

R

)Then ΦR (g) converges in law, as R →∞, to a normal randomvariable with variance

Qg =

∇∗i g(x)(∇∗i a∇i)

−1(x , y)∇∗i g(y) dxdy ,

where a = a(V )

In other words, ∇φ model converges in distribution to a(continuum) Gaussian free field with covariance(∇∗a∇)−1(x , y)

The Langevin dynamics

The ∇φ measure is invariant with respect to the Langevindynamics

dφt (x) =∑y∼x

V′(φt (y)− φt (x)) dt +√

2 dBt (x), x ∈ QL,

φt (x) = ξ(x), x ∈ ∂QL,

Hydrodynamic limit

Let hN(t , x) := 1NφN2t ([Nx ]).

Funaki and Spohn (97) proved that hN(t , x) converges in L2 tothe solution of the nonlinear PDE

∂th −∇ · (Dσ (∇h)) = 0 in (0,∞)× Rd .

To obtain the existence of a classical solution by the Schaudertheory, we need that ξ 7→ Dσ(ξ) is C1,α for some α > 0. Thatis, σ ∈ C2,α.

σ : Rd → R is the surface tension

Motion by mean curvature

Surface Tension

Finite volume surface tension is defined by

σL (p) :=1|QL|

logZL,p

ZL.

I σ (p) = limL→∞ σL (p) well defined (Funaki-Spohn 97,Dario 18).

I For Gaussian case V (x) = x2,

σ(p) = |p|2 + limL→∞

12Ld log det ∆L

I σ ∈ C1,1 and uniformly convex.I σ ∈ C2 “remains to be one of the important open problems”

(Funaki). Note that we need σ ∈ C2,α to apply theSchauder theory.

Equilibrium fluctuations

Giacomin, Olla and Spohn went to the next order in thisdescription, and consider the process

ζN(t ,dy) := N−d/2∑y∈Zd

(∇φN2t (x)− ξ)δx/N(dy).

They proved, using a parabolic version of Helffer-Sjostrand,that ζN converges in distribution to the solution of an the SPDEof the form

∂tζ −∇ · (a(ξ)∇ζ) =√

2W in (0,∞)× Rd ,

Conjecture: the Hessian D2σ(ξ) of the surface tension shouldcoincide with the diffusion matrix a.

Fluctuation-dissipation relation

Surface Tension

Theorem (Armstrong-W. 19)V ∈ C2,γ implies σ ∈ C2,β. Moreover, D2σ(ξ) = a(ξ).

Remark: the same proof works with logarithmic modulus∣∣V′′(s)− V′′(t)∣∣ ≤ ω (|s − t |)

where the modulus ω : [0,∞)→ [0,Λ] is an increasing,continuous function such that

lim supt→0

|log t |q ω(t) = 0.

CLT for nonlinear statisticsTheorem (Armstrong-W, in preparation)Let g ∈ L2 (Rd) and F ∈ C1,1(R;Rd ). Define

ΦR (F,g) := R−d/2∑x∈Zd

F(∇φ(x)) · g( x

R

)

− R−d/2

⟨∑x∈Zd

F(∇φ(x)) · g( x

R

)⟩µ

.

Then ΦR (F,g) converges in law, as R →∞, to a normalrandom variable with variance

QF,g := Q0 + F2

(∇∗g(x))T (∇∗a∇)−1(x , y)∇∗g(y) dxdy .

Moreover for large R and t √

log R,∣∣∣⟨ exp (tΦR (F,g))⟩µ− e

12 QF,g t2

∣∣∣ ≤ CeCt2R−α.

Two point functions

Theorem (Armstrong-W, in preparation)Let d = 2. There exists g > 0 and c ∈ R, depending only on V ,s.t.

|Varµ [φ(x)− φ(0)]− (g log |x |+ c)| ≤ C |x |−α

and, for every t ∈ R,∣∣∣∣log 〈exp (t (φ(x)− φ(0)))〉µ −12

(g log |x |) t2∣∣∣∣ ≤ Ct4. (1)

(1) improves a result of [Conlon-Spencer 14], with additionalassumption that Λ < 2λ and ‖V ′′′‖∞ <∞.

The Witten Laplacian

I Defining a derivative: for each ”suitable” function f : Ω→ R

∂x f (φ) := limh→0

f (φ+ h1x )− f (φ)

h.

I Defining the formal adjoint ∂∗x : for any ”suitable” pair offunctions f ,g : Ω→ R,

Ω∂x f (φ)g(φ)µ(dφ) =

Ω

f (φ)∂∗x g(φ)µ(dφ),

we have the explicit formula

∂∗x = −∂x +

(∑y∼x

V ′(φ(y)− φ(x))

)∂y .

The Witten Laplacian

I Defining a derivative: for each ”suitable” function f : Ω→ R

∂x f (φ) := limh→0

f (φ+ h1x )− f (φ)

h.

I Defining the formal adjoint ∂∗x : for any ”suitable” pair offunctions f ,g : Ω→ R,

Ω∂x f (φ)g(φ)µ(dφ) =

Ω

f (φ)∂∗x g(φ)µ(dφ),

we have the explicit formula

∂∗x = −∂x +

(∑y∼x

V ′(φ(y)− φ(x))

)∂y .

We can thus define the Witten-Laplacian

Lµ :=∑x∈Zd

∂∗x∂x

= −∑x∈Zd

∂2x +

∑x∈Zd

(∑y∼x

V ′(φ(y)− φ(x))

)∂x .

This operator satisfies, for any pair of functions f ,g : Ω→ R,

〈fLµg〉 =∑x∈Zd

〈∂x f∂xg〉 = 〈gLµf 〉 .

The Langevin dynamics

The ∇φ measure is invariant with respect to the Langevindynamics

dφt (x) =∑y∼x

V′(φt (y)− φt (x)) dt +√

2 dBt (x), x ∈ QL,

φt (x) = ξ(x), x ∈ ∂QL,

Its generator

LµLF (φ) :=∑

x∈QL

∂2x F (φ)−

∑x∈Q

L

∑y∼x

V′(φ(y)− φ(x))∂xF (φ)

:= −∑

x∈QL

∂∗x∂xF (φ).

is exactly the Witten Laplacian!

The Naddaf-Spencer ideas

For GFF, H =∑

(φ,∆φ), and ∆−1 encodes covariancestructure (and everything) about the field.

For general ∇φ model, recall the generator for the Langevindynamics is the Witten Laplacian L so that

〈FLG〉 =∑

x

〈∂xF∂xG〉 = 〈GLF 〉

We write, for each F = F (φ),⟨(F − 〈F 〉µ)2

⟩µ

= −∑x∈Zd

⟨(∂xF )

(∂x

(L−1µ (F − 〈F 〉µ)

))⟩µ.

LetLµv = F − 〈F 〉µ

The Naddaf-Spencer ideasTaking ∂x on both sides, and use the commutator identity

[∂y , ∂∗x ] = −1x∼yV′′ (φ(y)− φ(x)) + 1x=y

∑e3x

V′′ (∇φ(e))

We see that u = ∂xv solves the equation

−Lu +∇∗V ′′∇u = ∂xF , u ∈ H1(Zd × Ω(Zd ))

Definition (Helffer-Sjöstrand operator)The Helffer-Sjöstrand operator is defined by the formula

L := L+∇ · V ′′ (∇φ(e))∇

which acts on functions f : Ω× Zd → R.I The operator L is the Witten-Laplacian, it acts on the field

variable (infinite-dimensional);I The operator ∇ · V ′′∇ is a uniformly elliptic operator, it acts

on the space variable (dimension d).

Helffer-Sjöstrand representation

Theorem: H.-S. representation (Naddaf-Spencer 98)Given two random variables F ,G : Ω→ R, we denote by:I f (x , φ) = ∂xF (φ);I g(x , φ) = ∂xG(φ);I G : Ω× Zd → R the solution of the equation

LG = g in Ω× Zd ,

then we have

Cov[F ,G] =∑x∈Zd

〈f (x , φ)G(x , φ)〉µ .

The Naddaf-Spencer ideasLet u solves

−Lu +∇∗a∇u = ∇∗g, u ∈ H1(Zd × Ω(Zd ))

Applying the H.S. representation

Var[∑

x

∇φ(x)g(x)] =∑

x

〈∇∗g(x)u(x)〉

= −2∑

y

∑x

⟨(∂yu(x , ·))2

⟩− 2

∑e

⟨a(e)(∇u(e, ·))2

⟩Convergence of the variance↔ convergence of the energydensity of the PDE to the e.d. of

∇∗a∇u = ∇∗g

Naddaf-Spencer then adapt the soft homogenization technique(Papanicolaou-Varadhan) to prove the L2 homogenization.

Roadmap

Step 1: We quantify the Naddaf-Spencer idea and prove forsome α > 0, ∣∣∣D2σL(p)− D2σ(p)

∣∣∣ ≤ CL−α.

Step 2: Using probabilistic coupling arguments, we obtain forevery θ > 0 and L ≥ L0(θ),∣∣∣D2σL(p)− D2σL(p′)

∣∣∣ ≤ C(|p − p′|+ θ

)β.

Combine and send L→∞ we obtain∣∣∣D2σ(p)− D2σ(p′)∣∣∣ ≤ C

(|p − p′|

)β.

Quantify the Naddaf-Spencer ideasAdapting the variational approach for homogenization (alsoquantitative) from Armstrong, Kuusi, Mourrat ‘17.

Subadditive quantities:

ν(Q, f ,p) :=1|Q|

infv∈`p+H1

0 (Q,µ)EQ,f [v ] .

ν∗(Q, f ,q) :=1|Q|

supu∈H1(Q,µ)

∑e∈E(Q)

∇`q(e) 〈∇u(e, ·)〉µ − EQ,f [u]

.

where

EQ,f [w ] :=12

∑y∈Q

∑x∈Q

⟨(∂yw(x , ·))2

⟩µ

+12

∑e∈E(Q)

⟨a(e)(∇w(e, ·))2

⟩µ

−∑

x∈Q

〈f (x , ·)w(x , ·)〉µ .

Subadditive quantity

Finite volume surface tension is the quadratic part of the energyν(QL,0,p)!

D2σL(p) = a(QL),

if we write ν(Q,0,p) = 12p · a(Q)p − f(Q) · p − c(Q) ∀p ∈ Rd

Subadditivity implies ν(Q,0,p)→ 12p · ap − f · p − c.

Quanfitying the rate of convergence requires mixing.

Using probabilistic coupling arguments to show

ν(QL,0,p) ≈ ν2L(QL,0,p) + L−α

Quantify the Naddaf-Spencer ideasSubadditivity of the energy quantity

ν(QmL, f ,p) ≤ ν(QL, f ,p) + C (|p|+ K0)2 L−1

To quantify the rate of convergence, we study

J(Q,p,q) := ν(Q, f ,p) + ν∗(Q, f ,q)− p · q.and prove that for every p ∈ Rd

infq∈Rd

J(QL,p,q) ≤ C (|p|+ K0)2 L−β.

Use a multiscale argument to show

infq∈Rd

J(QL,p,q) ≤ C (|p|+ K0)2 (C′L−β+m∑

n=log L

3−(log L−n)τn).

where

τm := supp∈B1

(ν(Q3m , f ,p)− ν(Q3m+1 , f ,p))+

+ supq∈B1

(ν∗(Q3m , f ,q)− ν∗(Q3m+1 , f ,q))+ .

Mixing condition in Armstrong-Kuusi-Mourrat replaced byBrascamp-Lieb.

Open questions and outlook

I Higher regularity.

Conjecture (Sheffield):As long as V is convex and C2, the infinite volume surfacetension σ is infinitely differentiable

Working progress (with Armstrong and Kuusi): V ∈ Ck

implies σ ∈ Ck (and a bit more).

I Convergence to GFF

Conjecture (Sheffield):For d = 2, as long as V is convex and grows to infinity, thescaling limit is a GFF.

This includes e.g., V (x) =∞1|x |>1.

Open questions and outlook

I Optimal rate of convergence

Should have |σL(p)− σ(p)| ≤√

log LLd/2 .

I Application to statistical mechanics models

As long as the effective model has a convex action. Seerecent work of Dario-W. on Villain rotator models in d ≥ 3.