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.
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