Renormalization Group by ConditionalExpectations (Harmonic Extensions)

Hao Shen (Princeton University)

Harmonic analysis and Renormalization group Workshop

April 21, 2014

Gaussian measure

Measures on space of functions (distributions) {� : Rd ! R}

exp✓�1

2

ˆRd

(r�(x))2◆

D�.N

Gaussian measure + interactions

Measures on space of functions (distributions) {� : Rd ! R}

exp

�1

2

ˆRd

(r�(x))2 + �"

small

ˆRd

V (r�(x))!

D�.N

e.g.V (r�(x)) =

X

µ

cos(rµ�(x)) V (r�(x)) = �X

µ

(rµ�(x))4

Finite dimensional approximation: let ⇤ ⇢ Zd be a finitelattice,

exp

�1

2

X

x2⇤

(r�(x))2 + �X

x2⇤

V (r�(x))!Y

x2⇤

d�(x).N

The idea of renormalization group (RG)

“Coarse-graining”: let B be a block,

�(B) =1|B |X

x2B

�(x)

Integrating out the small scale details⇣(x) = �(x)� �(B)

F (�)D�! F 0(�)D�Iterate the above operations

F ! F 0 ! F 00 ! · · ·This is a transformation on the space of measures (models).

The study of measures =) Dynamical system

Heuristic statement of the result

At large scale,

exp✓�1

2

X(r�)2 + �

XV (r�)

◆

⇡ exp✓�1 + O(�)

2

X(r�)2

◆

Precise statement of the result

Recall the model on ⇤ = [�LN/2, LN/2]d \ Zd

exp

(�1

2

X

x2⇤

(r�(x))2 + �X

x2⇤

V (r�(x)))Y

x2⇤

d�(x)�

N

Squeeze ⇤ into a fixed (continuum) torus

⇤ ,! Td

Fix a mean zero smooth function f on Td .

Precise statement of the result

Moment generating functional (Laplace transform)

Z (f ) =⌦eP

x

f (x)�(x)↵

=

´e� 1

2P

⇤(r�)2+�P

⇤ V (r�)+P

x2⇤ f (x)�(x)Qx2⇤ d�(x)´

e� 12P

⇤(r�)2+�P

⇤ V (r�)Q

x2⇤ d�(x)

Here,f ( x

"point in ⇤

) := L�(d+2)N/2 f ( L�Nx"

point in Td

)

Precise statement of the result: scaling limit

Theorem

Assume that V (r�) = cos(r�), and f is sufficiently smooth,and |�| is sufficiently small. Then, there exists a constant �depending on � and

limN%1

Z (f ) = exp✓

12

ˆTd

f (x)(���)�1f (x)ddx◆

RG: If F (�) = e� 12 (r�)2+�V (r�) then

F ! F 0 ! F 00 ! · · · ! F ⇤

where F ⇤ is Gaussian.

Rigorous RG methods

The above idea of coarse-graining by “block averaging”was rigorously carried out Balaban, Gawedzki, Kupiainen (1980’s).Decomposition of Gaussian Abdessalem, Bauerschmidt, Brydges,

Dimock, Falco, Guadagni, Hurd, Slade, Yau etc. (1990’s): Let µC be theGaussian measure with covariance C = (��)�1,ˆ

F (�)dµC (�)

Decompose C into positive definite pieces C =P

j Cj s.t.

� =X

j

�j

Do integrations step by stepˆ· · ·ˆ

F (�) dµC1(�1)dµC2(�2) · · · dµCj

(�j) · · ·

The new method (conditional expectations)Write

´F (�)dµC (�) = E [F (�)]. Let ⌦ ⇢ ⇤,

EhF (�)

i= E

hE⇥F (�)

���(⌦c)⇤ i

where the conditional expectation is conditioned on all{�(x)|x /2 ⌦} being fixed.(Integrate out the field in ⌦, keeping the field out of ⌦ fixed.)Thinking of the Gaussian measure as e

12 (�,��),

�X

⇤

��� = Q(�(⌦)) + Q(�(⌦c)) + crossing term

= ” x2 + y 2 + xy ”

With ”y” fixed, shift ”x” to the minimizer of the quadraticform to cancel the crossing term.

The new method

The minimizer is P⌦� =P

y2@⌦ P⌦(·, y)�(y) i.e. theharmonic extension of � from ⌦c into ⌦,

� = P⌦�+ ⇣

where P⌦ is the Poisson kernel for domain ⌦. Then,

�X

x2⇤

�(x)��(x) = �X

x2X

⇣(x)�D⌦⇣(x)�

X

x2⇤

P⌦�(x)�P⌦�(x)

where �D⌦ is the Laplacian on ⌦ with Dirichlet b.c.

The conditional expectation = integrating out ⇣:

E⇥F (�)

���(⌦c)⇤= E⇣ [F (P⌦�+ ⇣)]

Covariance of ⇣ is the Dirichlet Green’s function of �� on ⌦.

The new method

Note that the functional

E⇣ [F (P⌦�+ ⇣)]

depends on � via P⌦�. Namely, the functional, after theintegration, depends on the field in a smoother way(smoothing effect of the Poisson kernel). Intuitively,

” P⌦ ⇡ the block averaging operator ”

The new method

The new method has the following advantages:the fluctuation field ⇣ automatically has finite rangesupport (In the block-average method, the fluctuation field

only has exponentially decaying correlation; in Brydges’

method, constructing the decomposition with finite range

properties is a nontrivial task.)

RG =) local elliptic PDE problems. (In the

non-translation-invariant models, the difficulty would be

merely estimating solutions to elliptic PDEs with non-constant

coefficients).

As we will see, the norm on the functionals will besimplified.

The a priori tuning of the GaussianBased on the above heuristics, we split the quadratic part:

�12(r�)2 =� �

2

X

x2⇤

(r�(x))2 + � � 12

X

x2⇤

(r�(x))2

�!�/p�

�����!� 12

X

x2⇤

(r�(x))2 + �

2

X

x2⇤

(r�(x))2

where � ⇡ 1 (will be determined in the end); � := 1 � ��1.

exp⇢�

2

X

x2⇤

(r�(x))2 + �X

x2⇤

W (r�(x))�

dµ(�)

dµ(�) = e� 12P

x2⇤(r�(x))2Y

x2⇤

d�(x)�N

An expansion over polymersTake f = 0 for simplicity, to focus on the main ideas of ourRG method. Let E be the Gaussian expectation.

Z = EY

x2⇤

e�2 (r�(x))2

✓�e�W (r�(x)) � 1| {z }

”O(�)”

�+ 1◆�

= EX

X✓⇤

I (⇤\X )K (X )

�

where for any sets X ,Y ,

I (Y ) =Y

x2Y

e�2 (r�(x))2 K (X ) =

Y

x2X

e�2 (r�(x))2

✓e�W (r�(x))�1

◆

Overview of the rest of the talk

EX

X✓⇤

I (⇤\X ,�)K (X ,�)

�I (Y ,�) =

Y

x2Y

e�2 (r�(x))2

|K (X ,�)| A�|X |eP

X

(r�)2 A � 1

Two questions:How to keep this algebraic structure at all the scales(sites ! blocks, � ! smoother fields)?How to maintain similar analytical bounds for all thescales?

Propagate algebraic structure to next scale

At scale j , one has Z =P

X2Pj

Ij(⇤\X ,�)Kj(X ,�)

Z = EX

X2Pj

Ij lives on the grey blocks, and Kj lives on the black regions.

Propagate algebraic structure to next scale

We would like to find a way to rewrite the expression onscale j + 1:

Z = E X

U2Pj+1

Ij+1(⇤\U)Kj+1(U)

�

while Ij+1,Kj+1 still have the above forms andfactorization properties.The idea is to postulate that Ij+1 will be exponential of anew quadratic form and then find Kj+1.

Propagate algebraic structure to next scale

E X

X2Pj

Ij(⇤\X )Kj(X )

�= E

X

X2Pj

Y

B2⇤\X

⇣Ij+1(B)+�Ij(B)

⌘Kj(X )

�

= E X

X2Pj

X

Y✓⇤\X

Ij+1(⇤\(X [ Y )) �Ij(Y ) Kj(X )

�

Consider the smallest j+1 scale polymer that contains X [ Y ,

written as U := X [ Y 2 Pj+1, and then resum:

E X

U2Pj+1

Ij+1(⇤\U)

K#j

(U)z }| {X

XqY=U

�Ij(Y )Kj(X )

�

Conditional expectation

E X

U2Pj+1

Ij+1(⇤\U)Y

V2c.c(U)

K#j (V )

�

= E X

U2Pj+1

Ij+1(⇤\U)

⇥Y

V2c.c(U)

E⇥K#

j (V )���(V+)c

⇤�

= E X

U2Pj+1

Ij+1(⇤\U)Kj+1(U)

�

Conditional expectation

Recall that “the conditional expectation = integration out ⇣”:

E⇥K (�)

���V c

⇤= E⇣ [K (PV�+ ⇣)]

where covariance of ⇣ is the Dirichlet Green’s function for V .After the integration, the resulting functional depends on � viaPV� which is expected to be “smoother” than �; however,

The Poisson kernel PU(x , y) for x 2 U, y 2 @U only hassmoothing effect for x sufficiently far from @U!If one directly took conditional expectation fixing the fieldoutside the “red line”, Ij+1 would also get involved in theintegration...

Conditional expectation (making a corridor)The idea to settle the above dilema is still “expand and resum”.

X

U2Pj+1

Y

B2Bj+1(⇤\U)

✓(Ij+1(B)� 1) + 1

◆K 0

j+1(U)

=X

U2Pj+1

X

D✓⇤\U

Y

B2Bj+1(D)

(Ij+1(B)� 1)K 0j+1(U)

Then we “glue” the Ij+1(B)� 1 which touch U onto K 0j+1

until the set U [ ([B) is surrounded by a “corridor” filledby only “1” ’s.After that, we resum up all the rest of Ij+1(B)� 1 out ofthe corridor, and we obtain

Z = E X

U2Pj+1

Ij+1(⇤\U)Kj+1(U)

�

where U is a set slightly larger than U.

Propagate analytical bound to next scale

We want:Kj(X ) . A�|X |

j 8X 2 Pj

) Kj+1(U) . A�|U|j+1 8U 2 Pj+1

However,

Kj+1(U) ⇡ EhX

B=U

�Ij(B) +X

X=U

Kj(X )��� · · ·

i

So many finer scale polymer X ,Y ’s are contained in afixed coarse scale polymer U!In fact, only O(Ld) terms...

The important scalingHow to deal with this O(Ld) growth of K? Let’s look at thecovariance of rPX�:Lemma: Let x 2 X , and C be the covariance of the Gaussianfield �. If d(x , @X ) � 1

3Lj , then

X

y1,y22@X

(rxPX )(x , y1)C (y1, y2)(rxPX )(x , y2) O(1)L�dj

This means that rPX� “scales down by O(L� d

2 )”.

” K = c0 + c2(rPX�)2 + O((rPX�)

4) ”

As long as we choose

Ij+1 = eEj+1+�

j+1(rPU

�)2

in the way that the c0 + c2(rPX�)2 part is absorbed into Ij+1,then O((rPX�)4) will scale down by O(L�2d).

The important scaling

Lemma: Let x 2 X , and C be the covariance of the Gaussianfield �. If d(x , @X ) � 1

3Lj , then

X

y1,y22@X

(rxPX )(x , y1)C (y1, y2)(rxPX )(x , y2) O(1)L�dj

Idea of proof:

LHS =X

y22@X

rxC (x , y2)| {z }.|x�y2|1�d=O(L�(d�1)j )

L�j

z}|{rx PX (x , y2)| {z }

O(L�(d�1)j )

Propagate analytical bound to next scale

Again, we want:

Kj(X ,�) . eP

X

(r�)2 8�

) Kj+1(U,�0) . eP

X

(r�0)2 8�0

In the field decomposition method,

E⇣eP

X

(r�)2 = E⇣eP

X

(r�0+r⇣)2 E⇣e2

PX

⇣(r�0)2+(r⇣)2

⌘

Bounds deteriorate (lose control) ... One has to put inmany extra terms.

The norm for K

The weight in “the method of decomposition of Gaussian field”was roughly

G (X ) ⇠ eP

X

(r�)2

but complete form is complicated

G (X ) = exp✓

c1X

X

(r�j+1)2 + c2

X

B2Bj

(X )

supB?

��L2jr2�j+1��2

+ c3X

@X

(r�j+1)2 + c4 max

0p2supB?

��Lpjrp⇣��2◆

where �j+1 = �j + ⇣.

Propagate analytical bound to next scale

We bound Kj(X ,�) . G (X ,X+) where

G (X ,Y ) := Ehe

2P

X

(r�)2���Y c

i �N(X ,Y )

for X ⇢ Y where

N(X ,Y ) := Ehe

2P

X

(r�)2���Y c = 0

i

An interesting property

exp

2

X

X

(@ 2)2

! G (X ,Y ) exp

2

X

X

(@ 1)2

!

where1 1 minimizes

PY (@�)2 �

PX (@�)

2with �Y c

fixed,

2and 2 minimizes

PY (@�)2 with �Y c

fixed.

Propagate analytical bound to next scale

The conditional expectation automatically takes care of theintegration of G :

E⇥G (X ,Y )

���Uc

⇤

=EhEhe

2P

X

(r�)2���Y c

i ���Uc

i �N(X ,Y )

=Ehe

2P

X

(r�)2���Uc

i �N(X ,Y )

=G (X ,U)N(X ,U)

N(X ,Y )| {z }cL

�dj |X |

Linearization of RG map

The map (�j , �j+1,Kj) ! Kj+1 is smooth (w.r.t. the norms wedefined).The linearized part of Kj+1(U) at (0, 0, 0) contains three parts:

The contributions from Kj(X ) where X ✓ U is “large” -which can be shown negaligible.For small X , the Taylor remainder after the second orderTaylor expansion of E

⇥Kj(X )

���(U+)c⇤

is negaligible.One can choose �j+1 so that the leading Taylor terms ofE⇥Kj(X )

���(U+)c⇤

are absorbed into Ij+1.

Determine �

The dynamical system

�j+1 = �j + ↵(Kj)

Kj+1 = LKj + f (�j ,Kj)(1)

satisfies kLk < 1. By standard stable manifold theorem, thereexists an initial tuning � so that |�j | ! 0, kKjkj ! 0.In other words,

Ehe

�2P

x2⇤(r�(x))2+zW (r�)i! ”econstE [1] ” (j ! 1)

Thank you!