FrancescoCaravenna Momentasymptoticsfor2ddirectedpolymerandStochasticHeatEquationinthecriticalwindow

Moment asymptotics for 2d directed polymer and Stochastic Heat Equation

in the critical window

Francesco Caravenna

Universit`a degli Studi di Milano-Bicocca

London, Imperial College ∼ 6 March 2018

Collaborators

Nikos Zygouras (Warwick) and Rongfeng Sun (NUS)

Outline

1. Directed Polymer and the Stochastic Heat Equation

2. Main Results

3. Renewal Theorems

4. Second Moment

5. Third Moment

Directed Polymer in Random Environment

Sn

N z

I (Sn)n≥0simple random walk on Z^d

I Disorder: i.i.d. random variables ω(n, x ) zero mean, unit variance

λ(β) := log E[e^{βω(n,x )}] < ∞

I (-)Hamiltonian HN,β(ω, S ) :=β

N

X

n=1

ω(n, Sn) − λ(β) N

Partition Functions

e^HN,β(ω,S) 

S0= zi

= 1

(2d )^N

X

(s0,...,sN) n.n.: s0=z

e^HN,β(ω,s)

Weak and strong disorder

Ford ≥ 3there isweak disorder:

forβ> 0 small: ZN(z)−−−−→^a.s.

N→∞ Z(z) > 0 Ford = 1, d = 2onlystrong disorder:

for any β> 0: ZN(z)−−−−→^a.s.

N→∞ 0 (despite E[ZN(z)] ≡ 1)

Intermediate disorder regime

Can wetune β = β_N → 0 to see an interesting regime ?

uN(t, x ) := ZNt(√ N x )

t∈[0,1], x ∈R^d

−−−−−→d

N→∞ u(t, x ) ?

[ Motivation: properties of the underlying polymer model (Gibbs measure) ]

Case d = 1

Theorem

Rescale

βN= βˆ N^1/4 Then

uN(t, x ) −−−−−→^d

N→∞ u(t, x ) (up to time rev.) Solution of 1dStochastic Heat Equation (SHE)







∂tu(t, x )=¹₂∆xu(t, x ) + ˆβ ˙W(t, x )u(t, x ) u(0, x )≡ 1

W = Gaussian white noise on [0, 1] × R

Directed polymer and SHE

I Partition functions solve alattice SHE

ZN+1(z) − Z_N(z) = ∆Z_N(z) + β ˜ω(N + 1, z) eZN(z)

I Space-mollified SHE: jδ(x ) :=_δ¹j ^√x

δ

(d = 2)

Generalized Feynman-Kac

uδ(t, x ) = E^{d BM}√x δ

 exp

 Z _δ^t

0

β (Wˆ ∗ j)(ds, Bs) − ¹₂βˆ²ds



Continuum directed polymer (Wiener sausage) with N = ¹_δ

uδ(t, x ) close to uN(t, x ) = ZNt(√ Nx )

Case d = 2

For d = 2 the right scaling is

βN ∼

r π

log N

βˆ [Lacoin ’10] [Berger, Lacoin ’15]

RN := E^rw

" _N X

n=1

1{Sn=S_n⁰}

#

∼ 1 πlog N

Logarithmic replica overlap ⇐⇒ disorder ismarginally relevant

Problem

uN(x ) = ZN(√

Nx ) −−−−−→^d

N→∞ u(x ) random field on R²?

Outline

1. Directed Polymer and the Stochastic Heat Equation

2. Main Results

3. Renewal Theorems

4. Second Moment

5. Third Moment

Subcritical regime ˆ β < 1

Rescale βN ∼

r π

log N

βˆ with β < 1ˆ

Theorem

I For every fixed x ∈ R² uN(x ) = ZN(

√

Nx ) −−−−−→^d

N→∞ expn

N(0,σ²) −¹₂σ²o withσ²= log ¹

1−βˆ²

I For any x 6= x⁰∈ R²

uN(x ) and uN(x⁰) become asymptotically independent (!)

[ Dependenceat all scales |x − x⁰| = o(1) ]

A different viewpoint

uN(x ) cannot converge in a space of functions

For continuous φ : R²→ R⁺define huN, φi :=

Z

R²

uN(x ) φ(x ) dx = _N¹ X

z∈Z²

ZN(z) φ ^√z

N

Look at uN(x ) as arandom distributionon R²(actuallyrandom measure)

Corollary (weak disorder)

Forβ < 1ˆ we have uN(x )−→ 1 as a random measure^d huN, φi −−−−−→^d

N→∞ h1 , φi = Z

R²

φ(x ) dx

Critical regime ˆ β = 1

Consider nowβ = 1, more generally theˆ critical window

βN= s

π log N

 1 + ϑ

log N



with ϑ∈ R

Key conjecture

I uN(x ) converges to anon-trivial random measure U (dx )on R² huN, φi −−−−−→^d

N→∞ hU, φi = Z

R²

φ(x )U (dx)

Theorem

For every fixed x ∈ R² we have uN(x ) −−−−→^d

N→∞ 0

Heuristic picture

x ∈ R² ZN(√

Nx )

1

Second moment in the critical window

What is known

Tightnessvia second moment bounds EhuN, φi

≡ 1, φ sup

N∈N

EhuN, φi²

< ∞

In fact EhuN, φi²

−−−−−→

N→∞ φ,Kφ

< ∞

with explicit kernel K x , x⁰
∼ C log 1

|x − x⁰| as |x − x⁰| → 0

Corollary

Subsequential limits hu_Nk, φi−−−−→^d

k→∞ hU, φi Trivial U ≡ 1 ?

Main result I. Third moment in the critical window

We determine the sharp asymptotics ofthird moment

Theorem

lim

N→∞ EhuN, φi³

=C (φ) < ∞

I Explicit expressionfor C (φ) (series of multiple integrals, see below)

Corollary

Any subsequential limit u_Nk −→^d U has covariance kernelK(x , x⁰) U 6≡ 1 is non-degenerate !

Point-to-point partition function

So far we have consideredpoint-to-planepartition functions Z_N(z)

Point-to-point partition function

ZN(z, w ) = E^rwh

e^HN,β(ω,S)1{SN=w }

S0= zi

This correspond to thefundamental solutionsof SHE (δ initial data) We determinesharp second moment asymptotics for diffusively rescaled

uN(t; x , y ) = ZNt(

√ Nx ,

√

Ny ) t ∈ [0, 1] , x , y ∈ R²

I Probabilistic approach (renewal) beyond [Bertini-Cancrini ’95]

I Key tool forthird moment asymptotics

Main result II. Second moment for point-to-point

Critical window βN = s

π log N

 1 + ϑ

log N



with ϑ∈ R

Theorem

As N → ∞

EuN(t; x , y )²

∼ (log N)²

πN² Gϑ(t)g^t

4(y − x )

I g_t(z) = 1

2πte^−|z|22t (standard Gaussian density on R²)

I Gϑ(t) = Z ∞

0

e^(ϑ−γ)st^s−1

Γ(s) ds (special renewal function)

Outline

1. Directed Polymer and the Stochastic Heat Equation

2. Main Results

3. Renewal Theorems

4. Second Moment

5. Third Moment

Heavy tailed renewal process

Fix N ∈ N and consider i.i.d. random variables T₁^(N), T₂^(N), . . . with P T_i^(N)= n
= 1

log N 1

n for n = 1, 2, . . . , N Consider the associated random walk (renewal process)

τ_k^(N) = T₁^(N) + T₂^(N) + . . . + T_k^(N)

Proposition

τ_{bs log Nc}^(N) N



s≥0

−−−−−→d

N→∞ Y = Ys

s≥0

Y is the subordinator (increasing L´evy process) with L´evy measure ν(dt) = 1

t 1(0,1)(t) dt

A special subordinator

The subordinator Y_s has anexplicit densityf_s(t)

Theorem

f_s(t) =







t^s−1e^−γs

Γ(s) if t ∈ (0, 1) . . . if t ≥ 1

Explicit formula for the exponentially weightedrenewal density

G_ϑ(t) = Z ∞

0

e^ϑsf_s(t) ds = Z ∞

0

e^(ϑ−γ)st^s−1 Γ(s) ds

∼ 1

t (log¹_t)² as t ↓ 0

Strong renewal theorem

Consider exponentially weighted renewal function for the random walk U_N(n) = X

k≥0

λ^kP τ_k^(N) = n

Renewal Theorem

Rescale (critical window)

λ = 1 + ϑ

log N + o _{log N}¹
Then

U_N(n) ∼ log N N G_{ϑ N}ⁿ

Space-time generalization

Random walk τ_k^(N), S_k^(N)
in N × Z²

I S_k^(N) = X₁^(N)+ . . . + X_k^(N)

I P(X₁^(N)= x | T₁^(N) = n) ∼ gn(x ) (n → ∞) Exponentially weighted renewal function

UN(n, x ) =X

k≥0

λ^kP τ_k^(N)= n , S_k^(N)= x

Space-time Renewal Theorem

UN(n, x ) ∼ log N N² Gϑ n

N,^√x_n

where Gϑ(t, z) :=Gϑ(t)gt(z)

Outline

1. Directed Polymer and the Stochastic Heat Equation

2. Main Results

3. Renewal Theorems

4. Second Moment

5. Third Moment

Partition function and polynomial chaos

ZN(0) = E^rwh

e^HN(ω,S)i

= E^rwh

e^{P1≤n≤NPx ∈Z2(βω}(n,x )−λ(β)) 1_{{Sn =x}}i

= E^rw

 Y

1≤n≤N

Y

x ∈Z²

e^(βω(n,x )−λ(β)) 1_{{Sn =x}}



= E^rw

 Y

1≤n≤N

Y

x ∈Z²

1 +X_n,x1{S_n=x }



= 1 + X

1≤n≤N x ∈Z²

P^rw(S_n= x )X_n,x

+ X

1≤n<m≤N x ,y ∈Z²

P^rw(Sn= x , Sm= y )Xn,xXm,y + . . .

ZN(0) multi-linear polynomialof RVs Xn,x:= eβω(n,x )−λ(β)− 1

Polynomial chaos

E[Xn,x] = 0 Var[Xn,x] ∼ β²

Let us pretend X_n,x = βY_n,x with Y_n,x

n,x i.i.d. N (0, 1)

Then ZN(0) ' 1 + β z_N⁽¹⁾ + β²z_N⁽²⁾ + . . .

z_N⁽¹⁾:= X

1≤n≤N x ∈Z²

P^rw(S_n= x )Y_n,x

z_N⁽²⁾:= X

1≤n≤m≤N x ,y ∈Z²

P^rw(Sn= x , Sm= y )Yn,xYm,y

The choice of β

z_N⁽¹⁾ is Gaussian with Var[z_N⁽¹⁾] = X

1≤n≤N

X

x ∈Z²

P^rw(Sn= x )² = X

1≤n≤N

P^rw(Sn= S_n⁰) = RN

where R_N = E^rw

" _N X

n=1

1{Sn=S_n⁰}

#

is thereplica overlap

Var[z_N⁽¹⁾] ∼ 1 π

X

1≤n≤N

1

n ∼ log N π

To normalize βz_N⁽¹⁾ we choose β = βN = βˆ qlog N

π

Sharp asymptotics

Varz_N^(k)

∼ 1 π^k

X

0<n1<...<nk≤N

1 n1

1 n2− n1

· · · 1 nk − nk−1

=  log N π

k

P

τ_k^(N)≤ N

∼  log N π

k

P Y k

log N ≤ 1 Variance of point-to-plane partition function

VarZN(0)

= X

k≥1

(βN)^kVarz_N^(k)

∼ X

k≥1

1 +_{log N}^ϑ
k

P Y k

log N ≤ 1

∼ (log N)

 Z ∞ 0

e^ϑsP(Ys ≤ 1)ds



Similar asymptotics for point-to-point partition function ZN(z, w ) Space-time Renewal Theorems

Outline

1. Directed Polymer and the Stochastic Heat Equation

2. Main Results

3. Renewal Theorems

4. Second Moment

5. Third Moment

Third moment in the critical window

hu_N, φi = hZ_N(√

N · ) , φi ismultilinear polynomialof i.i.d. RVsX_n,x huN, φi = X

I ⊆{1,...,N}×Z²

c(I ) Y

(n,x )∈I

Xn,x

for suitable coefficients c(I )

I Expand E[huN, φi³] in 3 sums

I X’s from different sums match in pairs or triples (by E[Xn,x] = 0)

I Triple matchings give negligible contribution

Pairwise matching of theX’s

non-trivial, yet manageablecombinatorial structure

Combinatorial structure

I Three copies of RVs X⁽¹⁾_{(n,x )}, X⁽²⁾_{(n,x )}, X⁽³⁾_{(n,x )} have tomatch in pairs

I Consecutive stretchesof pairwise matchings with same labels E.g. first (1, 2), then (1, 3), then (1, 2) again . . .

(a2, x2) (b2, y2)

(a3, x3) (b₃, y₃) (b1, y1)

(0, z1)

(0, z₂)

(a1, x1)

(0, z3)

(a₄, x₄) (b4, y4)

I Stretch Second moment of point-to-pointpartition function

Asymptotic third moment

Theorem

lim

N→∞Eh

huN, φi − E[huN, φi]
³i

= C (φ) =

∞

X

m=2

3 · 2^m−1Im(φ)

I m is the number of stretches

I 3 · 2^m−1 is the number of choices of stretch labels Im(φ) :=

Z

· · · Z

0<a₁<b₁<...<a_m<b_m<1 x1, y1, x2, y2, ..., xm, ym∈ R²

d~a d~b d~x d~y Φ²_a1(x1) Φa2(x2)

Gϑ(b₁− a1, y₁− x1)g_a2−b1

2

(x₂− y1)Gϑ(b₂− a2, y₂− x2)

m

Y

i =3

gai −bi−2 2

(xi− yi −2)gai −bi−1 2

(xi− yi −1)Gϑ(bi− ai, yi− xi)

Conclusion

I Non trivial to prove that C (φ) < ∞

I We need to show that Im(φ)decays super-exponentially as m → ∞

I We cannot factorize the integral using H¨older-type inequalities:

kernels gt(z) andGϑ(t, x ) are barely integrableas t ↓ 0

I We exploit the ordering a1< b1< a2< b2< . . . in order to prove sharp recursive bounds

Work in progress

I Uniqueness of subsequential limit U viacoarse-grainingarguments Existenceof the limit u_N −−−−−→^d

N→∞ U

I Properties of the limiting random measure U

(it looksnot so closeto Gaussian Multiplicative Chaos)

Thanks