Averaging principle and Shape Theoremfor growth with memory

Amir Dembo (Stanford)

Paris, June 19, 2018

Joint work with Ruojun Huang, Pablo Groisman,and Vladas Sidoravicius

Laplacian growth & motion in random media

Random growth processes arise in many physical and biologicalphenomena, going back to Eden (61’)/fpp (Hammersley-Welsh 65’).Math challenge to understand their evolution and pattern formation.

Laplacian growth models: growth at each portion of the boundarydetermined by the harmonic measure of the boundary from a source.

Ex.: Diffusion Limited Aggregation dla (Witten-Sander 81’),Dielectric Breakdown Model (dbl, Niemeyer et. al. 83’),idla (Meakin-Deuthch 86’; Diaconis-Fulton 91’),hl (Hastings-Levitov 98’); (see Miller-Sheffield 13’).

Also Abelian sandpiles (Bak et. al. 87’) & Rotor aggregation(see Levine-Peres 17’).

Related models of motion in random media:Once-Reinforced Random Walk orrw (Davis 90’),Origin-Excited Random Walk oerw (Kozma 06’).

Amir Dembo Random growth, Shape Theorem 1 / 16

Excited walk towards the center

Excited random walk: Benjamini-Wilson (04’), Kosygina-Zerner (12’).At first visit by rw to a vertex, it gets a (one-time) drift in a fixeddirection (e.g. ~e1), on subsequent visits to vertex perform srw.

Simplified model of orrw, where drift direction not fixed (one-timeedge/vertex conductance increase 1→ 1 + a).

Kozma (06’) studies oerw with normalized drift at −v direction:E[Xt+1 −Xt|Ft] = − δ

||v||2 v if Xt = v is first visit of v ∈ Zn.

Proving recurrence ∀n ≥ 1, δ > 0.

Kozma (07’, 13’): Conjectured Shape Theorem for “most” oerw-s.Sidoravicius: same for orrw if reinforcement strength a > 0 large.

Having a bulky limit shape connected to the conjectural recurrence ofsuch random walks (confined but not too trapped within a small region).By Shape Theorem get excursion probabilities and deduce recurrence.

Amir Dembo Random growth, Shape Theorem 2 / 16

Simulations of orrw

Vertex orrw on Z2.

Reinforcement strength a = 2 (left), a = 3 (middle), a = 100 (right)in a box of size 2000. Color proportional to

√· of vertex first visit time.

Amir Dembo Random growth, Shape Theorem 3 / 16

Simulations of oerw

oerw on Z2 with different excitation rules.

L: move one unit towards the origin in a coordinate chosenwith probability proportional to its absolute value.

M: move one unit towards the origin in the direction ofthe coordinate with largest absolute value.

R: move one unit towards the origin in each coordinate.

Sites colored according to the first visit time by the walk.

Amir Dembo Random growth, Shape Theorem 4 / 16

Shape Theorems: idla and generalizations

idla (Lawler-Bramson-Griffeath 92’): Each step runs srw till exitscurrent range R(t) ⊂ Zn. Then t← t+ 1 & particle back to the origin.

∀η > 0, a.s. B(o, (1− η)t) ⊂ R(cntn) ⊂ B(o, (1 + η)t) ∀t large.

Uniform idla (Benjamini-DuminilCopin-Kozma-Lucas 17’): Upon exitingR(t) jump to a uniformly chosen v ∈ R(t). Same conclusion as for idla.

idla with finitely fixed sources & particles moving simultaneously at rateε−1 (Gravner-Quastel 00’): particles empirical density on (εZ)n convergesweakly as ε→ 0 to solutions of a Stefan free boundary problem.

Amir Dembo Random growth, Shape Theorem 5 / 16

A random growth model in Rn

We construct a simplified growth model in Rn, n ≥ 2, to bypass sometechnical difficulties of the lattice models.

Family of domain processes (Dεt)t≥0 ⊂ Rn with scale & Poisson rate

parameter ε ∈ (0, 1]. Our rules keep domains star-shaped and compact,so represented by boundary Rεt : Sn−1 7→ R+, a pure jump (slow) processevolving by adding a small bump of approximate volume ε at randompoints on the boundary.

Growth location driven by a (fast) particle process (xεt)t≥0 via hittingprobability density F (Rεt , x

εt, ·) on Sn−1, specifying law of the angle ξt

where a bump is to be added. Mapping H(Rεt , ξt) says to where particleis transported upon hitting the boundary at angle ξt.

F (r, x, ·) : C(Sn−1)× Rn → L2(Sn−1), H(r, z) : C(Sn−1)× Sn−1 → Rn.

Example: F (r, x, ·) Poisson kernel of r, H(r, z) = αr(z)z, α ∈ [0, 1).

Amir Dembo Random growth, Shape Theorem 6 / 16

Spherical approximate identity & small bump on Sn−1

sai {gη(〈z, ·〉)}η>0 such that: gη ∈ C([−1, 1],R+), 1 ? gη = 1,

||f ? gη||2 ≤ ||f ||2 & ||f ? gη − f ||2 → 0 as η → 0

(where ? denotes spherical convolution).

E.g. gη(s) = cη−(n−1)φ(1− 1−s

η2

)for φ ∈ C([−1, 1],R+), φ(−1) = 0.

For a bump centered at angle ξ add damped sai ε1/nηn−1gη(〈ξ, ·〉)with η(ε, r, x) = ε1/ny

−1/(n−1)r,x . Such bump has height O(ε1/n)

& support on the spherical cap of Euclidean radius 2η around ξ.

-1 -0.5 0 0.5 1

L: gη(s) at different η. R: Adding gη(〈z, ·〉) to S2; z = (0, 0, 1).

Amir Dembo Random growth, Shape Theorem 7 / 16

Our random growth model

yr,x := ωn∫Sn−1 r(θ)

n−1F (r, x, θ)dσ(θ), ωn = σ(Sn−1)

=⇒ Each bump adds on average volume ε+ o(ε) to Dεt .

Hitting kernel: F (r, x, ·) : C(Sn−1)× Rn → L2(Sn−1).Transportation function: H(r, z) : C(Sn−1)× Sn−1 → Rn.

(Rεt , xεt)t≥0 updates at arrival times {T εi } of a rate ε−1 Poisson process.

At each t = T εi , conditional on Ft− := σ(Rεs, xεs, ξs : s ≤ t−), let

Rεt(θ) = Rεt−(θ) +

ε1/nη(ε,Rεt− ,x

εt− )n−1︷ ︸︸ ︷

ε

yRεt−,xεt−

gη(ε,Rεt−,xεt−

)(〈ξt, θ〉), θ ∈ Sn−1,

ξtd∼ F (Rεt− , x

εt− , ·) , xεt = H(Rεt− , ξt).

Amir Dembo Random growth, Shape Theorem 8 / 16

Simulations of the random growth model

L: H(r, ξ) = (r(ξ)− 1)+ξ. M: H(r, ξ) = (r(ξ)− |ξ|∞|ξ|2 )+ξ.

R: H(r, ξ) = (r(ξ)− |ξ|1|ξ|2 )+ξ. F (r, x, ·) harmonic measure on r (from x).

Snapshots t = 0.2 k2, 0 ≤ k ≤ 10, for ε = 10−4, φ = 1[0,1], c = 20,

Linear in k evolution ⇒ O(√t) asymptotic diameter growth.

Small t ≈ spherical shape (H = o, idla-like).

Large t⇒ Rεt “feels” the geometry ⇒ H-dependent limit shape(sphere, square, diamond; similar to different excitation for oerw).

Amir Dembo Random growth, Shape Theorem 9 / 16

Frozen domain & limiting ODE

Domain Rεt ≡ r frozen =⇒ (xε,rt )t≥0 Markov process.

Assume ∀r ∈ C(Sn−1), process {x1,rt } has a unique invariant law νr.

Then, limiting infinite-dimensional ode for Rεt is

r̄t(θ)= r̄0(θ) +

∫ t

0

b̄(r̄s)(θ)ds, θ ∈ Sn−1 (ode)

b̄(r)(θ) :=

∫Rnb(r, x)(θ)dνr(x) , b(r, x)(·) :=

ωnyr,x

F (r, x, ·)

mg decomposition of {Rεt} has bv-term (drift)

bε(r, x) := b(r, x) ? gη(ε,r,x) → b(r, x) as ε→ 0 (⇒ η → 0),

and mg term of O(ε) quadratic variation.

Leb(rt) = Leb(r0) + t in line with

E[Leb(Dεt)|Ft−]− Leb(Dε

t−) ≈ ε at each t = T εi

Amir Dembo Random growth, Shape Theorem 10 / 16

Assumptions on F & H

A1(a) :={r ∈ C(Sn−1) : inf

θr(θ) ≥ a, ||r||2 ≤ a−1

}A(a) :=

{(r, x) ∈ D(F ) : r ∈ A1(a), x ∈ Im(H(r, ·))

}Assumption (L)

∃K = K(a) <∞ so for all (r, x), (r′, x′) ∈ A(a), z, z′ ∈ Sn−1

||F (r, x, ·)− F (r′, x′, ·)||2 ≤ K(||r − r′||2 + |x− x′|

), (LF )

|H(r, z)−H(r′, z′)| ≤ K(||r − r′||2 + |z − z′|

), (LH)

||b̄(r)− b̄(r′)||2 ≤ K||r − r′||2 . (Lb̄)

Assumption (E)

∀r ∈ C(Sn−1), the invariant probability law νr of {x1,rt } exists & unique.

limt→∞

sup(r,x1,r

0 )∈A(a)

E[∣∣∣∣1

t

∫ t

0

[b(r, x1,rs )− b̄(r)]ds∣∣∣∣22

]= 0 .

For (E) suffices to have uniform minorization of jump kernel Pr of {x1,rTi }inf

(r,x)∈A(a){(Pr)

n0(x, ·)} ≥ m(·) .

Amir Dembo Random growth, Shape Theorem 11 / 16

Hydrodynamic limit by Averaging Principle

Theorem (DGHS,18’)

(a) Assume (L) & (E). Fix Rε0 = r0 ∈ C(Sn−1). Then,

limε→0

P(

sup0≤t≤T∧σε(δ)

||Rεt − r̄t||2 > ι)

= 0, ∀T, ι, δ > 0 (?)

with Ft-stopping time

σε(δ) := inf{t ≥ 0 : minθ{F (Rεt , x

εt, θ)} < δ}.

(b) No σε(δ) in (?) when inf{F (r, x, θ) : (r, x) ∈ A(a), θ ∈ Sn−1} > 0.

Proof: As ε→ 0, dynamic of {xεt} near equilibrium at a time scale where{Rεt} does not change macroscopically, namely we have here an

averaging principle.

Amir Dembo Random growth, Shape Theorem 12 / 16

Towards shape theorem: scale invariance

Assumption (I)

∀c > 0 if (r, x) ∈ D(F ) then (cr, cx) ∈ D(F ) and

F (r, x, ·) = F (cr, cx, ·) cH(r, ·) = H(cr, ·)

Example: (I) holds for F (r, x, ·) Poisson kernel at D (from x), r = ∂D;& H(r, z) = αr(z)z for z ∈ Sn−1, fixed α ∈ (0, 1).

Proposition (coupling)

Under Assumption (I), there exists coupling with(Rεt , x

εt) = (ε1/nR1

t/ε, ε1/n x1t/ε) for all t > 0, whenever holding for t = 0.

Amir Dembo Random growth, Shape Theorem 13 / 16

Candidate limiting shapes

ψ ∈ C(Sn−1) invariant (shape) for

r̄t(θ) = r̄0(θ) +

∫ t

0

b̄(r̄s)(θ)ds, θ ∈ Sn−1 (ode)

⇐⇒ r0 = ψ yields rt = ctψ in (ode); (ct)n = Leb(ψ) + t.

Recall: b̄(r)(θ) = ωn∫Rn y

−1r,xF (r, x, θ)dνr(x)

(I) ⇒ νcr(c ·) = νr(·), ycr,cx = cn−1yr,x ⇒ b̄(cr) = c−(n−1)b̄(r)

Under (I) shape ψ is invariant ⇐⇒ b̄(ψ)(θ) = 1nψ(θ) ∀θ ∈ Sn−1

Ex: F (r, x, ·) Poisson kernel & H(r, z) = α(z)r(z)z

ψ = c (Euclidean ball) invariant ⇐⇒ α(z) = α (constant).

Amir Dembo Random growth, Shape Theorem 14 / 16

Shape theorem

Theorem (DGHS 18’)

Assume (I) and hydrodynamic limit

limε→0

P(

sup0≤t≤T

||Rεt − r̄t||2 > ι)

= 0, ∀T, ι, δ > 0 (?)

If ψ invariant for ode (wlog Leb(ψ) = 1), then ∀T, c, ι > 0,

limN→∞

P(

sup1≤s≤T

∣∣∣∣(N(c+ s))−1/nRsN − ψ∣∣∣∣2> ι∣∣R0 = (Nc)1/nψ

)= 0

While ||F (r, x, ·)− F (r′, x, ·)||2 ≤ K(a)(||r − r′||2

)(LF )

fails for Poisson kernel F (r, x, ·), this is resolved upon regularizing F .

Amir Dembo Random growth, Shape Theorem 15 / 16

Thank you!

Generator & mg decomposition

The generator of (Rεt , xεt)t≥0 is(

Lεf)(r, x) =

1

ε

[ ∫Sn−1

f(r +

ε

yr,xgη(ε,r,x)(〈ξ, ·〉), H(r, ξ)

)F (r, x, ξ)dσ(ξ)− f(r, x)

].

Taking f(r, x) = r(θ) at fixed θ ∈ Sn−1 yields

Rεt(θ) = Rε0(θ) +

∫ t

0

bε(Rεs, xεs)(θ)ds+ Σεt(θ)

where Σεt(θ) a (small) mg.

Taking f(r, x) = x · ~ei, i = 1, . . . , n, yields

xεt = xε0 +

∫ t

0

ε−1h(Rεs, xεs)ds+M ε

t

h(r, x) :=

∫Sn−1

H(r, ξ)F (r, x, ξ)dσ(ξ)− x ,

for some Rn-valued mg M εt .

Amir Dembo Random growth, Shape Theorem 16 / 16