THE EXTREMAL PROCESS OF BRANCHING BROWNIAN …kistler/dateien/talks/bbm_zurich.pdfTHE EXTREMAL...

THE EXTREMAL PROCESS OFBRANCHING BROWNIAN MOTION

Nicola Kistler, University of Bonn

joint with

Louis-Pierre ARGUIN, University of Montreal

Anton BOVIER, University of Bonn

OUTLINE

I. Derrida’s Generalized Random Energy Models

II. Branching Brownian Motion & KPP equation

III. Other models?

IV. BBM and the evolution of the travelling wave

DERRIDA’S REM

Random energies XNk

iid∼ N (0, N), k = 1 . . .[

eN]

Largest values in the N →∞ limit?

Average level density at energy X ∼ eN√2πN

exp

(

−X2

2N

)

⇒ Order one iff X ∼√2N − 1

2√2logN =: r(N)

Thm. Extremal Process converges as N → ∞ to Poisson PP

•{

XNk − r(N)

}

k−→PPP

(

e−√2xdx

)

• P

[

maxkXNk − r(N) ≤ x

]

→ exp

(

−e−√2x

)

GUMBEL

I. GREM

DERRIDA’s GREM AT CRITICALITY d ∈ N FIXED!

• Configurations k = (k1, . . . , kd), kj = 1 . . . eN/d (♯k = eN)

• Energies: Xk = X{1}k1

+X{2}k1,k2

+ · · ·+X{d}k1,...,kd

X{j}·

iid∼ N(

0,N

d

)

independent for j 6= j′

Xk = X1k1

+X2k1,k2

+ · · ·+Xdk1,...,kd

(critical: ♯k1 = · · · = ♯kd, var(X1) = · · · = var(Xd) )

Number of levels = d

I. GREM

Correlations cov[

Xk, Xl

]

=N

d·maxj≤d

{k1 · · · kj = l1 · · · lj}

Thm (Bovier-Kurkova 2004). Extremal Process

{

Xk − r(N)}

k

N↑∞−→ PPP

(

Kde−√2xdx

)

• REM correction r(N) =√2N − 1

2√2logN

• deterministic shift Kd ∈ (0,1)

• P

[

maxkXk − r(N) ≤ x]

→ exp

(

−Kde−√2x

)

shifted Gumbel

crit GREM ≡ REM. What if d = d(N) = N?

I. GREM

BRANCHING BROWNIAN MOTION (BBM)

Particle performs BM, splits in two at expo-time (mean one, inde-

pendent of motion). Particles continue independently, under same

splitting rule. After time t, generated tree containing n(t) ≈ exp (t)

particles located at x1(t), . . . , xn(t)(t).

Picture by Matt Roberts, University of Bath

II. BBM & KPP

F-KPP EQUATION. Distribution of the maximum of BBM

u(t, x)def= P

[

maxk

xk(t) ≤ x

]

Thm. • (McKean 1975) u solves the F-KPP equation

∂tu =1

2uxx + u2 − u, u(0, x) =

1 x ≥ 0

0 x < 0

• (KPP 1932) u(

t, x+m(t))

→ ω(x)

12ωxx +

√2ωx + ω2 − ω = 0 shape of travelling wave

m(t) =√2 · t+ o(t)

Leading order of max of BBM ≡ REM

II. BBM & KPP

BUT LOWER ORDER IS DIFFERENT

Thm. (Bramson 1978) u(t, x+m(t)) → ω(x) for

m(t) =√2 · t− 3

2√2log t

Compare to REM

r(N) =√2 ·N − 1

2√2logN

• BBM is the critical model in which correlations just start to affect

the geometry of extremes

• more correlations affect the leading order, yet a classification of

extremal processes beyond the BBM-world is nowhere in sight

II. BBM & KPP

Limiting law ω of maximal displacement

Derivative martingale

Z(t) =∑

k

{√2t− xk(t)} exp

(

−√2{

√2t− xk(t)}

)

Thm. • (Lalley-Sellke 1987) Random shift of Gumbel

ω(x) = E exp

(

−Ze−√2x

)

Z(d)= limt→∞Z(t) delay of the wave

• (Bramson 1983) right-tail 1− ω(x) ∼ x · e−√2x x ↑ ∞

Compare to Gumbel 1− exp

(

−e−√2x

)

∼ 1 · e−√2x x ↑ ∞

II. BBM & KPP

BUT EXTREMAL PROCESS OF BBM

IS NOT RANDOM SHIFT OF REM

dn,n+1 ≡ distance of nth and (n+1)th rightmost particle

E

[

dn,n+1

]

REM=

1

n

Brunet-Derrida (2009). Evidence for

E

[

dn,n+1

]

BBM=

1

n−

1

n logn+ ...

...curiously ’close’ to that of PPP with density

−xe−√2xdx

on the negative

SO, WHAT IS IT?

II. BBM & KPP

Thm. A Poisson CLUSTER process.{

xk(t)−m(t), k ≤ n(t)

}

−→{

Pi +∆(i)k , i, k

}

• P = {Pi}i∈N ≡ PPP

(

Z · e−√2xdx

)

• Clusters are iid: BBM performing unusual displacements

∆ = {∆k} = limr→∞

{

xk(r)− xmax(r)

∣

∣

∣

∣

xmax(r) ≥√2r

}

k

PPP(Ze−√2xdx)

clusters of particles on the left of Poissonian points

II. BBM & KPP

MAIN STEPS IN THE PROOF

1. Characterize paths of extremal particles

2. Identify genealogical distances among particles at the edge

(correlations)

3. Identify process of cluster-extrema

4. Introduce auxiliary process of recent common ancestors

(”cavity dynamics”)

5. Bramson’s estimate on right tail

P

[

maxk

xk(t)−m(t) ≥ x

]

∼ xe−√2x, x ↑ ∞

yields IID and law of clusters

II. BBM & KPP

Step 1. Characterize PATHS of extremal particles.

m(t)

t

space

time

Can paths fluctuate wildlyin the upward direction ?

s 7→ stm(t)

unusually high values

II. BBM & KPP

Step 1.

m(t)

t

space

time

No! Below upper envelope Lt,γ

t/2

for most of the time

Lt,γ(s) = stm(t) +

sγ 0 ≤ s ≤ t/2

(t− s)γ t/2 ≤ s ≤ t

t− rr

0 < γ < 1/2

II. BBM & KPP

Step 1.

m(t)

t

space

time

t− r

Path of extremal particles?

r

x(0) = 0, x(s) BM, x(t) = m(t)

gaussian proc. with drift conditioned on endpoint

≡ zt(s) +stm(t)

zt Brownian bridge

zt(0) = zt(t) = 0

II. BBM & KPP

Step 1.

m(t)

t

space

time

t− rr

Reformulate

⇒ get rid of the drift

Condition on endpoint

rotate

II. BBM & KPP

Step 1.

space

Driftless B-bridgeconditioned to stay belowsymmetric, concave, power-law curve

time

r t− rt

s 7→

sγ 0 ≤ s ≤ t/2

(t− s)γ t/2 ≤ s ≤ t0 < γ < 1/2

Bbridge zt(s)

II. BBM & KPP

Step 1.

space

Dangerous fluctuations

time

t

s

t− rr

√

E[

zt(s)2] ≈ s1/2 ≫ sγ γ < 1/2

How to avoid upper envelope?

II. BBM & KPP

Step 1.

space

Go negative ! entropic repulsion

time

t

t− rr

s 7→

−sα 0 ≤ s ≤ t/2

−(t− s)α t/2 ≤ s ≤ t(α < 1/2)

II. BBM & KPP

Step 1.

t

space

time

⇒ Real picture

r t− r

m(t)O (−sα,−(t− s)α)

paths of extremal particlesare very low for most of their time

II. BBM & KPP

Step 2. Genealogies

r t− r

P [one path] = ♯part. at time s× P [jump] = esα × e−sα = O(1)

Ancestries in (r, t− r) ?

t

space

time

s

m(t)

P [2 paths] = ♯part. at time s× P [2 jumps] = esα × e−2sα → 0!!

II. BBM & KPP

Step 2. Genealogies (covariance between i and j)

Qt(i, j)def= sup{s ≤ t : xi(s) = xj(s)}

Thm.

limr→∞ sup

t>2rP

[

∃i,j extremal : Qt(i, j) ∈ (r, t− r)]

= 0

t

space

time

t− r

no branching

r

small branchesm(t)

II. BBM & KPP

Step 3. The process of cluster-extrema

r t− r t

space

time

Consider only subset of

extremal particles

which are also

small clusters

maximal inside≈ m(t)

II. BBM & KPP


r t− r t

space

time

Consider only subset of

extremal particles

which are also

maximal inside

small clusters

≈ m(t)

II. BBM & KPP


Thm. Such Point Process converges to

PPP

(

Z · exp(

−√2x

)

dx

)

,

random shift of PPP.

(Set w.l.o.g. Z ≡ 1)

This implies following picture for extremal process of BBM

PPP(Ze−√2xdx)

clusters of particles on the left of Poissonian points

What about the clusters?

II. BBM & KPP

Step 4. An auxiliary process & cavity dynamics

t

space

time

t− r

no branching

r

BBM {xk(r)−√2r}

independent√

II. BBM & KPP

Step 4. An auxiliary process & cavity dynamics

t

space

time

t− r

no branching

r

Why identically distributed

when seen from

cluster-extrema ?

II. BBM & KPP

Step 4.

t

space

time

t− r

no branching

r

Process of ancestors

II. BBM & KPP

Step 4.

t

space

time

t− r

no branching

r

Process of ancestors

should be close

to Poissonian.

⇒ DENSITY ??

II. BBM & KPP

Step 4.

t

space

time

t− r

no branching

r

E [♯ancestors ∼ X]

∼ −X · e−√2X

(X ≤ 0)

II. BBM & KPP

Thm. Extremal process of BBM, cavity dynamics:

{ξk} = limt→∞

{

xk(t)−m(t)}

= limt→∞

{

xk(t− r)−m(t− r) + x(k)i (r)−

√2r

}

t− r t

=

{

ξk + x(k)i (r)−

√2r

}

invariance for any r

←{

πi + x(i)k (r)−

√2r

}

for r → ∞

auxiliary process {πi, i ∈ N} = PPP

(

−x · e−√2x · dx, x ≤ 0

)

II. BBM & KPP

Step 5. Bramson’s estimate on right tail

time

space

0

r

Π = {πi, i ∈ N}=PPP

(

−x · e−√2x · dx, x ≤ 0

)

πi

II. BBM & KPP

Step 5.

time

space

0

r

Clusters = subcritical BBM

= {πi + x(i)k (r)−

√2r}

dynamics ↔ thinning of the π′s

πi

II. BBM & KPP

Step 5.

time

space

0

−πi +√2r

πi

πi survives thinning if superimposed

BBM performs unusual displacementr

II. BBM & KPP

Step 5.

time

space

0

πi

Contributing π′s? For r ↑ ∞P

[

maxx(i)k (r) ≥ −πi +

√2r

]

→ 0r

II. BBM & KPP

Step 5.

time

space

0

πi

r

But E [♯πi ∼ X] ∼ −X · e−√2X

→ +∞ as X → −∞

II. BBM & KPP

Step 5. Bramson’s estimate on right tail

time

space

0

πi

rContribution from the bulk: πi ≈ −√

r

√r

⇒ max of BBM vs. density of PPP

II. BBM & KPP

Step 5. Max conditioned on survival is expo distributed

P

[

maxxk(r) ≥ −π +√2r +X

∣

∣

∣

∣

maxxk(r) ≥ −π +√2r

]

→ e−√2X

Independent of π ⇒ seen from cluster-extrema, clusters IID �

Description of BBM conditioned on maxxk(r) ≥√2r, r → ∞ provided

by [Chauvin & Rouault 1990]

Conditioned BBM ??Standard BBM

Global drift?

II. BBM & KPP


P

[


∣

∣

∣

∣


]

→ e−√2X




Conditioned BBM ??Standard BBM

II. BBM & KPP


P

[


∣

∣

∣

∣


]

→ e−√2X




Conditioned BBMincrease: • birth rate,• ♯offspring. ONE

branch picks up

DRIFT

Standard BBM

II. BBM & KPP

Another approach [Aidekon, Beresticky, Brunet, Shi ’11]

Replace auxiliary process from Step 4 with description of the law of

paths of cluster-extrema (BM in a potential). Extremal proces ≡collection of such paths and generated BBMs

t

space

time

t− r

no branching

r

m(t)

II. BBM & KPP

Other models? Same picture expected to hold

• 2-dim Gaussian Free Field

• Spin Glasses with logarithmic correlated potentials

• Random Walk coverings in 2-d

(...)

III. Other models

...unifying feature?

• Right-tail X · e−√2X of the max (stiff class)

• entropic repulsion

• clustering of peaks/fractality

• Poissonianity of cluster-extrema

• clusters iid: model conditioned upon unusual displacement

III. Other models

BBM AND THE EVOLUTION OF THE TRAVELLING WAVE

t

space

time

front at time t


t

space

time

Initial branching


t

space

time

Go low ↔ huge jumps


t

space

time

Small branches


ONSET AND GEOMETRY OF THE WAVE FRONT

space

The derivative martingale

corresponds to a delay

of the wave



space

during this time

a certain speedthe wave gains



space

front =√2t− 3

2√2log t

speed =√2− 3

2√2

log ttand ”sets in”



space

front =√2t− 3

2√2log t

speed =√2− 3

2√2

log tt

front?

Mechanism behind the

displacement of the


��

��


space

front =√2t− 3

2√2log t

speed =√2− 3

2√2

log tt

√2(t− s)− C(t− s)α

√2+ C (t−s)α

s

higher speed

small wave with


��

��


travelling at

higher speed...

front


��

��


...the small wave

catches up...

front


��

��


...and splits into

”micro” waves

which hit the front

and push it further

”front + δ”


��

��


the micro waves

immediately die out

”front + δ”


��

��


is already rising

but a new wave

”front + δ”


THE EXTREMAL PROCESS OF BRANCHING BROWNIAN …kistler/dateien/talks/bbm_zurich.pdfTHE EXTREMAL...

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