The ubiquity of Branching random walks

Ofer Zeitouni

Weizmann & Courant

October 2, 2014

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 1 / 28

A Theorem

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c©S. Sheffield

Consider the (discrete) Gaussian Free Field on the (planar) box (ofside N) with Dirichlet boundary conditions.Let MN denote its maximum.

Theorem (Bramson-Ding-Z. ’13)MN − EMN converges in law: there exists a constant c > 0 and randomvariable Z > 0 so that

limN→∞

P(MN ≤ EMN + x) = E(e−cZe−√

2πx)

Shifted GumbelAlso, information on location of maximum, extremal process.. (later)

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 2 / 28

A Theorem

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20

30

40

50

10

20

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40

50

-4

-2

0

2

10

20

30

40

50

c©S. Sheffield

Consider the (discrete) Gaussian Free Field on the (planar) box (ofside N) with Dirichlet boundary conditions.Let MN denote its maximum.

Theorem (Bramson-Ding-Z. ’13)MN − EMN converges in law: there exists a constant c > 0 and randomvariable Z > 0 so that

limN→∞

P(MN ≤ EMN + x) = E(e−cZe−√

2πx)

Shifted GumbelAlso, information on location of maximum, extremal process.. (later)

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 2 / 28

A Theorem

10

20

30

40

50

10

20

30

40

50

-4

-2

0

2

10

20

30

40

50

c©S. Sheffield

Consider the (discrete) Gaussian Free Field on the (planar) box (ofside N) with Dirichlet boundary conditions.Let MN denote its maximum.

Theorem (Bramson-Ding-Z. ’13)MN − EMN converges in law: there exists a constant c > 0 and randomvariable Z > 0 so that

limN→∞

P(MN ≤ EMN + x) = E(e−cZe−√

2πx)

Shifted GumbelAlso, information on location of maximum, extremal process.. (later)

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 2 / 28

A Theorem

10

20

30

40

50

10

20

30

40

50

-4

-2

0

2

10

20

30

40

50

c©S. Sheffield

Consider the (discrete) Gaussian Free Field on the (planar) box (ofside N) with Dirichlet boundary conditions.Let MN denote its maximum.

Theorem (Bramson-Ding-Z. ’13)MN − EMN converges in law: there exists a constant c > 0 and randomvariable Z > 0 so that

limN→∞

P(MN ≤ EMN + x) = E(e−cZe−√

2πx)

Shifted GumbelAlso, information on location of maximum, extremal process.. (later)

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 2 / 28

A Theorem

10

20

30

40

50

10

20

30

40

50

-4

-2

0

2

10

20

30

40

50

c©S. Sheffield

Consider the (discrete) Gaussian Free Field on the (planar) box (ofside N) with Dirichlet boundary conditions.Let MN denote its maximum.

Theorem (Bramson-Ding-Z. ’13)MN − EMN converges in law: there exists a constant c > 0 and randomvariable Z > 0 so that

limN→∞

P(MN ≤ EMN + x) = E(e−cZe−√

2πx)

Shifted GumbelAlso, information on location of maximum, extremal process.. (later)

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 2 / 28

A Theorem

10

20

30

40

50

10

20

30

40

50

-4

-2

0

2

10

20

30

40

50

c©S. Sheffield

Consider the (discrete) Gaussian Free Field on the (planar) box (ofside N) with Dirichlet boundary conditions.Let MN denote its maximum.

Theorem (Bramson-Ding-Z. ’13)MN − EMN converges in law: there exists a constant c > 0 and randomvariable Z > 0 so that

limN→∞

P(MN ≤ EMN + x) = E(e−cZe−√

2πx)

Shifted GumbelAlso, information on location of maximum, extremal process.. (later)

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 2 / 28

Enters Random Walk

figure=rwwiki2d.eps,height=120pt

figure=half-mil.eps,height=100ptc©Y. Wataru c©Wikipedia c©R. D’Souza

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 3 / 28

Enters Random Walk

figure=half-mil.eps,height=100ptc©Y. Wataru c©Wikipedia c©R. D’Souza

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 3 / 28

Enters Random Walk

c©Y. Wataru c©Wikipedia c©R. D’Souza

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 3 / 28

Branching Brownian Motion

Model: At time 0, particle at 0 ∈ R.Particle moves by a Brownian motion until for time τ ∼ Exp(1)

P(τ > t) = e−t .

After the step, particle splits in 2. Repeat.

c©M. Roberts

figure=bbm7.eps,height=120pt

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 4 / 28

Branching Brownian Motion

Model: At time 0, particle at 0 ∈ R.Particle moves by a Brownian motion until for time τ ∼ Exp(1)

P(τ > t) = e−t .

After the step, particle splits in 2. Repeat.

c©M. Roberts

figure=bbm7.eps,height=120pt

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 4 / 28

Branching Brownian Motion

Model: At time 0, particle at 0 ∈ R.Particle moves by a Brownian motion until for time τ ∼ Exp(1)

P(τ > t) = e−t .

After the step, particle splits in 2. Repeat.

c©M. Roberts

figure=bbm7.eps,height=120pt

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 4 / 28

Branching Brownian Motion

Model: At time 0, particle at 0 ∈ R.Particle moves by a Brownian motion until for time τ ∼ Exp(1)

P(τ > t) = e−t .

After the step, particle splits in 2. Repeat.

c©M. Roberts

figure=bbm7.eps,height=120pt

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 4 / 28

Branching Brownian Motion

Model: At time 0, particle at 0 ∈ R.Particle moves by a Brownian motion until for time τ ∼ Exp(1)

P(τ > t) = e−t .

After the step, particle splits in 2. Repeat.

c©M. Roberts

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 4 / 28

Maximal displacement and the F-KPP

At time t , have Nt ∼ et particles, at locations ζ1(t), . . . , ζNt (t).Define the maximal displacement to be

Mt = max1≤i≤Nt

ζi(t) .

u(t , x) = P(Mt > x) satisfies

ut =12

uxx + u(1 − u), u(0, x) = 1x<0.

Kolmogorov-Petrovskii-Piskunov Equation ’37, Fisher ’37.Probabilistic interpretation due to McKean ’75: quadratic term due tobranching, Laplacian is Brownian motion.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 5 / 28

Maximal displacement and the F-KPP

At time t , have Nt ∼ et particles, at locations ζ1(t), . . . , ζNt (t).Define the maximal displacement to be

Mt = max1≤i≤Nt

ζi(t) .

u(t , x) = P(Mt > x) satisfies

ut =12

uxx + u(1 − u), u(0, x) = 1x<0.

Kolmogorov-Petrovskii-Piskunov Equation ’37, Fisher ’37.Probabilistic interpretation due to McKean ’75: quadratic term due tobranching, Laplacian is Brownian motion.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 5 / 28

Maximal displacement and the F-KPP

At time t , have Nt ∼ et particles, at locations ζ1(t), . . . , ζNt (t).Define the maximal displacement to be

Mt = max1≤i≤Nt

ζi(t) .

u(t , x) = P(Mt > x) satisfies

ut =12

uxx + u(1 − u), u(0, x) = 1x<0.

Kolmogorov-Petrovskii-Piskunov Equation ’37, Fisher ’37.Probabilistic interpretation due to McKean ’75: quadratic term due tobranching, Laplacian is Brownian motion.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 5 / 28

Maximal displacement and the F-KPP

At time t , have Nt ∼ et particles, at locations ζ1(t), . . . , ζNt (t).Define the maximal displacement to be

Mt = max1≤i≤Nt

ζi(t) .

u(t , x) = P(Mt > x) satisfies

ut =12

uxx + u(1 − u), u(0, x) = 1x<0.

Kolmogorov-Petrovskii-Piskunov Equation ’37, Fisher ’37.Probabilistic interpretation due to McKean ’75: quadratic term due tobranching, Laplacian is Brownian motion.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 5 / 28

Maximal displacement and the F-KPP

At time t , have Nt ∼ et particles, at locations ζ1(t), . . . , ζNt (t).Define the maximal displacement to be

Mt = max1≤i≤Nt

ζi(t) .

u(t , x) = P(Mt > x) satisfies

ut =12

uxx + u(1 − u), u(0, x) = 1x<0.

Kolmogorov-Petrovskii-Piskunov Equation ’37, Fisher ’37.Probabilistic interpretation due to McKean ’75: quadratic term due tobranching, Laplacian is Brownian motion.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 5 / 28

Maximal displacement and the F-KPP

F-KPP ’37: u(t , x + m(t)) → w(x) travelling wave, and m(t)/t →√

2.In fact, Bramson ’78

m(t) = ct − 32c

log t + O(1), t → ∞ , (c =√

2).

Proof by Probabilistic methods.Heart of proof is a variation of the “second moment method”, andstandard estimates on Brownian bridges. Explain next.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 6 / 28

Maximal displacement and the F-KPP

F-KPP ’37: u(t , x + m(t)) → w(x) travelling wave, and m(t)/t →√

2.In fact, Bramson ’78

m(t) = ct − 32c

log t + O(1), t → ∞ , (c =√

2).

Proof by Probabilistic methods.Heart of proof is a variation of the “second moment method”, andstandard estimates on Brownian bridges. Explain next.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 6 / 28

Maximal displacement and the F-KPP

F-KPP ’37: u(t , x + m(t)) → w(x) travelling wave, and m(t)/t →√

2.In fact, Bramson ’78

m(t) = ct − 32c

log t + O(1), t → ∞ , (c =√

2).

Proof by Probabilistic methods.Heart of proof is a variation of the “second moment method”, andstandard estimates on Brownian bridges. Explain next.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 6 / 28

Maximal displacement and the F-KPP

F-KPP ’37: u(t , x + m(t)) → w(x) travelling wave, and m(t)/t →√

2.In fact, Bramson ’78

m(t) = ct − 32c

log t + O(1), t → ∞ , (c =√

2).

Proof by Probabilistic methods.Heart of proof is a variation of the “second moment method”, andstandard estimates on Brownian bridges. Explain next.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 6 / 28

Back of envelope - role of atypical particles

How far can particles go? Typical particle ∼√

n (Gaussian).A specific particle gets to xt with probability p(t , xt) =

ct1/2 e−(xt )

2/2t .There are et particles, so maximal should be such that etp(t , xt) ∼ 1Gives xt ∼

√2t − 1

2√

2log t .

This is correct answer for et independent particles.But Bramson’s result is that location of maximal particle is only

xt ∼√

2t − 3

2√

2log t

Why?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 7 / 28

Back of envelope - role of atypical particles

How far can particles go? Typical particle ∼√

n (Gaussian).A specific particle gets to xt with probability p(t , xt) =

ct1/2 e−(xt )

2/2t .There are et particles, so maximal should be such that etp(t , xt) ∼ 1Gives xt ∼

√2t − 1

2√

2log t .

This is correct answer for et independent particles.But Bramson’s result is that location of maximal particle is only

xt ∼√

2t − 3

2√

2log t

Why?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 7 / 28

Back of envelope - role of atypical particles

How far can particles go? Typical particle ∼√

n (Gaussian).A specific particle gets to xt with probability p(t , xt) =

ct1/2 e−(xt )

2/2t .There are et particles, so maximal should be such that etp(t , xt) ∼ 1Gives xt ∼

√2t − 1

2√

2log t .

This is correct answer for et independent particles.But Bramson’s result is that location of maximal particle is only

xt ∼√

2t − 3

2√

2log t

Why?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 7 / 28

Back of envelope - role of atypical particles

How far can particles go? Typical particle ∼√

n (Gaussian).A specific particle gets to xt with probability p(t , xt) =

ct1/2 e−(xt )

2/2t .There are et particles, so maximal should be such that etp(t , xt) ∼ 1Gives xt ∼

√2t − 1

2√

2log t .

This is correct answer for et independent particles.But Bramson’s result is that location of maximal particle is only

xt ∼√

2t − 3

2√

2log t

Why?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 7 / 28

Back of envelope - role of atypical particles

How far can particles go? Typical particle ∼√

n (Gaussian).A specific particle gets to xt with probability p(t , xt) =

ct1/2 e−(xt )

2/2t .There are et particles, so maximal should be such that etp(t , xt) ∼ 1Gives xt ∼

√2t − 1

2√

2log t .

This is correct answer for et independent particles.But Bramson’s result is that location of maximal particle is only

xt ∼√

2t − 3

2√

2log t

Why?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 7 / 28

Back of envelope - role of atypical particles

How far can particles go? Typical particle ∼√

n (Gaussian).A specific particle gets to xt with probability p(t , xt) =

ct1/2 e−(xt )

2/2t .There are et particles, so maximal should be such that etp(t , xt) ∼ 1Gives xt ∼

√2t − 1

2√

2log t .

This is correct answer for et independent particles.But Bramson’s result is that location of maximal particle is only

xt ∼√

2t − 3

2√

2log t

Why?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 7 / 28

Back of envelope - role of atypical particles

How far can particles go? Typical particle ∼√

n (Gaussian).A specific particle gets to xt with probability p(t , xt) =

ct1/2 e−(xt )

2/2t .There are et particles, so maximal should be such that etp(t , xt) ∼ 1Gives xt ∼

√2t − 1

2√

2log t .

This is correct answer for et independent particles.But Bramson’s result is that location of maximal particle is only

xt ∼√

2t − 3

2√

2log t

Why?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 7 / 28

Back of envelope - role of atypical particles

How far can particles go? Typical particle ∼√

n (Gaussian).A specific particle gets to xt with probability p(t , xt) =

ct1/2 e−(xt )

2/2t .There are et particles, so maximal should be such that etp(t , xt) ∼ 1Gives xt ∼

√2t − 1

2√

2log t .

This is correct answer for et independent particles.But Bramson’s result is that location of maximal particle is only

xt ∼√

2t − 3

2√

2log t

Why?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 7 / 28

Back of envelope - role of atypical particles - ct’d

Reason: Genealogy of particles - tree structure.At time s < t , there are only es particles, so maximal one cannot befarther than

√2s.

But a Brownian motion reaches xt at time t and stays below the linesxt/t for s ≤ t with probability

p(t , xt) ∼c

t3/2e−(xt )

2/2t .

Compare with

p(t , xt) =c

t1/2e−(xt )

2/2t

et p(t , xt) ∼ 1 gives Bramson’s result.Justification Z =

∑

i∈Nt1i th particle is good; EZ 2/(EZ )2 ∼ 1 implies, by

Cauchy-Schwarz, that P(Z > 0) ∼ 1.Second moment method, probabilistic method.Convergence proof uses F-KPP!!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 8 / 28

Back of envelope - role of atypical particles - ct’d

Reason: Genealogy of particles - tree structure.At time s < t , there are only es particles, so maximal one cannot befarther than

√2s.

But a Brownian motion reaches xt at time t and stays below the linesxt/t for s ≤ t with probability

p(t , xt) ∼c

t3/2e−(xt )

2/2t .

Compare with

p(t , xt) =c

t1/2e−(xt )

2/2t

et p(t , xt) ∼ 1 gives Bramson’s result.Justification Z =

∑

i∈Nt1i th particle is good; EZ 2/(EZ )2 ∼ 1 implies, by

Cauchy-Schwarz, that P(Z > 0) ∼ 1.Second moment method, probabilistic method.Convergence proof uses F-KPP!!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 8 / 28

Back of envelope - role of atypical particles - ct’d

Reason: Genealogy of particles - tree structure.At time s < t , there are only es particles, so maximal one cannot befarther than

√2s.

But a Brownian motion reaches xt at time t and stays below the linesxt/t for s ≤ t with probability

p(t , xt) ∼c

t3/2e−(xt )

2/2t .

Compare with

p(t , xt) =c

t1/2e−(xt )

2/2t

et p(t , xt) ∼ 1 gives Bramson’s result.Justification Z =

∑

i∈Nt1i th particle is good; EZ 2/(EZ )2 ∼ 1 implies, by

Cauchy-Schwarz, that P(Z > 0) ∼ 1.Second moment method, probabilistic method.Convergence proof uses F-KPP!!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 8 / 28

Back of envelope - role of atypical particles - ct’d

Reason: Genealogy of particles - tree structure.At time s < t , there are only es particles, so maximal one cannot befarther than

√2s.

But a Brownian motion reaches xt at time t and stays below the linesxt/t for s ≤ t with probability

p(t , xt) ∼c

t3/2e−(xt )

2/2t .

Compare with

p(t , xt) =c

t1/2e−(xt )

2/2t

et p(t , xt) ∼ 1 gives Bramson’s result.Justification Z =

∑

i∈Nt1i th particle is good; EZ 2/(EZ )2 ∼ 1 implies, by

Cauchy-Schwarz, that P(Z > 0) ∼ 1.Second moment method, probabilistic method.Convergence proof uses F-KPP!!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 8 / 28

Back of envelope - role of atypical particles - ct’d

Reason: Genealogy of particles - tree structure.At time s < t , there are only es particles, so maximal one cannot befarther than

√2s.

But a Brownian motion reaches xt at time t and stays below the linesxt/t for s ≤ t with probability

p(t , xt) ∼c

t3/2e−(xt )

2/2t .

Compare with

p(t , xt) =c

t1/2e−(xt )

2/2t

et p(t , xt) ∼ 1 gives Bramson’s result.Justification Z =

∑

i∈Nt1i th particle is good; EZ 2/(EZ )2 ∼ 1 implies, by

Cauchy-Schwarz, that P(Z > 0) ∼ 1.Second moment method, probabilistic method.Convergence proof uses F-KPP!!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 8 / 28

Back of envelope - role of atypical particles - ct’d

Reason: Genealogy of particles - tree structure.At time s < t , there are only es particles, so maximal one cannot befarther than

√2s.

But a Brownian motion reaches xt at time t and stays below the linesxt/t for s ≤ t with probability

p(t , xt) ∼c

t3/2e−(xt )

2/2t .

Compare with

p(t , xt) =c

t1/2e−(xt )

2/2t

et p(t , xt) ∼ 1 gives Bramson’s result.Justification Z =

∑

i∈Nt1i th particle is good; EZ 2/(EZ )2 ∼ 1 implies, by

Cauchy-Schwarz, that P(Z > 0) ∼ 1.Second moment method, probabilistic method.Convergence proof uses F-KPP!!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 8 / 28

Back of envelope - role of atypical particles - ct’d

Reason: Genealogy of particles - tree structure.At time s < t , there are only es particles, so maximal one cannot befarther than

√2s.

But a Brownian motion reaches xt at time t and stays below the linesxt/t for s ≤ t with probability

p(t , xt) ∼c

t3/2e−(xt )

2/2t .

Compare with

p(t , xt) =c

t1/2e−(xt )

2/2t

et p(t , xt) ∼ 1 gives Bramson’s result.Justification Z =

∑

i∈Nt1i th particle is good; EZ 2/(EZ )2 ∼ 1 implies, by

Cauchy-Schwarz, that P(Z > 0) ∼ 1.Second moment method, probabilistic method.Convergence proof uses F-KPP!!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 8 / 28

Back of envelope - role of atypical particles - ct’d

Reason: Genealogy of particles - tree structure.At time s < t , there are only es particles, so maximal one cannot befarther than

√2s.

But a Brownian motion reaches xt at time t and stays below the linesxt/t for s ≤ t with probability

p(t , xt) ∼c

t3/2e−(xt )

2/2t .

Compare with

p(t , xt) =c

t1/2e−(xt )

2/2t

et p(t , xt) ∼ 1 gives Bramson’s result.Justification Z =

∑

i∈Nt1i th particle is good; EZ 2/(EZ )2 ∼ 1 implies, by

Cauchy-Schwarz, that P(Z > 0) ∼ 1.Second moment method, probabilistic method.Convergence proof uses F-KPP!!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 8 / 28

More recently..

Consider the point process Lt =∑

v∈Dtδζi (t)−m(t)

Theorem (ABBS;ABK ’13)[Aidekon,Berestycki,Brunet,Shi; Arguin,Bovier Kistler ’13]

Lt → SDPPP

SDPPP - (randomly) shifted decorated Poisson Point ProcessIntensity of PPP: e−cx

Decoration: neighborhood of leader.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 9 / 28

More recently..

Consider the point process Lt =∑

v∈Dtδζi (t)−m(t)

Theorem (ABBS;ABK ’13)[Aidekon,Berestycki,Brunet,Shi; Arguin,Bovier Kistler ’13]

Lt → SDPPP

SDPPP - (randomly) shifted decorated Poisson Point ProcessIntensity of PPP: e−cx

Decoration: neighborhood of leader.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 9 / 28

More recently..

Consider the point process Lt =∑

v∈Dtδζi (t)−m(t)

Theorem (ABBS;ABK ’13)[Aidekon,Berestycki,Brunet,Shi; Arguin,Bovier Kistler ’13]

Lt → SDPPP

SDPPP - (randomly) shifted decorated Poisson Point ProcessIntensity of PPP: e−cx

Decoration: neighborhood of leader.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 9 / 28

More recently..

Consider the point process Lt =∑

v∈Dtδζi (t)−m(t)

Theorem (ABBS;ABK ’13)[Aidekon,Berestycki,Brunet,Shi; Arguin,Bovier Kistler ’13]

Lt → SDPPP

SDPPP - (randomly) shifted decorated Poisson Point ProcessIntensity of PPP: e−cx

Decoration: neighborhood of leader.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 9 / 28

Remarks on SDPPP

A (randomly shifted) PPP is characterized by:Add i.i.d. random variable, then shift (deterministically) and get sameprocess. (Liggett; Aizenman-Ruzmaikina; Arguin)A decorated PPP is characterized by the following property:Exponentially-c-stable (P. Maillard; Davydov-Molchanov-Zuyev):

ea + eb = c ⇒ ξd= θaξ1 + θbξ2.

Identifying the random shift reduces to proving DPPP.For SDPPP, a (less clean) characterization exists: say ξ exhibitsfreezing if, for f ≥ 0 continuous, bounded support,

E(e−〈ξ,θy f 〉) = g(y − cf ) .

Motivated by freezing of free energy (Derrida-Spohn,Fyodorov-LeDoussal,..).

Theorem (Subag-Z. ’14)Under technical conditions, freezing ⇔ SDPPP.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 10 / 28

Remarks on SDPPP

A (randomly shifted) PPP is characterized by:Add i.i.d. random variable, then shift (deterministically) and get sameprocess. (Liggett; Aizenman-Ruzmaikina; Arguin)A decorated PPP is characterized by the following property:Exponentially-c-stable (P. Maillard; Davydov-Molchanov-Zuyev):

ea + eb = c ⇒ ξd= θaξ1 + θbξ2.

Identifying the random shift reduces to proving DPPP.For SDPPP, a (less clean) characterization exists: say ξ exhibitsfreezing if, for f ≥ 0 continuous, bounded support,

E(e−〈ξ,θy f 〉) = g(y − cf ) .

Motivated by freezing of free energy (Derrida-Spohn,Fyodorov-LeDoussal,..).

Theorem (Subag-Z. ’14)Under technical conditions, freezing ⇔ SDPPP.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 10 / 28

Remarks on SDPPP

A (randomly shifted) PPP is characterized by:Add i.i.d. random variable, then shift (deterministically) and get sameprocess. (Liggett; Aizenman-Ruzmaikina; Arguin)A decorated PPP is characterized by the following property:Exponentially-c-stable (P. Maillard; Davydov-Molchanov-Zuyev):

ea + eb = c ⇒ ξd= θaξ1 + θbξ2.

Identifying the random shift reduces to proving DPPP.For SDPPP, a (less clean) characterization exists: say ξ exhibitsfreezing if, for f ≥ 0 continuous, bounded support,

E(e−〈ξ,θy f 〉) = g(y − cf ) .

Motivated by freezing of free energy (Derrida-Spohn,Fyodorov-LeDoussal,..).

Theorem (Subag-Z. ’14)Under technical conditions, freezing ⇔ SDPPP.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 10 / 28

Remarks on SDPPP

A (randomly shifted) PPP is characterized by:Add i.i.d. random variable, then shift (deterministically) and get sameprocess. (Liggett; Aizenman-Ruzmaikina; Arguin)A decorated PPP is characterized by the following property:Exponentially-c-stable (P. Maillard; Davydov-Molchanov-Zuyev):

ea + eb = c ⇒ ξd= θaξ1 + θbξ2.

Identifying the random shift reduces to proving DPPP.For SDPPP, a (less clean) characterization exists: say ξ exhibitsfreezing if, for f ≥ 0 continuous, bounded support,

E(e−〈ξ,θy f 〉) = g(y − cf ) .

Motivated by freezing of free energy (Derrida-Spohn,Fyodorov-LeDoussal,..).

Theorem (Subag-Z. ’14)Under technical conditions, freezing ⇔ SDPPP.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 10 / 28

Remarks on SDPPP

A (randomly shifted) PPP is characterized by:Add i.i.d. random variable, then shift (deterministically) and get sameprocess. (Liggett; Aizenman-Ruzmaikina; Arguin)A decorated PPP is characterized by the following property:Exponentially-c-stable (P. Maillard; Davydov-Molchanov-Zuyev):

ea + eb = c ⇒ ξd= θaξ1 + θbξ2.

Identifying the random shift reduces to proving DPPP.For SDPPP, a (less clean) characterization exists: say ξ exhibitsfreezing if, for f ≥ 0 continuous, bounded support,

E(e−〈ξ,θy f 〉) = g(y − cf ) .

Motivated by freezing of free energy (Derrida-Spohn,Fyodorov-LeDoussal,..).

Theorem (Subag-Z. ’14)Under technical conditions, freezing ⇔ SDPPP.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 10 / 28

Remarks on SDPPP

A (randomly shifted) PPP is characterized by:Add i.i.d. random variable, then shift (deterministically) and get sameprocess. (Liggett; Aizenman-Ruzmaikina; Arguin)A decorated PPP is characterized by the following property:Exponentially-c-stable (P. Maillard; Davydov-Molchanov-Zuyev):

ea + eb = c ⇒ ξd= θaξ1 + θbξ2.

Identifying the random shift reduces to proving DPPP.For SDPPP, a (less clean) characterization exists: say ξ exhibitsfreezing if, for f ≥ 0 continuous, bounded support,

E(e−〈ξ,θy f 〉) = g(y − cf ) .

Motivated by freezing of free energy (Derrida-Spohn,Fyodorov-LeDoussal,..).

Theorem (Subag-Z. ’14)Under technical conditions, freezing ⇔ SDPPP.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 10 / 28

Related model: Branching Random Walks

Discrete Model: At time 0, particle at 0 ∈ R.Particle makes a step, random of law Q.After the step, particle splits in 2. Then repeat.(Probabilistic) analysis similar to BBM.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 11 / 28

Related model: Branching Random Walks

Discrete Model: At time 0, particle at 0 ∈ R.Particle makes a step, random of law Q.After the step, particle splits in 2. Then repeat.(Probabilistic) analysis similar to BBM.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 11 / 28

Related model: Branching Random Walks

Discrete Model: At time 0, particle at 0 ∈ R.Particle makes a step, random of law Q.After the step, particle splits in 2. Then repeat.(Probabilistic) analysis similar to BBM.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 11 / 28

Related model: Branching Random Walks

Discrete Model: At time 0, particle at 0 ∈ R.Particle makes a step, random of law Q.After the step, particle splits in 2. Then repeat.(Probabilistic) analysis similar to BBM.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 11 / 28

Related model: Branching Random Walks

Discrete Model: At time 0, particle at 0 ∈ R.Particle makes a step, random of law Q.After the step, particle splits in 2. Then repeat.(Probabilistic) analysis similar to BBM.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 11 / 28

Related model: Branching Random Walks

Discrete Model: At time 0, particle at 0 ∈ R.Particle makes a step, random of law Q.After the step, particle splits in 2. Then repeat.(Probabilistic) analysis similar to BBM.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 11 / 28

Branching Random Walks

Similar behavior when particles live in Euclidean space.c©Pascal Maillard

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 12 / 28

Branching Random Walks

Similar behavior when particles live in Euclidean space.c©Pascal Maillard

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 12 / 28

Branching random walks - remarks

Estimates refined to allow for non-Gaussian distributions (tightness,EMn) - Adarrio-Berry and Reed ’09Recursion-based proofs: Bramson ’82, Lui ’82, Bachman ’00,Bramson-Z ’09Limit law for general BRW: Aidekon ’11, Bramson-Ding-Z. ’13 - tailestimates by (nonstandard) application of second moment method, noF-KPP.Limit is the same as for maximum of independent particles (Gumbel),except for a random shift that is due to a neighborhood of the root ofthe tree.Extremal process for BRW: Madaule ’13Remark Interesting effects when allow time-inhomogenuity in jumpdistribution: logarithmic correction may change, phase transitions, T 1/3

corrections. Fang-Z. ’12, Maillard-Z. ’13 (F-KPP setup).

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 13 / 28

Branching random walks - remarks

Estimates refined to allow for non-Gaussian distributions (tightness,EMn) - Adarrio-Berry and Reed ’09Recursion-based proofs: Bramson ’82, Lui ’82, Bachman ’00,Bramson-Z ’09Limit law for general BRW: Aidekon ’11, Bramson-Ding-Z. ’13 - tailestimates by (nonstandard) application of second moment method, noF-KPP.Limit is the same as for maximum of independent particles (Gumbel),except for a random shift that is due to a neighborhood of the root ofthe tree.Extremal process for BRW: Madaule ’13Remark Interesting effects when allow time-inhomogenuity in jumpdistribution: logarithmic correction may change, phase transitions, T 1/3

corrections. Fang-Z. ’12, Maillard-Z. ’13 (F-KPP setup).

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 13 / 28

Branching random walks - remarks

Estimates refined to allow for non-Gaussian distributions (tightness,EMn) - Adarrio-Berry and Reed ’09Recursion-based proofs: Bramson ’82, Lui ’82, Bachman ’00,Bramson-Z ’09Limit law for general BRW: Aidekon ’11, Bramson-Ding-Z. ’13 - tailestimates by (nonstandard) application of second moment method, noF-KPP.Limit is the same as for maximum of independent particles (Gumbel),except for a random shift that is due to a neighborhood of the root ofthe tree.Extremal process for BRW: Madaule ’13Remark Interesting effects when allow time-inhomogenuity in jumpdistribution: logarithmic correction may change, phase transitions, T 1/3

corrections. Fang-Z. ’12, Maillard-Z. ’13 (F-KPP setup).

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 13 / 28

Branching random walks - remarks

Estimates refined to allow for non-Gaussian distributions (tightness,EMn) - Adarrio-Berry and Reed ’09Recursion-based proofs: Bramson ’82, Lui ’82, Bachman ’00,Bramson-Z ’09Limit law for general BRW: Aidekon ’11, Bramson-Ding-Z. ’13 - tailestimates by (nonstandard) application of second moment method, noF-KPP.Limit is the same as for maximum of independent particles (Gumbel),except for a random shift that is due to a neighborhood of the root ofthe tree.Extremal process for BRW: Madaule ’13Remark Interesting effects when allow time-inhomogenuity in jumpdistribution: logarithmic correction may change, phase transitions, T 1/3

corrections. Fang-Z. ’12, Maillard-Z. ’13 (F-KPP setup).

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 13 / 28

Branching random walks - remarks

Estimates refined to allow for non-Gaussian distributions (tightness,EMn) - Adarrio-Berry and Reed ’09Recursion-based proofs: Bramson ’82, Lui ’82, Bachman ’00,Bramson-Z ’09Limit law for general BRW: Aidekon ’11, Bramson-Ding-Z. ’13 - tailestimates by (nonstandard) application of second moment method, noF-KPP.Limit is the same as for maximum of independent particles (Gumbel),except for a random shift that is due to a neighborhood of the root ofthe tree.Extremal process for BRW: Madaule ’13Remark Interesting effects when allow time-inhomogenuity in jumpdistribution: logarithmic correction may change, phase transitions, T 1/3

corrections. Fang-Z. ’12, Maillard-Z. ’13 (F-KPP setup).

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 13 / 28

Brownian motion - multiple points

P. Levy ’40: For d = 2, almost every path of BM has special points that arevisited at least twice.S. Kakutani ’44: Not true for d ≥ 5.

Theorem (Dvoretzky-Erdos-Kakutani ’50–’58, DEK-Taylor ’57)For d = 2 there are points of arbitrarily large multiplicity: for any k,there are points of multiplicity exactly k (Adelman-Dvoretzky ’85)

For d = 3 there are double points but not triple points.

For d ≥ 4 there are no double points.

Quantitatively: LeGall ’87 Hausdorff dimension of multiple points is 2; ∃ pointsof infinite multiplicity.Other quantitative results (exact Hausdorff measure, etc.:) Bass, Burdzy,Khoshnevisan, Lawler, LeGall, Werner early 90’s.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 14 / 28

Brownian motion - multiple points

P. Levy ’40: For d = 2, almost every path of BM has special points that arevisited at least twice.S. Kakutani ’44: Not true for d ≥ 5.

Theorem (Dvoretzky-Erdos-Kakutani ’50–’58, DEK-Taylor ’57)For d = 2 there are points of arbitrarily large multiplicity: for any k,there are points of multiplicity exactly k (Adelman-Dvoretzky ’85)

For d = 3 there are double points but not triple points.

For d ≥ 4 there are no double points.

Quantitatively: LeGall ’87 Hausdorff dimension of multiple points is 2; ∃ pointsof infinite multiplicity.Other quantitative results (exact Hausdorff measure, etc.:) Bass, Burdzy,Khoshnevisan, Lawler, LeGall, Werner early 90’s.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 14 / 28

Brownian motion - multiple points

P. Levy ’40: For d = 2, almost every path of BM has special points that arevisited at least twice.S. Kakutani ’44: Not true for d ≥ 5.

Theorem (Dvoretzky-Erdos-Kakutani ’50–’58, DEK-Taylor ’57)For d = 2 there are points of arbitrarily large multiplicity: for any k,there are points of multiplicity exactly k (Adelman-Dvoretzky ’85)

For d = 3 there are double points but not triple points.

For d ≥ 4 there are no double points.

Quantitatively: LeGall ’87 Hausdorff dimension of multiple points is 2; ∃ pointsof infinite multiplicity.Other quantitative results (exact Hausdorff measure, etc.:) Bass, Burdzy,Khoshnevisan, Lawler, LeGall, Werner early 90’s.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 14 / 28

Brownian motion - multiple points

P. Levy ’40: For d = 2, almost every path of BM has special points that arevisited at least twice.S. Kakutani ’44: Not true for d ≥ 5.

Theorem (Dvoretzky-Erdos-Kakutani ’50–’58, DEK-Taylor ’57)For d = 2 there are points of arbitrarily large multiplicity: for any k,there are points of multiplicity exactly k (Adelman-Dvoretzky ’85)

For d = 3 there are double points but not triple points.

For d ≥ 4 there are no double points.

Quantitatively: LeGall ’87 Hausdorff dimension of multiple points is 2; ∃ pointsof infinite multiplicity.Other quantitative results (exact Hausdorff measure, etc.:) Bass, Burdzy,Khoshnevisan, Lawler, LeGall, Werner early 90’s.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 14 / 28

Brownian motion - multiple points

P. Levy ’40: For d = 2, almost every path of BM has special points that arevisited at least twice.S. Kakutani ’44: Not true for d ≥ 5.

Theorem (Dvoretzky-Erdos-Kakutani ’50–’58, DEK-Taylor ’57)For d = 2 there are points of arbitrarily large multiplicity: for any k,there are points of multiplicity exactly k (Adelman-Dvoretzky ’85)

For d = 3 there are double points but not triple points.

For d ≥ 4 there are no double points.

Quantitatively: LeGall ’87 Hausdorff dimension of multiple points is 2; ∃ pointsof infinite multiplicity.Other quantitative results (exact Hausdorff measure, etc.:) Bass, Burdzy,Khoshnevisan, Lawler, LeGall, Werner early 90’s.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 14 / 28

Brownian motion - multiple points

P. Levy ’40: For d = 2, almost every path of BM has special points that arevisited at least twice.S. Kakutani ’44: Not true for d ≥ 5.

Theorem (Dvoretzky-Erdos-Kakutani ’50–’58, DEK-Taylor ’57)For d = 2 there are points of arbitrarily large multiplicity: for any k,there are points of multiplicity exactly k (Adelman-Dvoretzky ’85)

For d = 3 there are double points but not triple points.

For d ≥ 4 there are no double points.

Quantitatively: LeGall ’87 Hausdorff dimension of multiple points is 2; ∃ pointsof infinite multiplicity.Other quantitative results (exact Hausdorff measure, etc.:) Bass, Burdzy,Khoshnevisan, Lawler, LeGall, Werner early 90’s.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 14 / 28

Brownian motion - multiple points

P. Levy ’40: For d = 2, almost every path of BM has special points that arevisited at least twice.S. Kakutani ’44: Not true for d ≥ 5.

Theorem (Dvoretzky-Erdos-Kakutani ’50–’58, DEK-Taylor ’57)For d = 2 there are points of arbitrarily large multiplicity: for any k,there are points of multiplicity exactly k (Adelman-Dvoretzky ’85)

For d = 3 there are double points but not triple points.

For d ≥ 4 there are no double points.

Quantitatively: LeGall ’87 Hausdorff dimension of multiple points is 2; ∃ pointsof infinite multiplicity.Other quantitative results (exact Hausdorff measure, etc.:) Bass, Burdzy,Khoshnevisan, Lawler, LeGall, Werner early 90’s.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 14 / 28

Planar Brownian motion - thick points

If x is on a planar Brownian path, T (x , ǫ) is total time spent in ball of radius ǫ

around x before time 1; for “typical” x , T (x , ǫ) ∼ ǫ2 log(1/ǫ).A point x is thick if

lim supǫ→0

T (x , ǫ)ǫ2(log 1/ǫ)2 > 0 .

Discrete analogue: for simple random walk on Z2 run up to time n,

occupation time Tn(x) of a point x on path is of order log n. A point x isthick if Tn(x) is of order (log n)2.

Theorem (Erdos-Taylor ’60)With T ∗

n = maxx Tn(x),

14π

≤ lim infT ∗

n

(log n)2 ≤ lim supT ∗

n

(log n)2 ≤ 1π

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 15 / 28

Planar Brownian motion - thick points

If x is on a planar Brownian path, T (x , ǫ) is total time spent in ball of radius ǫ

around x before time 1; for “typical” x , T (x , ǫ) ∼ ǫ2 log(1/ǫ).A point x is thick if

lim supǫ→0

T (x , ǫ)ǫ2(log 1/ǫ)2 > 0 .

Discrete analogue: for simple random walk on Z2 run up to time n,

occupation time Tn(x) of a point x on path is of order log n. A point x isthick if Tn(x) is of order (log n)2.

Theorem (Erdos-Taylor ’60)With T ∗

n = maxx Tn(x),

14π

≤ lim infT ∗

n

(log n)2 ≤ lim supT ∗

n

(log n)2 ≤ 1π

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 15 / 28

Planar Brownian motion - thick points

If x is on a planar Brownian path, T (x , ǫ) is total time spent in ball of radius ǫ

around x before time 1; for “typical” x , T (x , ǫ) ∼ ǫ2 log(1/ǫ).A point x is thick if

lim supǫ→0

T (x , ǫ)ǫ2(log 1/ǫ)2 > 0 .

Discrete analogue: for simple random walk on Z2 run up to time n,

occupation time Tn(x) of a point x on path is of order log n. A point x isthick if Tn(x) is of order (log n)2.

Theorem (Erdos-Taylor ’60)With T ∗

n = maxx Tn(x),

14π

≤ lim infT ∗

n

(log n)2 ≤ lim supT ∗

n

(log n)2 ≤ 1π

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 15 / 28

Planar Brownian motion - thick points

If x is on a planar Brownian path, T (x , ǫ) is total time spent in ball of radius ǫ

around x before time 1; for “typical” x , T (x , ǫ) ∼ ǫ2 log(1/ǫ).A point x is thick if

lim supǫ→0

T (x , ǫ)ǫ2(log 1/ǫ)2 > 0 .

Discrete analogue: for simple random walk on Z2 run up to time n,

occupation time Tn(x) of a point x on path is of order log n. A point x isthick if Tn(x) is of order (log n)2.

Theorem (Erdos-Taylor ’60)With T ∗

n = maxx Tn(x),

14π

≤ lim infT ∗

n

(log n)2 ≤ lim supT ∗

n

(log n)2 ≤ 1π

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 15 / 28

Thick points (ct’d)

(ET)1

4π≤ lim inf

T ∗

n

(log n)2 ≤ lim supT ∗

n

(log n)2 ≤ 1π

Erdos-Taylor conjectured that upper bound is correct, i.e thickestpoints achieve 1/π.

Theorem (Dembo-Peres-Rosen-Zeitouni ’02)Erdos-Taylor conjecture is true:

limT ∗n /(log n)2 = 1/π

Similarly, for Brownian motion,

limǫ

supx

T (x , ǫ)ǫ2(log ǫ)2 = 2 .

BM part resolves a conjecture of Perkins and Taylor; DPRZ get quantitativeresults: Hausdorff dimensions of thick points at level a is 2 − a.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 16 / 28

Thick points (ct’d)

(ET)1

4π≤ lim inf

T ∗

n

(log n)2 ≤ lim supT ∗

n

(log n)2 ≤ 1π

Erdos-Taylor conjectured that upper bound is correct, i.e thickestpoints achieve 1/π.

Theorem (Dembo-Peres-Rosen-Zeitouni ’02)Erdos-Taylor conjecture is true:

limT ∗n /(log n)2 = 1/π

Similarly, for Brownian motion,

limǫ

supx

T (x , ǫ)ǫ2(log ǫ)2 = 2 .

BM part resolves a conjecture of Perkins and Taylor; DPRZ get quantitativeresults: Hausdorff dimensions of thick points at level a is 2 − a.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 16 / 28

Thick points (ct’d)

(ET)1

4π≤ lim inf

T ∗

n

(log n)2 ≤ lim supT ∗

n

(log n)2 ≤ 1π

Erdos-Taylor conjectured that upper bound is correct, i.e thickestpoints achieve 1/π.

Theorem (Dembo-Peres-Rosen-Zeitouni ’02)Erdos-Taylor conjecture is true:

limT ∗n /(log n)2 = 1/π

Similarly, for Brownian motion,

limǫ

supx

T (x , ǫ)ǫ2(log ǫ)2 = 2 .

BM part resolves a conjecture of Perkins and Taylor; DPRZ get quantitativeresults: Hausdorff dimensions of thick points at level a is 2 − a.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 16 / 28

Thick points (ct’d)

(ET)1

4π≤ lim inf

T ∗

n

(log n)2 ≤ lim supT ∗

n

(log n)2 ≤ 1π

Erdos-Taylor conjectured that upper bound is correct, i.e thickestpoints achieve 1/π.

Theorem (Dembo-Peres-Rosen-Zeitouni ’02)Erdos-Taylor conjecture is true:

limT ∗n /(log n)2 = 1/π

Similarly, for Brownian motion,

limǫ

supx

T (x , ǫ)ǫ2(log ǫ)2 = 2 .

BM part resolves a conjecture of Perkins and Taylor; DPRZ get quantitativeresults: Hausdorff dimensions of thick points at level a is 2 − a.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 16 / 28

Thick points (ct’d)

(ET)1

4π≤ lim inf

T ∗

n

(log n)2 ≤ lim supT ∗

n

(log n)2 ≤ 1π

Erdos-Taylor conjectured that upper bound is correct, i.e thickestpoints achieve 1/π.

Theorem (Dembo-Peres-Rosen-Zeitouni ’02)Erdos-Taylor conjecture is true:

limT ∗n /(log n)2 = 1/π

Similarly, for Brownian motion,

limǫ

supx

T (x , ǫ)ǫ2(log ǫ)2 = 2 .

BM part resolves a conjecture of Perkins and Taylor; DPRZ get quantitativeresults: Hausdorff dimensions of thick points at level a is 2 − a.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 16 / 28

Thick points - enters tree structure

DPRZ proof (mainly of the lower bound) introduces an approximatetree structure by looking at excursions between a (sparsely chosen)collection of points in the plane.Toy problem: thick points for random walk on binary tree. With properadjustement of parameters, answer (and proof!) is same as for planarBM/SRW.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 17 / 28

Thick points - enters tree structure

DPRZ proof (mainly of the lower bound) introduces an approximatetree structure by looking at excursions between a (sparsely chosen)collection of points in the plane.Toy problem: thick points for random walk on binary tree. With properadjustement of parameters, answer (and proof!) is same as for planarBM/SRW.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 17 / 28

Thick points - enters tree structure

DPRZ proof (mainly of the lower bound) introduces an approximatetree structure by looking at excursions between a (sparsely chosen)collection of points in the plane.Toy problem: thick points for random walk on binary tree. With properadjustement of parameters, answer (and proof!) is same as for planarBM/SRW.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 17 / 28

Thick points - enters tree structure II

Idea of proofUpper bound: as for SRW.Lower bound: difficulty is in correlation structure. Consider

ε

ε = ek

k

-k

A point x is k successful if # of excursions of BM between εk and εk+1

disc is ak2.Remark: if ε = εk+1, for x succesful the time spent at inner ball isak2 · ǫ2 ∼ aǫ2(log 1

ǫ )2 with overwhelming probability.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 18 / 28

Thick points - enters tree structure II

Idea of proofUpper bound: as for SRW.Lower bound: difficulty is in correlation structure. Consider

ε

ε = ek

k

-k

A point x is k successful if # of excursions of BM between εk and εk+1

disc is ak2.Remark: if ε = εk+1, for x succesful the time spent at inner ball isak2 · ǫ2 ∼ aǫ2(log 1

ǫ )2 with overwhelming probability.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 18 / 28

Thick points - enters tree structure II

Idea of proofUpper bound: as for SRW.Lower bound: difficulty is in correlation structure. Consider

ε

ε = ek

k

-k

A point x is k successful if # of excursions of BM between εk and εk+1

disc is ak2.Remark: if ε = εk+1, for x succesful the time spent at inner ball isak2 · ǫ2 ∼ aǫ2(log 1

ǫ )2 with overwhelming probability.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 18 / 28

Thick points - enters tree structure III

Now, for points x 6= y , we have

⇒ tree structure

x y

0

ak excursions2

and same technique as for tree works.Technical issues: Many!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 19 / 28

Thick points - enters tree structure III

Now, for points x 6= y , we have

⇒ tree structure

x y

0

ak excursions2

and same technique as for tree works.Technical issues: Many!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 19 / 28

Thick points - enters tree structure III

Now, for points x 6= y , we have

⇒ tree structure

x y

0

ak excursions2

and same technique as for tree works.Technical issues: Many!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 19 / 28

Cover times

Similar issues: Cover time problemConsider SRW on 2D torus of side n. Let Cn denote time to cover torus.

Theorem (DPRZ ’02)In probability,

Cn

n2(log n)2 → 4π

Resolves questions of Aldous, Kesten, Lawler and Revesz.Again, answer (and proof) are closely related to cover problem forSRW on binary tree, which was solved by Aldous using recursions.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 20 / 28

Cover times

Similar issues: Cover time problemConsider SRW on 2D torus of side n. Let Cn denote time to cover torus.

Theorem (DPRZ ’02)In probability,

Cn

n2(log n)2 → 4π

Resolves questions of Aldous, Kesten, Lawler and Revesz.Again, answer (and proof) are closely related to cover problem forSRW on binary tree, which was solved by Aldous using recursions.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 20 / 28

Cover times

Similar issues: Cover time problemConsider SRW on 2D torus of side n. Let Cn denote time to cover torus.

Theorem (DPRZ ’02)In probability,

Cn

n2(log n)2 → 4π

Resolves questions of Aldous, Kesten, Lawler and Revesz.Again, answer (and proof) are closely related to cover problem forSRW on binary tree, which was solved by Aldous using recursions.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 20 / 28

Cover times

Similar issues: Cover time problemConsider SRW on 2D torus of side n. Let Cn denote time to cover torus.

Theorem (DPRZ ’02)In probability,

Cn

n2(log n)2 → 4π

Resolves questions of Aldous, Kesten, Lawler and Revesz.Again, answer (and proof) are closely related to cover problem forSRW on binary tree, which was solved by Aldous using recursions.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 20 / 28

Gaussian Free Field - Definitions

10

20

30

40

50

10

20

30

40

50

-4

-2

0

2

10

20

30

40

50

VN = ([0,N − 1] ∩ Z)2; {wm}m≥0 - simple random walk killed atτ = min{m : wm ∈ ∂VN}.

GN(x , y) = Ex(τ

∑

m=0

1wm=y )

GFF is the zero-mean Gaussian field {XNz }z∈VN

, covariance GN .Alternatively, density is

1ZN

e−∑x∼y (Xx−Xy )2/2 .

GFF on tree = BRW.Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 21 / 28

Gaussian Free Field - Definitions

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50

-4

-2

0

2

10

20

30

40

50

VN = ([0,N − 1] ∩ Z)2; {wm}m≥0 - simple random walk killed atτ = min{m : wm ∈ ∂VN}.

GN(x , y) = Ex(τ

∑

m=0

1wm=y )

GFF is the zero-mean Gaussian field {XNz }z∈VN

, covariance GN .Alternatively, density is

1ZN

e−∑x∼y (Xx−Xy )2/2 .

GFF on tree = BRW.Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 21 / 28

Gaussian Free Field - Definitions

10

20

30

40

50

10

20

30

40

50

-4

-2

0

2

10

20

30

40

50

VN = ([0,N − 1] ∩ Z)2; {wm}m≥0 - simple random walk killed atτ = min{m : wm ∈ ∂VN}.

GN(x , y) = Ex(τ

∑

m=0

1wm=y )

GFF is the zero-mean Gaussian field {XNz }z∈VN

, covariance GN .Alternatively, density is

1ZN

e−∑x∼y (Xx−Xy )2/2 .

GFF on tree = BRW.Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 21 / 28

Gaussian Free Field - Definitions

10

20

30

40

50

10

20

30

40

50

-4

-2

0

2

10

20

30

40

50

VN = ([0,N − 1] ∩ Z)2; {wm}m≥0 - simple random walk killed atτ = min{m : wm ∈ ∂VN}.

GN(x , y) = Ex(τ

∑

m=0

1wm=y )

GFF is the zero-mean Gaussian field {XNz }z∈VN

, covariance GN .Alternatively, density is

1ZN

e−∑x∼y (Xx−Xy )2/2 .

GFF on tree = BRW.Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 21 / 28

Gaussian Free Field - Definitions

10

20

30

40

50

10

20

30

40

50

-4

-2

0

2

10

20

30

40

50

VN = ([0,N − 1] ∩ Z)2; {wm}m≥0 - simple random walk killed atτ = min{m : wm ∈ ∂VN}.

GN(x , y) = Ex(τ

∑

m=0

1wm=y )

GFF is the zero-mean Gaussian field {XNz }z∈VN

, covariance GN .Alternatively, density is

1ZN

e−∑x∼y (Xx−Xy )2/2 .

GFF on tree = BRW.Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 21 / 28

Gaussian Free Field - Definitions

10

20

30

40

50

10

20

30

40

50

-4

-2

0

2

10

20

30

40

50

VN = ([0,N − 1] ∩ Z)2; {wm}m≥0 - simple random walk killed atτ = min{m : wm ∈ ∂VN}.

GN(x , y) = Ex(τ

∑

m=0

1wm=y )

GFF is the zero-mean Gaussian field {XNz }z∈VN

, covariance GN .Alternatively, density is

1ZN

e−∑x∼y (Xx−Xy )2/2 .

GFF on tree = BRW.Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 21 / 28

Law of large numbers

MN = maxz∈VNXN

z

Theorem (Bolthausen-Deuschel-Giacomin ’01)

MN

log N→ 2

√

2/π .

Value of limit is the same as if correlations were ignored (i.e., maximumof N2 Gaussian variables of variance GN(x , x) ∼ (2/π) log N).Upper bound by first moment computation (as for BRW).Lower bound by introducing branching structure via averaging field ondifferent scales - roughly embedding a BRW in GFF.Truncated second moment method can also be used for the embeddedBRW Daviaud ’06 and for the analogous continuous GFFHu-Miller-Peres ’10Fluctuations?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 22 / 28

Law of large numbers

MN = maxz∈VNXN

z

Theorem (Bolthausen-Deuschel-Giacomin ’01)

MN

log N→ 2

√

2/π .

Value of limit is the same as if correlations were ignored (i.e., maximumof N2 Gaussian variables of variance GN(x , x) ∼ (2/π) log N).Upper bound by first moment computation (as for BRW).Lower bound by introducing branching structure via averaging field ondifferent scales - roughly embedding a BRW in GFF.Truncated second moment method can also be used for the embeddedBRW Daviaud ’06 and for the analogous continuous GFFHu-Miller-Peres ’10Fluctuations?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 22 / 28

Law of large numbers

MN = maxz∈VNXN

z

Theorem (Bolthausen-Deuschel-Giacomin ’01)

MN

log N→ 2

√

2/π .

Value of limit is the same as if correlations were ignored (i.e., maximumof N2 Gaussian variables of variance GN(x , x) ∼ (2/π) log N).Upper bound by first moment computation (as for BRW).Lower bound by introducing branching structure via averaging field ondifferent scales - roughly embedding a BRW in GFF.Truncated second moment method can also be used for the embeddedBRW Daviaud ’06 and for the analogous continuous GFFHu-Miller-Peres ’10Fluctuations?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 22 / 28

Law of large numbers

MN = maxz∈VNXN

z

Theorem (Bolthausen-Deuschel-Giacomin ’01)

MN

log N→ 2

√

2/π .

Value of limit is the same as if correlations were ignored (i.e., maximumof N2 Gaussian variables of variance GN(x , x) ∼ (2/π) log N).Upper bound by first moment computation (as for BRW).Lower bound by introducing branching structure via averaging field ondifferent scales - roughly embedding a BRW in GFF.Truncated second moment method can also be used for the embeddedBRW Daviaud ’06 and for the analogous continuous GFFHu-Miller-Peres ’10Fluctuations?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 22 / 28

Law of large numbers

MN = maxz∈VNXN

z

Theorem (Bolthausen-Deuschel-Giacomin ’01)

MN

log N→ 2

√

2/π .

Value of limit is the same as if correlations were ignored (i.e., maximumof N2 Gaussian variables of variance GN(x , x) ∼ (2/π) log N).Upper bound by first moment computation (as for BRW).Lower bound by introducing branching structure via averaging field ondifferent scales - roughly embedding a BRW in GFF.Truncated second moment method can also be used for the embeddedBRW Daviaud ’06 and for the analogous continuous GFFHu-Miller-Peres ’10Fluctuations?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 22 / 28

Law of large numbers

MN = maxz∈VNXN

z

Theorem (Bolthausen-Deuschel-Giacomin ’01)

MN

log N→ 2

√

2/π .

Value of limit is the same as if correlations were ignored (i.e., maximumof N2 Gaussian variables of variance GN(x , x) ∼ (2/π) log N).Upper bound by first moment computation (as for BRW).Lower bound by introducing branching structure via averaging field ondifferent scales - roughly embedding a BRW in GFF.Truncated second moment method can also be used for the embeddedBRW Daviaud ’06 and for the analogous continuous GFFHu-Miller-Peres ’10Fluctuations?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 22 / 28

Law of large numbers

MN = maxz∈VNXN

z

Theorem (Bolthausen-Deuschel-Giacomin ’01)

MN

log N→ 2

√

2/π .

Value of limit is the same as if correlations were ignored (i.e., maximumof N2 Gaussian variables of variance GN(x , x) ∼ (2/π) log N).Upper bound by first moment computation (as for BRW).Lower bound by introducing branching structure via averaging field ondifferent scales - roughly embedding a BRW in GFF.Truncated second moment method can also be used for the embeddedBRW Daviaud ’06 and for the analogous continuous GFFHu-Miller-Peres ’10Fluctuations?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 22 / 28

Fluctuations - preliminaries

GFF can be defined on any graph; in particular, sequence of boxes inan infinite graphs, e.g. ZZ d .

Theorem (Borell-Tsirelson ’75)For a centered Gaussian field {Xz}z∈T and X ∗ = maxz∈T Xz ,

P(|X ∗ − EX ∗| > δ) ≤ 2e−δ2/2σ2∗

where σ2∗ = maxz∈T EX 2

z .

Thus, if SRW on an infinite graph is transient, then fluctuations ofmaximum of GFF are of order 1.On the other hand, GFF on ZZ is just discrete analogue of BrownianBridge - and fluctuations of maximum are of order

√N (also, maximum

does not converge in probability).d = 2 is critical case!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 23 / 28

Fluctuations - preliminaries

GFF can be defined on any graph; in particular, sequence of boxes inan infinite graphs, e.g. ZZ d .

Theorem (Borell-Tsirelson ’75)For a centered Gaussian field {Xz}z∈T and X ∗ = maxz∈T Xz ,

P(|X ∗ − EX ∗| > δ) ≤ 2e−δ2/2σ2∗

where σ2∗ = maxz∈T EX 2

z .

Thus, if SRW on an infinite graph is transient, then fluctuations ofmaximum of GFF are of order 1.On the other hand, GFF on ZZ is just discrete analogue of BrownianBridge - and fluctuations of maximum are of order

√N (also, maximum

does not converge in probability).d = 2 is critical case!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 23 / 28

Fluctuations - preliminaries

GFF can be defined on any graph; in particular, sequence of boxes inan infinite graphs, e.g. ZZ d .

Theorem (Borell-Tsirelson ’75)For a centered Gaussian field {Xz}z∈T and X ∗ = maxz∈T Xz ,

P(|X ∗ − EX ∗| > δ) ≤ 2e−δ2/2σ2∗

where σ2∗ = maxz∈T EX 2

z .

Thus, if SRW on an infinite graph is transient, then fluctuations ofmaximum of GFF are of order 1.On the other hand, GFF on ZZ is just discrete analogue of BrownianBridge - and fluctuations of maximum are of order

√N (also, maximum

does not converge in probability).d = 2 is critical case!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 23 / 28

Fluctuations - preliminaries

GFF can be defined on any graph; in particular, sequence of boxes inan infinite graphs, e.g. ZZ d .

Theorem (Borell-Tsirelson ’75)For a centered Gaussian field {Xz}z∈T and X ∗ = maxz∈T Xz ,

P(|X ∗ − EX ∗| > δ) ≤ 2e−δ2/2σ2∗

where σ2∗ = maxz∈T EX 2

z .

Thus, if SRW on an infinite graph is transient, then fluctuations ofmaximum of GFF are of order 1.On the other hand, GFF on ZZ is just discrete analogue of BrownianBridge - and fluctuations of maximum are of order

√N (also, maximum

does not converge in probability).d = 2 is critical case!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 23 / 28

Fluctuations - preliminaries

GFF can be defined on any graph; in particular, sequence of boxes inan infinite graphs, e.g. ZZ d .

Theorem (Borell-Tsirelson ’75)For a centered Gaussian field {Xz}z∈T and X ∗ = maxz∈T Xz ,

P(|X ∗ − EX ∗| > δ) ≤ 2e−δ2/2σ2∗

where σ2∗ = maxz∈T EX 2

z .

Thus, if SRW on an infinite graph is transient, then fluctuations ofmaximum of GFF are of order 1.On the other hand, GFF on ZZ is just discrete analogue of BrownianBridge - and fluctuations of maximum are of order

√N (also, maximum

does not converge in probability).d = 2 is critical case!

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 23 / 28

Gaussian free field maximum

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20

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50

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20

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50

-4

-2

0

2

10

20

30

40

50

Consider the maximum MN of the (discrete) Gaussian Free Field onthe (planar) torus (of side N) with Dirichlet boundary conditions.

Theorem (Bramson-Z. ’11, after Bolthausen-Deuschel-Z. ’10)The fluctuations of MN are of order 1, and

EMN = 2√

2/π log N − (3/4)√

2/π log log N + O(1) .

But in fact...

Theorem (Bramson-Ding-Z. ’13)MN − EMN converges in distribution as N → ∞.

Some universality over the class of logarithmically correlated fields.Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 24 / 28

Gaussian free field maximum

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20

30

40

50

10

20

30

40

50

-4

-2

0

2

10

20

30

40

50

Consider the maximum MN of the (discrete) Gaussian Free Field onthe (planar) torus (of side N) with Dirichlet boundary conditions.

Theorem (Bramson-Z. ’11, after Bolthausen-Deuschel-Z. ’10)The fluctuations of MN are of order 1, and

EMN = 2√

2/π log N − (3/4)√

2/π log log N + O(1) .

But in fact...

Theorem (Bramson-Ding-Z. ’13)MN − EMN converges in distribution as N → ∞.

Some universality over the class of logarithmically correlated fields.Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 24 / 28

Gaussian free field maximum

10

20

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40

50

10

20

30

40

50

-4

-2

0

2

10

20

30

40

50

Consider the maximum MN of the (discrete) Gaussian Free Field onthe (planar) torus (of side N) with Dirichlet boundary conditions.

Theorem (Bramson-Z. ’11, after Bolthausen-Deuschel-Z. ’10)The fluctuations of MN are of order 1, and

EMN = 2√

2/π log N − (3/4)√

2/π log log N + O(1) .

But in fact...

Theorem (Bramson-Ding-Z. ’13)MN − EMN converges in distribution as N → ∞.

Some universality over the class of logarithmically correlated fields.Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 24 / 28

Gaussian free field maximum

10

20

30

40

50

10

20

30

40

50

-4

-2

0

2

10

20

30

40

50

Consider the maximum MN of the (discrete) Gaussian Free Field onthe (planar) torus (of side N) with Dirichlet boundary conditions.

Theorem (Bramson-Z. ’11, after Bolthausen-Deuschel-Z. ’10)The fluctuations of MN are of order 1, and

EMN = 2√

2/π log N − (3/4)√

2/π log log N + O(1) .

But in fact...

Theorem (Bramson-Ding-Z. ’13)MN − EMN converges in distribution as N → ∞.

Some universality over the class of logarithmically correlated fields.Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 24 / 28

Gaussian free field maximum

10

20

30

40

50

10

20

30

40

50

-4

-2

0

2

10

20

30

40

50

Consider the maximum MN of the (discrete) Gaussian Free Field onthe (planar) torus (of side N) with Dirichlet boundary conditions.

Theorem (Bramson-Z. ’11, after Bolthausen-Deuschel-Z. ’10)The fluctuations of MN are of order 1, and

EMN = 2√

2/π log N − (3/4)√

2/π log log N + O(1) .

But in fact...

Theorem (Bramson-Ding-Z. ’13)MN − EMN converges in distribution as N → ∞.

Some universality over the class of logarithmically correlated fields.Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 24 / 28

Gaussian free field maximum

10

20

30

40

50

10

20

30

40

50

-4

-2

0

2

10

20

30

40

50

Consider the maximum MN of the (discrete) Gaussian Free Field onthe (planar) torus (of side N) with Dirichlet boundary conditions.

Theorem (Bramson-Z. ’11, after Bolthausen-Deuschel-Z. ’10)The fluctuations of MN are of order 1, and

EMN = 2√

2/π log N − (3/4)√

2/π log log N + O(1) .

But in fact...

Theorem (Bramson-Ding-Z. ’13)MN − EMN converges in distribution as N → ∞.

Some universality over the class of logarithmically correlated fields.Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 24 / 28

GFF and BRW

Proof by comparing GFF to branching random walk and an appropriatemodified BRW.Heart of proof: modified second moment (with a variance reductionstep) for obtaining precise tail estimates on the maximum.Basic decomposition: Write

Xv = E(Xv |F) + (Xv − E(Xv |F)) = X cv + X f

v

where F is boundary of boxes.

30mm-5mmfigure=boxes.eps,height=92pt

Basic facts: X fv is GFF in smaller box, and it is impossible to have two

large values of Xv for v ,w that are mesoscopically separated.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 25 / 28

GFF and BRW

Proof by comparing GFF to branching random walk and an appropriatemodified BRW.Heart of proof: modified second moment (with a variance reductionstep) for obtaining precise tail estimates on the maximum.Basic decomposition: Write

Xv = E(Xv |F) + (Xv − E(Xv |F)) = X cv + X f

v

where F is boundary of boxes.

30mm-5mmfigure=boxes.eps,height=92pt

Basic facts: X fv is GFF in smaller box, and it is impossible to have two

large values of Xv for v ,w that are mesoscopically separated.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 25 / 28

GFF and BRW

Proof by comparing GFF to branching random walk and an appropriatemodified BRW.Heart of proof: modified second moment (with a variance reductionstep) for obtaining precise tail estimates on the maximum.Basic decomposition: Write

Xv = E(Xv |F) + (Xv − E(Xv |F)) = X cv + X f

v

where F is boundary of boxes.

30mm-5mmfigure=boxes.eps,height=92pt

Basic facts: X fv is GFF in smaller box, and it is impossible to have two

large values of Xv for v ,w that are mesoscopically separated.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 25 / 28

GFF and BRW

Proof by comparing GFF to branching random walk and an appropriatemodified BRW.Heart of proof: modified second moment (with a variance reductionstep) for obtaining precise tail estimates on the maximum.Basic decomposition: Write

Xv = E(Xv |F) + (Xv − E(Xv |F)) = X cv + X f

v

where F is boundary of boxes.

Basic facts: X fv is GFF in smaller box, and it is impossible to have two

large values of Xv for v ,w that are mesoscopically separated.Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 25 / 28

GFF and BRW

Proof by comparing GFF to branching random walk and an appropriatemodified BRW.Heart of proof: modified second moment (with a variance reductionstep) for obtaining precise tail estimates on the maximum.Basic decomposition: Write

Xv = E(Xv |F) + (Xv − E(Xv |F)) = X cv + X f

v

where F is boundary of boxes.

Basic facts: X fv is GFF in smaller box, and it is impossible to have two

large values of Xv for v ,w that are mesoscopically separated.Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 25 / 28

GFF and BRW

A version of the theorem holds for continuous GFF, where N isreplaced by smoothing at scale 1/N, and for certain star-scalelogarithmically correlated Gaussian fields Acosta ’13,’14, Madaule ’13.Analogy goes deeper than maxima: structure of point process of nearmaxima.Let Xvi be maximum in i th box (of fixed but large size K ), and setζN =

∑

i δXvi −mN.

Theorem (Biskup-Louidor ’13)ζN converges to a SPPP process.

Expect full convergence to SDPPP.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 26 / 28

GFF and BRW

A version of the theorem holds for continuous GFF, where N isreplaced by smoothing at scale 1/N, and for certain star-scalelogarithmically correlated Gaussian fields Acosta ’13,’14, Madaule ’13.Analogy goes deeper than maxima: structure of point process of nearmaxima.Let Xvi be maximum in i th box (of fixed but large size K ), and setζN =

∑

i δXvi −mN.

Theorem (Biskup-Louidor ’13)ζN converges to a SPPP process.

Expect full convergence to SDPPP.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 26 / 28

GFF and BRW

A version of the theorem holds for continuous GFF, where N isreplaced by smoothing at scale 1/N, and for certain star-scalelogarithmically correlated Gaussian fields Acosta ’13,’14, Madaule ’13.Analogy goes deeper than maxima: structure of point process of nearmaxima.Let Xvi be maximum in i th box (of fixed but large size K ), and setζN =

∑

i δXvi −mN.

Theorem (Biskup-Louidor ’13)ζN converges to a SPPP process.

Expect full convergence to SDPPP.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 26 / 28

GFF and BRW

A version of the theorem holds for continuous GFF, where N isreplaced by smoothing at scale 1/N, and for certain star-scalelogarithmically correlated Gaussian fields Acosta ’13,’14, Madaule ’13.Analogy goes deeper than maxima: structure of point process of nearmaxima.Let Xvi be maximum in i th box (of fixed but large size K ), and setζN =

∑

i δXvi −mN.

Theorem (Biskup-Louidor ’13)ζN converges to a SPPP process.

Expect full convergence to SDPPP.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 26 / 28

GFF and BRW

A version of the theorem holds for continuous GFF, where N isreplaced by smoothing at scale 1/N, and for certain star-scalelogarithmically correlated Gaussian fields Acosta ’13,’14, Madaule ’13.Analogy goes deeper than maxima: structure of point process of nearmaxima.Let Xvi be maximum in i th box (of fixed but large size K ), and setζN =

∑

i δXvi −mN.

Theorem (Biskup-Louidor ’13)ζN converges to a SPPP process.

Expect full convergence to SDPPP.

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 26 / 28

Cover Times

For a given finite graph G, the cover time tcov is the time it takes arandom walk to visit every vertex of G.Aldous ’91 showed that if Etcov is much larger than the expectedhitting time of any vertex, then tcov/Etcov converges in probability to 1,leading to the question of fluctuations of tcov.Isomorphism theorems: fix o, a continuous time random walk Xt on G,and consider Lt(x) =

∫ t0 1Xs=xds. Let τt = inf{s > 0 : Ls(o) > t}. Then

{Lxτt+

12

G2x}x

d= {1

2(Gx +

√2t)2}x .

(Ray-Knight, Dynkin; this version: Eisenbaum, Kaspi, Marcus, Rosen,Shi ’00).Ding, Lee, Peres ’10, Ding ’12: For bounded degree graphs/trees,

√

tcov(Gn)/|En| = (1 + o(1))maxx

Gx

Refinements: Ding, A. Zhai (continuous versions of isomorphism).Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 27 / 28

Cover Times

For a given finite graph G, the cover time tcov is the time it takes arandom walk to visit every vertex of G.Aldous ’91 showed that if Etcov is much larger than the expectedhitting time of any vertex, then tcov/Etcov converges in probability to 1,leading to the question of fluctuations of tcov.Isomorphism theorems: fix o, a continuous time random walk Xt on G,and consider Lt(x) =

∫ t0 1Xs=xds. Let τt = inf{s > 0 : Ls(o) > t}. Then

{Lxτt+

12

G2x}x

d= {1

2(Gx +

√2t)2}x .

(Ray-Knight, Dynkin; this version: Eisenbaum, Kaspi, Marcus, Rosen,Shi ’00).Ding, Lee, Peres ’10, Ding ’12: For bounded degree graphs/trees,

√

tcov(Gn)/|En| = (1 + o(1))maxx

Gx

Refinements: Ding, A. Zhai (continuous versions of isomorphism).Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 27 / 28

Cover Times

For a given finite graph G, the cover time tcov is the time it takes arandom walk to visit every vertex of G.Aldous ’91 showed that if Etcov is much larger than the expectedhitting time of any vertex, then tcov/Etcov converges in probability to 1,leading to the question of fluctuations of tcov.Isomorphism theorems: fix o, a continuous time random walk Xt on G,and consider Lt(x) =

∫ t0 1Xs=xds. Let τt = inf{s > 0 : Ls(o) > t}. Then

{Lxτt+

12

G2x}x

d= {1

2(Gx +

√2t)2}x .

(Ray-Knight, Dynkin; this version: Eisenbaum, Kaspi, Marcus, Rosen,Shi ’00).Ding, Lee, Peres ’10, Ding ’12: For bounded degree graphs/trees,

√

tcov(Gn)/|En| = (1 + o(1))maxx

Gx

Refinements: Ding, A. Zhai (continuous versions of isomorphism).Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 27 / 28

Cover Times

For a given finite graph G, the cover time tcov is the time it takes arandom walk to visit every vertex of G.Aldous ’91 showed that if Etcov is much larger than the expectedhitting time of any vertex, then tcov/Etcov converges in probability to 1,leading to the question of fluctuations of tcov.Isomorphism theorems: fix o, a continuous time random walk Xt on G,and consider Lt(x) =

∫ t0 1Xs=xds. Let τt = inf{s > 0 : Ls(o) > t}. Then

{Lxτt+

12

G2x}x

d= {1

2(Gx +

√2t)2}x .

(Ray-Knight, Dynkin; this version: Eisenbaum, Kaspi, Marcus, Rosen,Shi ’00).Ding, Lee, Peres ’10, Ding ’12: For bounded degree graphs/trees,

√

tcov(Gn)/|En| = (1 + o(1))maxx

Gx

Refinements: Ding, A. Zhai (continuous versions of isomorphism).Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 27 / 28

Cover Times

For a given finite graph G, the cover time tcov is the time it takes arandom walk to visit every vertex of G.Aldous ’91 showed that if Etcov is much larger than the expectedhitting time of any vertex, then tcov/Etcov converges in probability to 1,leading to the question of fluctuations of tcov.Isomorphism theorems: fix o, a continuous time random walk Xt on G,and consider Lt(x) =

∫ t0 1Xs=xds. Let τt = inf{s > 0 : Ls(o) > t}. Then

{Lxτt+

12

G2x}x

d= {1

2(Gx +

√2t)2}x .

(Ray-Knight, Dynkin; this version: Eisenbaum, Kaspi, Marcus, Rosen,Shi ’00).Ding, Lee, Peres ’10, Ding ’12: For bounded degree graphs/trees,

√

tcov(Gn)/|En| = (1 + o(1))maxx

Gx

Refinements: Ding, A. Zhai (continuous versions of isomorphism).Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 27 / 28

Cover Times

For regular trees,√

tcov/|En| ∼ O(1)

(proof similar to BRW tightness) Bramson-Z. ’09.Correction term is again logarithmic, but constant is 1 instead of 3/2.Ding-Z ’12.As we saw, law of large numbers for cover time of the torus wasestablished by Dembo, Peres, Rosen, Z. ’04Logarithmic correction term for torus is as for the tree Belius-Kistler,’14: ingenious refinement of DPRZ.Limit laws?Thick points?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 28 / 28

Cover Times

For regular trees,√

tcov/|En| ∼ O(1)

(proof similar to BRW tightness) Bramson-Z. ’09.Correction term is again logarithmic, but constant is 1 instead of 3/2.Ding-Z ’12.As we saw, law of large numbers for cover time of the torus wasestablished by Dembo, Peres, Rosen, Z. ’04Logarithmic correction term for torus is as for the tree Belius-Kistler,’14: ingenious refinement of DPRZ.Limit laws?Thick points?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 28 / 28

Cover Times

For regular trees,√

tcov/|En| ∼ O(1)

(proof similar to BRW tightness) Bramson-Z. ’09.Correction term is again logarithmic, but constant is 1 instead of 3/2.Ding-Z ’12.As we saw, law of large numbers for cover time of the torus wasestablished by Dembo, Peres, Rosen, Z. ’04Logarithmic correction term for torus is as for the tree Belius-Kistler,’14: ingenious refinement of DPRZ.Limit laws?Thick points?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 28 / 28

Cover Times

For regular trees,√

tcov/|En| ∼ O(1)

(proof similar to BRW tightness) Bramson-Z. ’09.Correction term is again logarithmic, but constant is 1 instead of 3/2.Ding-Z ’12.As we saw, law of large numbers for cover time of the torus wasestablished by Dembo, Peres, Rosen, Z. ’04Logarithmic correction term for torus is as for the tree Belius-Kistler,’14: ingenious refinement of DPRZ.Limit laws?Thick points?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 28 / 28

Cover Times

For regular trees,√

tcov/|En| ∼ O(1)

(proof similar to BRW tightness) Bramson-Z. ’09.Correction term is again logarithmic, but constant is 1 instead of 3/2.Ding-Z ’12.As we saw, law of large numbers for cover time of the torus wasestablished by Dembo, Peres, Rosen, Z. ’04Logarithmic correction term for torus is as for the tree Belius-Kistler,’14: ingenious refinement of DPRZ.Limit laws?Thick points?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 28 / 28

Cover Times

For regular trees,√

tcov/|En| ∼ O(1)

(proof similar to BRW tightness) Bramson-Z. ’09.Correction term is again logarithmic, but constant is 1 instead of 3/2.Ding-Z ’12.As we saw, law of large numbers for cover time of the torus wasestablished by Dembo, Peres, Rosen, Z. ’04Logarithmic correction term for torus is as for the tree Belius-Kistler,’14: ingenious refinement of DPRZ.Limit laws?Thick points?

Ofer Zeitouni (Oxford talk) Trees everywhere October 2, 2014 28 / 28