Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d...

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Natural parametrization for the scaling limit of loop-erased random walk in three dimensions Daisuke Shiraishi, Kyoto University joint work with Xinyi Li (University of Chicago) November 2018, Nagoya University 1 / 16

Transcript of Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d...

Page 1: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Natural parametrization for the scaling limit ofloop-erased random walk in three dimensions

Daisuke Shiraishi, Kyoto University

joint work with Xinyi Li (University of Chicago)

November 2018, Nagoya University

1 / 16

Page 2: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I Random continuous curves γ : [0,∞)→ Rd are importantobjects.

I Typical example: Brownian motion (BM). But it’s a bit ideal.

I The curves I want to consider have strong interaction with thepast.

I Interest: Curves with self-repulsion, which are forbidden tovisit any place that has already been visited.

I Dimension dependent behavior:I d = 1, Nothing interesting.I d ≥ 4, Effect of “self-avoidance” on the curve becomes weak

so that the curve ends up with BM.(Recall that BM is a simple curve iff d ≥ 4.)

I Interesting dimensions are d = 2, 3.

2 / 16

Page 3: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I Random continuous curves γ : [0,∞)→ Rd are importantobjects.

I Typical example: Brownian motion (BM). But it’s a bit ideal.

I The curves I want to consider have strong interaction with thepast.

I Interest: Curves with self-repulsion, which are forbidden tovisit any place that has already been visited.

I Dimension dependent behavior:I d = 1, Nothing interesting.I d ≥ 4, Effect of “self-avoidance” on the curve becomes weak

so that the curve ends up with BM.(Recall that BM is a simple curve iff d ≥ 4.)

I Interesting dimensions are d = 2, 3.

2 / 16

Page 4: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I Random continuous curves γ : [0,∞)→ Rd are importantobjects.

I Typical example: Brownian motion (BM). But it’s a bit ideal.

I The curves I want to consider have strong interaction with thepast.

I Interest: Curves with self-repulsion, which are forbidden tovisit any place that has already been visited.

I Dimension dependent behavior:I d = 1, Nothing interesting.I d ≥ 4, Effect of “self-avoidance” on the curve becomes weak

so that the curve ends up with BM.(Recall that BM is a simple curve iff d ≥ 4.)

I Interesting dimensions are d = 2, 3.

2 / 16

Page 5: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I Random continuous curves γ : [0,∞)→ Rd are importantobjects.

I Typical example: Brownian motion (BM). But it’s a bit ideal.

I The curves I want to consider have strong interaction with thepast.

I Interest: Curves with self-repulsion, which are forbidden tovisit any place that has already been visited.

I Dimension dependent behavior:I d = 1, Nothing interesting.I d ≥ 4, Effect of “self-avoidance” on the curve becomes weak

so that the curve ends up with BM.(Recall that BM is a simple curve iff d ≥ 4.)

I Interesting dimensions are d = 2, 3.

2 / 16

Page 6: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I Random continuous curves γ : [0,∞)→ Rd are importantobjects.

I Typical example: Brownian motion (BM). But it’s a bit ideal.

I The curves I want to consider have strong interaction with thepast.

I Interest: Curves with self-repulsion, which are forbidden tovisit any place that has already been visited.

I Dimension dependent behavior:

I d = 1, Nothing interesting.I d ≥ 4, Effect of “self-avoidance” on the curve becomes weak

so that the curve ends up with BM.(Recall that BM is a simple curve iff d ≥ 4.)

I Interesting dimensions are d = 2, 3.

2 / 16

Page 7: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I Random continuous curves γ : [0,∞)→ Rd are importantobjects.

I Typical example: Brownian motion (BM). But it’s a bit ideal.

I The curves I want to consider have strong interaction with thepast.

I Interest: Curves with self-repulsion, which are forbidden tovisit any place that has already been visited.

I Dimension dependent behavior:I d = 1, Nothing interesting.

I d ≥ 4, Effect of “self-avoidance” on the curve becomes weakso that the curve ends up with BM.(Recall that BM is a simple curve iff d ≥ 4.)

I Interesting dimensions are d = 2, 3.

2 / 16

Page 8: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I Random continuous curves γ : [0,∞)→ Rd are importantobjects.

I Typical example: Brownian motion (BM). But it’s a bit ideal.

I The curves I want to consider have strong interaction with thepast.

I Interest: Curves with self-repulsion, which are forbidden tovisit any place that has already been visited.

I Dimension dependent behavior:I d = 1, Nothing interesting.I d ≥ 4, Effect of “self-avoidance” on the curve becomes weak

so that the curve ends up with BM.

(Recall that BM is a simple curve iff d ≥ 4.)

I Interesting dimensions are d = 2, 3.

2 / 16

Page 9: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I Random continuous curves γ : [0,∞)→ Rd are importantobjects.

I Typical example: Brownian motion (BM). But it’s a bit ideal.

I The curves I want to consider have strong interaction with thepast.

I Interest: Curves with self-repulsion, which are forbidden tovisit any place that has already been visited.

I Dimension dependent behavior:I d = 1, Nothing interesting.I d ≥ 4, Effect of “self-avoidance” on the curve becomes weak

so that the curve ends up with BM.(Recall that BM is a simple curve iff d ≥ 4.)

I Interesting dimensions are d = 2, 3.

2 / 16

Page 10: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I Random continuous curves γ : [0,∞)→ Rd are importantobjects.

I Typical example: Brownian motion (BM). But it’s a bit ideal.

I The curves I want to consider have strong interaction with thepast.

I Interest: Curves with self-repulsion, which are forbidden tovisit any place that has already been visited.

I Dimension dependent behavior:I d = 1, Nothing interesting.I d ≥ 4, Effect of “self-avoidance” on the curve becomes weak

so that the curve ends up with BM.(Recall that BM is a simple curve iff d ≥ 4.)

I Interesting dimensions are d = 2, 3.

2 / 16

Page 11: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I The most interesting case is d = 3.

I We live in three dimensions.I Recent big progress via conformal field theory for d = 2. The

techniques used there are not available for d = 3.

I Focus on d = 3. What can we do?

I Discretization: Consider a discrete random path and take itsscaling limit.

I Our choice of the discretization is loop-erased random walk(LERW).

I LERW is the only model with self-repulsion that we cananalyze rigorously for d = 3.

3 / 16

Page 12: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I The most interesting case is d = 3.I We live in three dimensions.

I Recent big progress via conformal field theory for d = 2. Thetechniques used there are not available for d = 3.

I Focus on d = 3. What can we do?

I Discretization: Consider a discrete random path and take itsscaling limit.

I Our choice of the discretization is loop-erased random walk(LERW).

I LERW is the only model with self-repulsion that we cananalyze rigorously for d = 3.

3 / 16

Page 13: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I The most interesting case is d = 3.I We live in three dimensions.I Recent big progress via conformal field theory for d = 2. The

techniques used there are not available for d = 3.

I Focus on d = 3. What can we do?

I Discretization: Consider a discrete random path and take itsscaling limit.

I Our choice of the discretization is loop-erased random walk(LERW).

I LERW is the only model with self-repulsion that we cananalyze rigorously for d = 3.

3 / 16

Page 14: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I The most interesting case is d = 3.I We live in three dimensions.I Recent big progress via conformal field theory for d = 2. The

techniques used there are not available for d = 3.

I Focus on d = 3. What can we do?

I Discretization: Consider a discrete random path and take itsscaling limit.

I Our choice of the discretization is loop-erased random walk(LERW).

I LERW is the only model with self-repulsion that we cananalyze rigorously for d = 3.

3 / 16

Page 15: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I The most interesting case is d = 3.I We live in three dimensions.I Recent big progress via conformal field theory for d = 2. The

techniques used there are not available for d = 3.

I Focus on d = 3. What can we do?

I Discretization: Consider a discrete random path and take itsscaling limit.

I Our choice of the discretization is loop-erased random walk(LERW).

I LERW is the only model with self-repulsion that we cananalyze rigorously for d = 3.

3 / 16

Page 16: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I The most interesting case is d = 3.I We live in three dimensions.I Recent big progress via conformal field theory for d = 2. The

techniques used there are not available for d = 3.

I Focus on d = 3. What can we do?

I Discretization: Consider a discrete random path and take itsscaling limit.

I Our choice of the discretization is loop-erased random walk(LERW).

I LERW is the only model with self-repulsion that we cananalyze rigorously for d = 3.

3 / 16

Page 17: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

General Overview

I The most interesting case is d = 3.I We live in three dimensions.I Recent big progress via conformal field theory for d = 2. The

techniques used there are not available for d = 3.

I Focus on d = 3. What can we do?

I Discretization: Consider a discrete random path and take itsscaling limit.

I Our choice of the discretization is loop-erased random walk(LERW).

I LERW is the only model with self-repulsion that we cananalyze rigorously for d = 3.

3 / 16

Page 18: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Setting and Question

I Loop-erased random walk (LERW) is the random simple pathobtained by erasing all loops chronologically from a simplerandom walk path, which was originally introduced by GregLawler in 1980.

I Focus on 3D LERW in the talk, although the model alsoenjoys interesting properties in other dimensions, especially in2D via conformal field theory. Many elementary problems for3D LERW still remains open.

I Interest: Scaling limit of 3D LERW.

I No similar procedure to erase loops from Brownianmotion!

4 / 16

Page 19: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Setting and Question

I Loop-erased random walk (LERW) is the random simple pathobtained by erasing all loops chronologically from a simplerandom walk path, which was originally introduced by GregLawler in 1980.

I Focus on 3D LERW in the talk, although the model alsoenjoys interesting properties in other dimensions, especially in2D via conformal field theory. Many elementary problems for3D LERW still remains open.

I Interest: Scaling limit of 3D LERW.

I No similar procedure to erase loops from Brownianmotion!

4 / 16

Page 20: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Setting and Question

I Loop-erased random walk (LERW) is the random simple pathobtained by erasing all loops chronologically from a simplerandom walk path, which was originally introduced by GregLawler in 1980.

I Focus on 3D LERW in the talk, although the model alsoenjoys interesting properties in other dimensions, especially in2D via conformal field theory. Many elementary problems for3D LERW still remains open.

I Interest: Scaling limit of 3D LERW.

I No similar procedure to erase loops from Brownianmotion!

4 / 16

Page 21: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Setting and Question

I Loop-erased random walk (LERW) is the random simple pathobtained by erasing all loops chronologically from a simplerandom walk path, which was originally introduced by GregLawler in 1980.

I Focus on 3D LERW in the talk, although the model alsoenjoys interesting properties in other dimensions, especially in2D via conformal field theory. Many elementary problems for3D LERW still remains open.

I Interest: Scaling limit of 3D LERW.

I No similar procedure to erase loops from Brownianmotion!

4 / 16

Page 22: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Picture credit: Fredrik Viklund.

5 / 16

Page 23: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Setting and Question

I S (n) =(S (n)(k)

)k≥0: SRW on 1

nZ3 started at the origin.

I D = {x ∈ R3 : |x | < 1} : unit ball, D: its closure.

I T (n) = inf{k ≥ 0 : S (n)(k) /∈ D}: first exit time from D.

I γ(n) =(γ(n)(k)

)k≥0 = LE

(S (n)[0,T (n)]

): LERW on 1

nZ3

started at the origin and stopped at ∂D.

I Another interpretation: LERW = Laplacian random walk.Namely, for x ∼ γ(n)(k).

P(γ(n)(k + 1) = x

∣∣∣ γ(n)[0, k])

=f (x)∑

y∼γ(n)(k) f (y).

Here f is discrete harmonic on(D ∩ 1

nZ3)\ γ(n)[0, k] with

f ≡ 1 on ∂(D ∩ 1

nZ3)

and f ≡ 0 on γ(n)[0, k].

I Question: What is the scaling limit of γ(n) as n→∞? Whatis the topology for the limit?

6 / 16

Page 24: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Setting and Question

I S (n) =(S (n)(k)

)k≥0: SRW on 1

nZ3 started at the origin.

I D = {x ∈ R3 : |x | < 1} : unit ball, D: its closure.

I T (n) = inf{k ≥ 0 : S (n)(k) /∈ D}: first exit time from D.

I γ(n) =(γ(n)(k)

)k≥0 = LE

(S (n)[0,T (n)]

): LERW on 1

nZ3

started at the origin and stopped at ∂D.

I Another interpretation: LERW = Laplacian random walk.Namely, for x ∼ γ(n)(k).

P(γ(n)(k + 1) = x

∣∣∣ γ(n)[0, k])

=f (x)∑

y∼γ(n)(k) f (y).

Here f is discrete harmonic on(D ∩ 1

nZ3)\ γ(n)[0, k] with

f ≡ 1 on ∂(D ∩ 1

nZ3)

and f ≡ 0 on γ(n)[0, k].

I Question: What is the scaling limit of γ(n) as n→∞? Whatis the topology for the limit?

6 / 16

Page 25: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Setting and Question

I S (n) =(S (n)(k)

)k≥0: SRW on 1

nZ3 started at the origin.

I D = {x ∈ R3 : |x | < 1} : unit ball, D: its closure.

I T (n) = inf{k ≥ 0 : S (n)(k) /∈ D}: first exit time from D.

I γ(n) =(γ(n)(k)

)k≥0 = LE

(S (n)[0,T (n)]

): LERW on 1

nZ3

started at the origin and stopped at ∂D.

I Another interpretation: LERW = Laplacian random walk.Namely, for x ∼ γ(n)(k).

P(γ(n)(k + 1) = x

∣∣∣ γ(n)[0, k])

=f (x)∑

y∼γ(n)(k) f (y).

Here f is discrete harmonic on(D ∩ 1

nZ3)\ γ(n)[0, k] with

f ≡ 1 on ∂(D ∩ 1

nZ3)

and f ≡ 0 on γ(n)[0, k].

I Question: What is the scaling limit of γ(n) as n→∞? Whatis the topology for the limit?

6 / 16

Page 26: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Setting and Question

I S (n) =(S (n)(k)

)k≥0: SRW on 1

nZ3 started at the origin.

I D = {x ∈ R3 : |x | < 1} : unit ball, D: its closure.

I T (n) = inf{k ≥ 0 : S (n)(k) /∈ D}: first exit time from D.

I γ(n) =(γ(n)(k)

)k≥0 = LE

(S (n)[0,T (n)]

): LERW on 1

nZ3

started at the origin and stopped at ∂D.

I Another interpretation: LERW = Laplacian random walk.Namely, for x ∼ γ(n)(k).

P(γ(n)(k + 1) = x

∣∣∣ γ(n)[0, k])

=f (x)∑

y∼γ(n)(k) f (y).

Here f is discrete harmonic on(D ∩ 1

nZ3)\ γ(n)[0, k] with

f ≡ 1 on ∂(D ∩ 1

nZ3)

and f ≡ 0 on γ(n)[0, k].

I Question: What is the scaling limit of γ(n) as n→∞? Whatis the topology for the limit?

6 / 16

Page 27: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Setting and Question

I S (n) =(S (n)(k)

)k≥0: SRW on 1

nZ3 started at the origin.

I D = {x ∈ R3 : |x | < 1} : unit ball, D: its closure.

I T (n) = inf{k ≥ 0 : S (n)(k) /∈ D}: first exit time from D.

I γ(n) =(γ(n)(k)

)k≥0 = LE

(S (n)[0,T (n)]

): LERW on 1

nZ3

started at the origin and stopped at ∂D.

I Another interpretation: LERW = Laplacian random walk.Namely, for x ∼ γ(n)(k).

P(γ(n)(k + 1) = x

∣∣∣ γ(n)[0, k])

=f (x)∑

y∼γ(n)(k) f (y).

Here f is discrete harmonic on(D ∩ 1

nZ3)\ γ(n)[0, k] with

f ≡ 1 on ∂(D ∩ 1

nZ3)

and f ≡ 0 on γ(n)[0, k].

I Question: What is the scaling limit of γ(n) as n→∞? Whatis the topology for the limit?

6 / 16

Page 28: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Setting and Question

I S (n) =(S (n)(k)

)k≥0: SRW on 1

nZ3 started at the origin.

I D = {x ∈ R3 : |x | < 1} : unit ball, D: its closure.

I T (n) = inf{k ≥ 0 : S (n)(k) /∈ D}: first exit time from D.

I γ(n) =(γ(n)(k)

)k≥0 = LE

(S (n)[0,T (n)]

): LERW on 1

nZ3

started at the origin and stopped at ∂D.

I Another interpretation: LERW = Laplacian random walk.Namely, for x ∼ γ(n)(k).

P(γ(n)(k + 1) = x

∣∣∣ γ(n)[0, k])

=f (x)∑

y∼γ(n)(k) f (y).

Here f is discrete harmonic on(D ∩ 1

nZ3)\ γ(n)[0, k] with

f ≡ 1 on ∂(D ∩ 1

nZ3)

and f ≡ 0 on γ(n)[0, k].

I Question: What is the scaling limit of γ(n) as n→∞? Whatis the topology for the limit?

6 / 16

Page 29: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Known results for 3D LERW

I Mn = len(γ(n)

): length (number of steps) of γ(n).

I (Lawler ’99): ∃c, ε > 0 s.t. for all n ≥ 1

cn1+ε ≤ E (Mn) ≤ 1

cn

53 .

I(H(D), dHaus

): space of all non-empty compact subsets of D

with the Hausdorff distance dHaus.

I (Kozma ’07): As n→∞, γ(2n) converges weakly to some

random compact set K w,r,t, the Hausdorff distance dHaus.

I Kozma shows that γ(2n) is a Cauchy sequence w.r.t. the

Prokhorov metric. No nice tool like SLE to describe K!

7 / 16

Page 30: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Known results for 3D LERW

I Mn = len(γ(n)

): length (number of steps) of γ(n).

I (Lawler ’99): ∃c, ε > 0 s.t. for all n ≥ 1

cn1+ε ≤ E (Mn) ≤ 1

cn

53 .

I(H(D), dHaus

): space of all non-empty compact subsets of D

with the Hausdorff distance dHaus.

I (Kozma ’07): As n→∞, γ(2n) converges weakly to some

random compact set K w,r,t, the Hausdorff distance dHaus.

I Kozma shows that γ(2n) is a Cauchy sequence w.r.t. the

Prokhorov metric. No nice tool like SLE to describe K!

7 / 16

Page 31: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Known results for 3D LERW

I Mn = len(γ(n)

): length (number of steps) of γ(n).

I (Lawler ’99): ∃c, ε > 0 s.t. for all n ≥ 1

cn1+ε ≤ E (Mn) ≤ 1

cn

53 .

I(H(D), dHaus

): space of all non-empty compact subsets of D

with the Hausdorff distance dHaus.

I (Kozma ’07): As n→∞, γ(2n) converges weakly to some

random compact set K w,r,t, the Hausdorff distance dHaus.

I Kozma shows that γ(2n) is a Cauchy sequence w.r.t. the

Prokhorov metric. No nice tool like SLE to describe K!

7 / 16

Page 32: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Known results for 3D LERW

I Mn = len(γ(n)

): length (number of steps) of γ(n).

I (Lawler ’99): ∃c, ε > 0 s.t. for all n ≥ 1

cn1+ε ≤ E (Mn) ≤ 1

cn

53 .

I(H(D), dHaus

): space of all non-empty compact subsets of D

with the Hausdorff distance dHaus.

I (Kozma ’07): As n→∞, γ(2n) converges weakly to some

random compact set K w,r,t, the Hausdorff distance dHaus.

I Kozma shows that γ(2n) is a Cauchy sequence w.r.t. the

Prokhorov metric. No nice tool like SLE to describe K!

7 / 16

Page 33: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Known results for 3D LERW

I Mn = len(γ(n)

): length (number of steps) of γ(n).

I (Lawler ’99): ∃c, ε > 0 s.t. for all n ≥ 1

cn1+ε ≤ E (Mn) ≤ 1

cn

53 .

I(H(D), dHaus

): space of all non-empty compact subsets of D

with the Hausdorff distance dHaus.

I (Kozma ’07): As n→∞, γ(2n) converges weakly to some

random compact set K w,r,t, the Hausdorff distance dHaus.

I Kozma shows that γ(2n) is a Cauchy sequence w.r.t. the

Prokhorov metric. No nice tool like SLE to describe K!

7 / 16

Page 34: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Known results for 3D LERW

I Problems:

– Is K a simple curve a.s.?– What is the Hausdorff dimension of K?– Time rescaled LERW

w−→ Time parametrized Kw.r.t. the uniform norm? How to parametrize K?

I (S. ’14) ∃β ∈ (1, 53 ] s.t.

E (Mn) = nβ+o(1) as n→∞.

Moreover, MnE(Mn)

is tight.

I (Wilson ’10) Simulation: β = 1.624± 0.001.

I (Sapozhnikov - S. ’15) W.p.1, K is homeomorphic to [0, 1].

I (S. ’16) W.p.1, dim(K) = β.

8 / 16

Page 35: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Known results for 3D LERW

I Problems:

– Is K a simple curve a.s.?

– What is the Hausdorff dimension of K?– Time rescaled LERW

w−→ Time parametrized Kw.r.t. the uniform norm? How to parametrize K?

I (S. ’14) ∃β ∈ (1, 53 ] s.t.

E (Mn) = nβ+o(1) as n→∞.

Moreover, MnE(Mn)

is tight.

I (Wilson ’10) Simulation: β = 1.624± 0.001.

I (Sapozhnikov - S. ’15) W.p.1, K is homeomorphic to [0, 1].

I (S. ’16) W.p.1, dim(K) = β.

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Known results for 3D LERW

I Problems:

– Is K a simple curve a.s.?– What is the Hausdorff dimension of K?

– Time rescaled LERWw−→ Time parametrized K

w.r.t. the uniform norm? How to parametrize K?

I (S. ’14) ∃β ∈ (1, 53 ] s.t.

E (Mn) = nβ+o(1) as n→∞.

Moreover, MnE(Mn)

is tight.

I (Wilson ’10) Simulation: β = 1.624± 0.001.

I (Sapozhnikov - S. ’15) W.p.1, K is homeomorphic to [0, 1].

I (S. ’16) W.p.1, dim(K) = β.

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Known results for 3D LERW

I Problems:

– Is K a simple curve a.s.?– What is the Hausdorff dimension of K?– Time rescaled LERW

w−→ Time parametrized Kw.r.t. the uniform norm? How to parametrize K?

I (S. ’14) ∃β ∈ (1, 53 ] s.t.

E (Mn) = nβ+o(1) as n→∞.

Moreover, MnE(Mn)

is tight.

I (Wilson ’10) Simulation: β = 1.624± 0.001.

I (Sapozhnikov - S. ’15) W.p.1, K is homeomorphic to [0, 1].

I (S. ’16) W.p.1, dim(K) = β.

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Known results for 3D LERW

I Problems:

– Is K a simple curve a.s.?– What is the Hausdorff dimension of K?– Time rescaled LERW

w−→ Time parametrized Kw.r.t. the uniform norm? How to parametrize K?

I (S. ’14) ∃β ∈ (1, 53 ] s.t.

E (Mn) = nβ+o(1) as n→∞.

Moreover, MnE(Mn)

is tight.

I (Wilson ’10) Simulation: β = 1.624± 0.001.

I (Sapozhnikov - S. ’15) W.p.1, K is homeomorphic to [0, 1].

I (S. ’16) W.p.1, dim(K) = β.

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Known results for 3D LERW

I Problems:

– Is K a simple curve a.s.?– What is the Hausdorff dimension of K?– Time rescaled LERW

w−→ Time parametrized Kw.r.t. the uniform norm? How to parametrize K?

I (S. ’14) ∃β ∈ (1, 53 ] s.t.

E (Mn) = nβ+o(1) as n→∞.

Moreover, MnE(Mn)

is tight.

I (Wilson ’10) Simulation: β = 1.624± 0.001.

I (Sapozhnikov - S. ’15) W.p.1, K is homeomorphic to [0, 1].

I (S. ’16) W.p.1, dim(K) = β.

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Known results for 3D LERW

I Problems:

– Is K a simple curve a.s.?– What is the Hausdorff dimension of K?– Time rescaled LERW

w−→ Time parametrized Kw.r.t. the uniform norm? How to parametrize K?

I (S. ’14) ∃β ∈ (1, 53 ] s.t.

E (Mn) = nβ+o(1) as n→∞.

Moreover, MnE(Mn)

is tight.

I (Wilson ’10) Simulation: β = 1.624± 0.001.

I (Sapozhnikov - S. ’15) W.p.1, K is homeomorphic to [0, 1].

I (S. ’16) W.p.1, dim(K) = β.

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Known results for 3D LERW

I Problems:

– Is K a simple curve a.s.?– What is the Hausdorff dimension of K?– Time rescaled LERW

w−→ Time parametrized Kw.r.t. the uniform norm? How to parametrize K?

I (S. ’14) ∃β ∈ (1, 53 ] s.t.

E (Mn) = nβ+o(1) as n→∞.

Moreover, MnE(Mn)

is tight.

I (Wilson ’10) Simulation: β = 1.624± 0.001.

I (Sapozhnikov - S. ’15) W.p.1, K is homeomorphic to [0, 1].

I (S. ’16) W.p.1, dim(K) = β.

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Main Results

Theorem (Li - S. ’18)

There exist universal c0 > 0, δ > 0 such that

E (M2n) = c02βn{

1 + O(2−δn

)}as n→∞.

I From this theorem, M2n/2βn is tight.

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Main Results

Theorem (Li - S. ’18)

There exist universal c0 > 0, δ > 0 such that

E (M2n) = c02βn{

1 + O(2−δn

)}as n→∞.

I From this theorem, M2n/2βn is tight.

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Main Results

I Let ηn be the time rescaled γ(2n) defined by

ηn(t) = γ(2n)(2βnt

)for 0 ≤ t ≤ M2n/2βn,

where we assume the linear interpolation for γ(2n).

I Want to show ηn converges weakly to suitably parametrized Kw.r.t. the uniform norm. How to parametrize K?

I Letµn = 2−βn

∑x∈γ(2n)∩2−nZ3

δx

be the normalized counting measure on γ(2n) where δx stands

for the Dirac measure at x .

I ηn =(γ(2

n) parametrized by µn)

.

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Main Results

I Let ηn be the time rescaled γ(2n) defined by

ηn(t) = γ(2n)(2βnt

)for 0 ≤ t ≤ M2n/2βn,

where we assume the linear interpolation for γ(2n).

I Want to show ηn converges weakly to suitably parametrized Kw.r.t. the uniform norm. How to parametrize K?

I Letµn = 2−βn

∑x∈γ(2n)∩2−nZ3

δx

be the normalized counting measure on γ(2n) where δx stands

for the Dirac measure at x .

I ηn =(γ(2

n) parametrized by µn)

.

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Page 46: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Main Results

I Let ηn be the time rescaled γ(2n) defined by

ηn(t) = γ(2n)(2βnt

)for 0 ≤ t ≤ M2n/2βn,

where we assume the linear interpolation for γ(2n).

I Want to show ηn converges weakly to suitably parametrized Kw.r.t. the uniform norm. How to parametrize K?

I Letµn = 2−βn

∑x∈γ(2n)∩2−nZ3

δx

be the normalized counting measure on γ(2n) where δx stands

for the Dirac measure at x .

I ηn =(γ(2

n) parametrized by µn)

.

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Page 47: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Main Results

I Let ηn be the time rescaled γ(2n) defined by

ηn(t) = γ(2n)(2βnt

)for 0 ≤ t ≤ M2n/2βn,

where we assume the linear interpolation for γ(2n).

I Want to show ηn converges weakly to suitably parametrized Kw.r.t. the uniform norm. How to parametrize K?

I Letµn = 2−βn

∑x∈γ(2n)∩2−nZ3

δx

be the normalized counting measure on γ(2n) where δx stands

for the Dirac measure at x .

I ηn =(γ(2

n) parametrized by µn)

.

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Main Results

I Recall(H(D), dHaus

)stands for the space of compact subsets

of D with the Hausdorff distance dHaus.

I Let M(D) be the space of finite measures on D endowed withthe weak convergence topology.

Theorem (Li - S. ’18 work in progress)

As n→∞, (γ(2n), µn) converges weakly to some (K, µ) w.r.t. the

product topology of H(D) andM(D), where K is Kozma’s scalinglimit. Furthermore, the measure µ is a measurable function of K.

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Main Results

I Recall(H(D), dHaus

)stands for the space of compact subsets

of D with the Hausdorff distance dHaus.

I Let M(D) be the space of finite measures on D endowed withthe weak convergence topology.

Theorem (Li - S. ’18 work in progress)

As n→∞, (γ(2n), µn) converges weakly to some (K, µ) w.r.t. the

product topology of H(D) andM(D), where K is Kozma’s scalinglimit. Furthermore, the measure µ is a measurable function of K.

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Main Results

I Recall(H(D), dHaus

)stands for the space of compact subsets

of D with the Hausdorff distance dHaus.

I Let M(D) be the space of finite measures on D endowed withthe weak convergence topology.

Theorem (Li - S. ’18 work in progress)

As n→∞, (γ(2n), µn) converges weakly to some (K, µ) w.r.t. the

product topology of H(D) andM(D), where K is Kozma’s scalinglimit. Furthermore, the measure µ is a measurable function of K.

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Main Results

Theorem (Li - S. ’18 work in progress)

W.p.1, supp(µ) = K. Moreover, it follows that w.p.1, for eachx ∈ K

limy∈Ky→x

µ(Ky ) = µ(Kx),

where Ky is the curve in K between the origin and y ∈ K.

I Recall ηn is the time rescaled γ(2n) defined by

ηn(t) = γ(2n)(2βnt

)for 0 ≤ t ≤ M2n/2βn.

I For each t ∈ [0, µ(K)], there exists unique xt ∈ K s.t.t = µ(Kxt ). Define η(t) = xt for t ∈ [0, µ(K)].

I Take two continuous curves λi : [0, ti ]→ D (i = 1, 2). Let

ρ(λ1, λ2) = |t1 − t2|+ max0≤s≤1

∣∣λ1(st1)− λ2(st2)∣∣

be the supremum distance of them.

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Main Results

Theorem (Li - S. ’18 work in progress)

W.p.1, supp(µ) = K. Moreover, it follows that w.p.1, for eachx ∈ K

limy∈Ky→x

µ(Ky ) = µ(Kx),

where Ky is the curve in K between the origin and y ∈ K.I Recall ηn is the time rescaled γ(2

n) defined by

ηn(t) = γ(2n)(2βnt

)for 0 ≤ t ≤ M2n/2βn.

I For each t ∈ [0, µ(K)], there exists unique xt ∈ K s.t.t = µ(Kxt ). Define η(t) = xt for t ∈ [0, µ(K)].

I Take two continuous curves λi : [0, ti ]→ D (i = 1, 2). Let

ρ(λ1, λ2) = |t1 − t2|+ max0≤s≤1

∣∣λ1(st1)− λ2(st2)∣∣

be the supremum distance of them.

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Main Results

Theorem (Li - S. ’18 work in progress)

W.p.1, supp(µ) = K. Moreover, it follows that w.p.1, for eachx ∈ K

limy∈Ky→x

µ(Ky ) = µ(Kx),

where Ky is the curve in K between the origin and y ∈ K.I Recall ηn is the time rescaled γ(2

n) defined by

ηn(t) = γ(2n)(2βnt

)for 0 ≤ t ≤ M2n/2βn.

I For each t ∈ [0, µ(K)], there exists unique xt ∈ K s.t.t = µ(Kxt ). Define η(t) = xt for t ∈ [0, µ(K)].

I Take two continuous curves λi : [0, ti ]→ D (i = 1, 2). Let

ρ(λ1, λ2) = |t1 − t2|+ max0≤s≤1

∣∣λ1(st1)− λ2(st2)∣∣

be the supremum distance of them.

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Page 54: Natural parametrization for the scaling limit of loop ...€¦ · I d = 1, Nothing interesting. I d 4, E ect of \self-avoidance" on the curve becomes weak so that the curve ends up

Main Results

Theorem (Li - S. ’18 work in progress)

W.p.1, supp(µ) = K. Moreover, it follows that w.p.1, for eachx ∈ K

limy∈Ky→x

µ(Ky ) = µ(Kx),

where Ky is the curve in K between the origin and y ∈ K.I Recall ηn is the time rescaled γ(2

n) defined by

ηn(t) = γ(2n)(2βnt

)for 0 ≤ t ≤ M2n/2βn.

I For each t ∈ [0, µ(K)], there exists unique xt ∈ K s.t.t = µ(Kxt ). Define η(t) = xt for t ∈ [0, µ(K)].

I Take two continuous curves λi : [0, ti ]→ D (i = 1, 2). Let

ρ(λ1, λ2) = |t1 − t2|+ max0≤s≤1

∣∣λ1(st1)− λ2(st2)∣∣

be the supremum distance of them.

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Main Results

Theorem (Li - S. ’18 work in progress)

As n→∞, ηn converges weakly to η w.r.t. the metric ρ.

I Remark: The same convergence results w.r.t. the supremumdistance ρ hold for other dimensions:

I (Lawler 1991) For d ≥ 5, ∃cd > 0 s.t. γ(n)(cdn

2t)→ BM.

I (Lawler 1993) For d = 4, ∃c4 > 0 s.t.

γ(n)(c4n

2(log n)−13 t)→ BM.

I (Lawler-Viklund 2017) For d = 2, ∃c2 > 0 s.t.

γ(n)(c2n

54 t)→ γ where γ is SLE2 parametrized by

5/4-dimensional Minkowski content.

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Main Results

Theorem (Li - S. ’18 work in progress)

As n→∞, ηn converges weakly to η w.r.t. the metric ρ.

I Remark: The same convergence results w.r.t. the supremumdistance ρ hold for other dimensions:

I (Lawler 1991) For d ≥ 5, ∃cd > 0 s.t. γ(n)(cdn

2t)→ BM.

I (Lawler 1993) For d = 4, ∃c4 > 0 s.t.

γ(n)(c4n

2(log n)−13 t)→ BM.

I (Lawler-Viklund 2017) For d = 2, ∃c2 > 0 s.t.

γ(n)(c2n

54 t)→ γ where γ is SLE2 parametrized by

5/4-dimensional Minkowski content.

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Main Results

Theorem (Li - S. ’18 work in progress)

As n→∞, ηn converges weakly to η w.r.t. the metric ρ.

I Remark: The same convergence results w.r.t. the supremumdistance ρ hold for other dimensions:

I (Lawler 1991) For d ≥ 5, ∃cd > 0 s.t. γ(n)(cdn

2t)→ BM.

I (Lawler 1993) For d = 4, ∃c4 > 0 s.t.

γ(n)(c4n

2(log n)−13 t)→ BM.

I (Lawler-Viklund 2017) For d = 2, ∃c2 > 0 s.t.

γ(n)(c2n

54 t)→ γ where γ is SLE2 parametrized by

5/4-dimensional Minkowski content.

13 / 16

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Main Results

Theorem (Li - S. ’18 work in progress)

As n→∞, ηn converges weakly to η w.r.t. the metric ρ.

I Remark: The same convergence results w.r.t. the supremumdistance ρ hold for other dimensions:

I (Lawler 1991) For d ≥ 5, ∃cd > 0 s.t. γ(n)(cdn

2t)→ BM.

I (Lawler 1993) For d = 4, ∃c4 > 0 s.t.

γ(n)(c4n

2(log n)−13 t)→ BM.

I (Lawler-Viklund 2017) For d = 2, ∃c2 > 0 s.t.

γ(n)(c2n

54 t)→ γ where γ is SLE2 parametrized by

5/4-dimensional Minkowski content.

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Main Results

Theorem (Li - S. ’18 work in progress)

As n→∞, ηn converges weakly to η w.r.t. the metric ρ.

I Remark: The same convergence results w.r.t. the supremumdistance ρ hold for other dimensions:

I (Lawler 1991) For d ≥ 5, ∃cd > 0 s.t. γ(n)(cdn

2t)→ BM.

I (Lawler 1993) For d = 4, ∃c4 > 0 s.t.

γ(n)(c4n

2(log n)−13 t)→ BM.

I (Lawler-Viklund 2017) For d = 2, ∃c2 > 0 s.t.

γ(n)(c2n

54 t)→ γ where γ is SLE2 parametrized by

5/4-dimensional Minkowski content.

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Main Results

I Let λn := LE(S (2n)[0,∞)

)be the infinite loop-erased

random walk on 2−nZ3 started at the origin.

I For two continuous curves ζi : [0,∞)→ R3 (i = 1, 2), define

χ(ζ1, ζ2) =∞∑k=1

2−k max0≤t≤k

min{∣∣ζ1(t)− ζ2(t)

∣∣, 1}.Theorem (Li - S. ’18 work in progress)

There exists a random continuous curve λ : [0,∞)→ R3 such thatas n→∞, λn

(2βn ·

)converges weakly to λ with respect to the

metric χ.

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Main Results

I Let λn := LE(S (2n)[0,∞)

)be the infinite loop-erased

random walk on 2−nZ3 started at the origin.

I For two continuous curves ζi : [0,∞)→ R3 (i = 1, 2), define

χ(ζ1, ζ2) =∞∑k=1

2−k max0≤t≤k

min{∣∣ζ1(t)− ζ2(t)

∣∣, 1}.

Theorem (Li - S. ’18 work in progress)

There exists a random continuous curve λ : [0,∞)→ R3 such thatas n→∞, λn

(2βn ·

)converges weakly to λ with respect to the

metric χ.

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Main Results

I Let λn := LE(S (2n)[0,∞)

)be the infinite loop-erased

random walk on 2−nZ3 started at the origin.

I For two continuous curves ζi : [0,∞)→ R3 (i = 1, 2), define

χ(ζ1, ζ2) =∞∑k=1

2−k max0≤t≤k

min{∣∣ζ1(t)− ζ2(t)

∣∣, 1}.Theorem (Li - S. ’18 work in progress)

There exists a random continuous curve λ : [0,∞)→ R3 such thatas n→∞, λn

(2βn ·

)converges weakly to λ with respect to the

metric χ.

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What are η and λ?

A big challenging problem is to find a “nice” way to describe η andλ.

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Thank you for your attention!

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