Natural Parametrization for the Scaling Limit ofLoop-Erased Random Walk in Three Dimensions‡

Xinyi Li‡ , the University of Chicago −→ Peking University
joint work with Daisuke Shiraishi (Kyoto University)

August 2019, Fukuoka

1 / 19Chronological loop-erasure of a path
Loop-erased random walk (LERW) is the random simple pathobtained by erasing all loops chronologically from a simple random
walk path. In other words, we erase a loop immediately when it iscreated.

2 / 19Precise definition of loop erasure (but don’t look at it)
I Given a path λ = [λ(0), λ(1), · · · , λ(m)] ⊂ Zd , we define its

loop-erasure LE(λ) as follows. Let
s0 := max{t λ(t) = λ(0)},and for i ≥ 1, letsi := max{t λ(t) = λ(si−1 + 1)}.
We write n = min{i si = m}. Then we define LE(λ) by
LE(λ) = [λ(s0 ), λ(s1 ), · · · , λ(sn )].

3 / 19Precise definition of loop erasure (but don’t look at it)
I Given a path λ = [λ(0), λ(1), · · · , λ(m)] ⊂ Zd , we define its

loop-erasure LE(λ) as follows. Let
s0 := max{t λ(t) = λ(0)},and for i ≥ 1, letsi := max{t λ(t) = λ(si−1 + 1)}.
We write n = min{i si = m}. Then we define LE(λ) by
LE(λ) = [λ(s0 ), λ(s1 ), · · · , λ(sn )].
I However, to save brainpower (and time!!), just imagine a kindof self-repulsive motion on the lattice.

3 / 19Precise definition of loop erasure (but don’t look at it)
I Given a path λ = [λ(0), λ(1), · · · , λ(m)] ⊂ Zd , we define its

loop-erasure LE(λ) as follows. Let
s0 := max{t λ(t) = λ(0)},and for i ≥ 1, letsi := max{t λ(t) = λ(si−1 + 1)}.
We write n = min{i si = m}. Then we define LE(λ) by
LE(λ) = [λ(s0 ), λ(s1 ), · · · , λ(sn )].
I However, to save brainpower (and time!!), just imagine a kindof self-repulsive motion on the lattice.
I In fact, it is equivalent to a special case of Laplacian b−walk(again don’t search for the definition on Google for the
moment).

3 / 19Setup
For the sake of simplicity in this talk we always work under thefollowing setup.

I Let D be the open unit ball in Rd , and consider the rescaled
lattice N1 Zd . Let DN = D ∩ N1 Zd be the discretized unit ball.

4 / 19Setup
For the sake of simplicity in this talk we always work under thefollowing setup.

I Let D be the open unit ball in Rd , and consider the rescaled
lattice N1 Zd . Let DN = D ∩ N1 Zd be the discretized unit ball.
I Let SN be the simple random walk from 0 stopped at exiting

DN , and write γN = LE(SN ) for the LERW in DN .

4 / 19Setup
For the sake of simplicity in this talk we always work under thefollowing setup.

I Let D be the open unit ball in Rd , and consider the rescaled
lattice N1 Zd . Let DN = D ∩ N1 Zd be the discretized unit ball.
I Let SN be the simple random walk from 0 stopped at exiting

DN , and write γN = LE(SN ) for the LERW in DN .
I For x ∈ D, let xN be its discretization. LetaN,x = P xN ∈ γN

be the one-point function (or Green’s function) of LERW.
4 / 19Behaviour of LERW on Zd in different dimensions
I LERW on Zd enjoys a Gaussian behavior if d is large.
I
I

The scaling limit of LERW is the Brownian motion if d ≥ 4.
Considerthe one-point function aN,x for a fixed x ∈ D. It is
O N 2−d for d ≥ 5 and O N −2 (log N)−1/3 for d = 4.

1

the Schramm-Loewner evolution with parameter 2, which is a random
fractal with Hausdorff dimension 5/4.
5 / 19Behaviour of LERW on Zd in different dimensions
I LERW on Zd enjoys a Gaussian behavior if d is large.

The scaling limit of LERW is the Brownian motion if d ≥ 4.
Considerthe one-point function aN,x for a fixed x ∈ D. It is
O N 2−d for d ≥ 5 and O N −2 (log N)−1/3 for d = 4.
I An intuitive explanation: in high dimensions, it is very difficult
for SRW to intersect itself, hence not much is erased.
I
I

1

the Schramm-Loewner evolution with parameter 2, which is a random
fractal with Hausdorff dimension 5/4.
5 / 19Behaviour of LERW on Zd in different dimensions
I LERW on Zd enjoys a Gaussian behavior if d is large.

The scaling limit of LERW is the Brownian motion if d ≥ 4.
Considerthe one-point function aN,x for a fixed x ∈ D. It is
O N 2−d for d ≥ 5 and O N −2 (log N)−1/3 for d = 4.
I An intuitive explanation: in high dimensions, it is very difficult
for SRW to intersect itself, hence not much is erased.
I
I

I When d = 2, the scaling limit of LERW is SLE2 1(Lawler-Schramm-Werner).

1

the Schramm-Loewner evolution with parameter 2, which is a random
fractal with Hausdorff dimension 5/4.
5 / 19Behaviour of LERW on Zd in different dimensions
I LERW on Zd enjoys a Gaussian behavior if d is large.

The scaling limit of LERW is the Brownian motion if d ≥ 4.
Considerthe one-point function aN,x for a fixed x ∈ D. It is
O N 2−d for d ≥ 5 and O N −2 (log N)−1/3 for d = 4.
I An intuitive explanation: in high dimensions, it is very difficult
for SRW to intersect itself, hence not much is erased.
I
I

I When d = 2, the scaling limit of LERW is SLE2 1(Lawler-Schramm-Werner). Furthermore, we have thefollowing asymptotics of the one-point function:
aN,x = cx N −3/4 1 + O(N −c ) . (Beneš-Lawler-Viklund)1

the Schramm-Loewner evolution with parameter 2, which is a random
fractal with Hausdorff dimension 5/4.
5 / 19

A simulation for 2D LERW

Picture credit: Fredrik Viklund.
6 / 19

Previously known facts on 3D LERWI In contrast to other dimensions, relatively little is known for

LERW in three dimensions. The fundamental difficulty is that
we have no a priori description of the scaling limit.

7 / 19

Previously known facts on 3D LERWI In contrast to other dimensions, relatively little is known for

LERW in three dimensions. The fundamental difficulty is that
we have no a priori description of the scaling limit.
I Let KN stand for the trace of γN (a broken line), understoodas a random subset of D. Kozma proved in ’07 that there
exists K, a random subset of D, s.t.
w
K2n −→ K w.r.t. the Hausdorff metric dH .

7 / 19

Previously known facts on 3D LERWI In contrast to other dimensions, relatively little is known for

LERW in three dimensions. The fundamental difficulty is that
we have no a priori description of the scaling limit.
I Let KN stand for the trace of γN (a broken line), understoodas a random subset of D. Kozma proved in ’07 that there
exists K, a random subset of D, s.t.
w
K2n −→ K w.r.t. the Hausdorff metric dH .
Note that for technically fundamental reasons the convergenceis only along dyadic scales.
I A.s., K is indeed a simple curve (Sapozhnikov-Shiraishi, ’15).
I

7 / 19

Previously known facts on 3D LERWI In contrast to other dimensions, relatively little is known for

LERW in three dimensions. The fundamental difficulty is that
we have no a priori description of the scaling limit.
I Let KN stand for the trace of γN (a broken line), understoodas a random subset of D. Kozma proved in ’07 that there
exists K, a random subset of D, s.t.
w
K2n −→ K w.r.t. the Hausdorff metric dH .
Note that for technically fundamental reasons the convergenceis only along dyadic scales.
I A.s., K is indeed a simple curve (Sapozhnikov-Shiraishi, ’15).
I

I Shiraishi proved in ’13 that∃ α ∈ [ 31 , 1) such that aN,x = N −1−α+o(1) .

7 / 19

Previously known facts on 3D LERWI In contrast to other dimensions, relatively little is known for

LERW in three dimensions. The fundamental difficulty is that
we have no a priori description of the scaling limit.
I Let KN stand for the trace of γN (a broken line), understoodas a random subset of D. Kozma proved in ’07 that there
exists K, a random subset of D, s.t.
w
K2n −→ K w.r.t. the Hausdorff metric dH .
Note that for technically fundamental reasons the convergenceis only along dyadic scales.
I A.s., K is indeed a simple curve (Sapozhnikov-Shiraishi, ’15).
I

I Shiraishi proved in ’13 that∃ α ∈ [ 31 , 1) such that aN,x = N −1−α+o(1) .
IThis implies (with a lot of work, see Shiraishi ’16) that theHausdorff dimension of LERW and K is equal to β = 2 − α.

7 / 19

Previously known facts on 3D LERWI In contrast to other dimensions, relatively little is known for

LERW in three dimensions. The fundamental difficulty is that
we have no a priori description of the scaling limit.
I Let KN stand for the trace of γN (a broken line), understoodas a random subset of D. Kozma proved in ’07 that there
exists K, a random subset of D, s.t.
w
K2n −→ K w.r.t. the Hausdorff metric dH .
Note that for technically fundamental reasons the convergenceis only along dyadic scales.
I A.s., K is indeed a simple curve (Sapozhnikov-Shiraishi, ’15).
I

I Shiraishi proved in ’13 that∃ α ∈ [ 31 , 1) such that aN,x = N −1−α+o(1) .

This implies (with a lot of work, see Shiraishi ’16) that theHausdorff dimension of LERW and K is equal to β = 2 − α.
I Numerical experiments and field-theoretical prediction suggestthat β = 1.62 ± 0.01, but there is no reason to believe that β
is any nice number.
I

7 / 19Results in 3D

Theorem (L.-Shiraishi ’18)
There exist universal δ > 0 and c : D \ {0} → R+ such that∀n ∈ Z+ and x ∈ D \ {0},i
−(1+α) h
4a2n ,x = P x2n ∈ γ2n = c(x) 2n
1 + dx−δ O 2−δnwhere dx = min |x|, 1 − |x| .

8 / 19Results in 3D

Theorem (L.-Shiraishi ’18)
There exist universal δ > 0 and c : D \ {0} → R+ such that∀n ∈ Z+ and x ∈ D \ {0},i
−(1+α) h
4a2n ,x = P x2n ∈ γ2n = c(x) 2n
1 + dx−δ O 2−δnwhere dx = min |x|, 1 − |x| .

Remark
Our result also only works on dyadic scales, for relatively lessfundamentally technical reasons. However, recently, we have foundsome tricks to extend the above theorem to any mesh size.

8 / 19Non-intersection probability
I In parallel to the one-point function estimates, we also

investigate the following non-intersection probability of simple
random walk and LERW.

9 / 19Non-intersection probability
I In parallel to the one-point function estimates, we also

investigate the following non-intersection probability of simple
random walk and LERW.

I Recall that SN is the random walk from 0 stopped at exiting
DN and γN = LE(SN ). Let SN0 be an i.i.d. copy of SN , endingat time T 0 . We are interested in4Es(N) = P γN ∩ SN0 [1, T 0 ] = ∅the probability that LERW γN and simple random walk SN0 do
not intersect except at the origin.
9 / 19Non-intersection probability
I In parallel to the one-point function estimates, we also

investigate the following non-intersection probability of simple
random walk and LERW.
I Recall that SN is the random walk from 0 stopped at exiting
DN and γN = LE(SN ). Let SN0 be an i.i.d. copy of SN , endingat time T 0 . We are interested in


4
0
0
Es(N) = P γN ∩ SN [1, T ] = ∅the probability that LERW γN and simple random walk SN0 do
not intersect except at the origin.

Theorem (L.-Shiraishi 18’)
There exist c > 0 and δ > 0 such that for all n ∈ Z+ ,
Es(2n ) = c2−αn 1 + O(2−δn ) .
10 / 19

Proof strategy for the non-intersection probability
Theorem (L.-Shiraishi ’18)
There exist c > 0 and δ > 0 such that for all n ∈ Z+ ,
Es(2n ) = c2−αn 1 + O(2−δn ) .
I Replacement of discrete objects by continuous objects:- SRW −→ Brownian motion;- LERW −→ K, the scaling limit provided by Kozma.
I However, lattice effects around the origin and error bounds

from Kozma’s result do not allow us to do this directly.

11 / 19

Proof strategy for the non-intersection probability
Theorem (L.-Shiraishi ’18)
There exist c > 0 and δ > 0 such that for all n ∈ Z+ ,
Es(2n ) = c2−αn 1 + O(2−δn ) .
I Replacement of discrete objects by continuous objects:- SRW −→ Brownian motion;- LERW −→ K, the scaling limit provided by Kozma.
I However, lattice effects around the origin and error bounds

from Kozma’s result do not allow us to do this directly.
I Solution: Replace the starting points by two points far away

through conditioning and modifying a recent coupling result
from Greg Lawler (The infinite two-sided loop-erased random
walk, arXiv:1802.06667).
11 / 19

“Asymptotically Markov” property of LERWI The input of the argument is that LERW is in some sense

asymptotically independent of its initial configuration.

12 / 19

“Asymptotically Markov” property of LERWI The input of the argument is that LERW is in some sense

asymptotically independent of its initial configuration.
I Let γ, γ 0 be LERWs starting from different locations in B().

I Then it is possible to couple γ and γ 0 such that their “outer”

part agree with high probability.

12 / 19

“Asymptotically Markov” property of LERWI The input of the argument is that LERW is in some sense

asymptotically independent of its initial configuration.
I Let γ, γ 0 be LERWs starting from different locations in B().

I Then it is possible to couple γ and γ 0 such that their “outer”

part agree with high probability.
I This allows us to separate the “local” behavior and the“global” behaviour of LERW.
12 / 19

Let bn = Es(2n )/Es(2n−1 ). Want to show bn+1 /bn = 1 + O(2−δn ).

13 / 19

Convergence of 3D LERW to its scaling limit in naturalparametrization
I Recall that DN is the discretized unit ball with mesh size 1/N,
and let γN is the LERW on N −1 Z3 , from the origin andstopped at exiting DN . Recall that β ∈ (1, 5/3] is theHausdorff dimension of K.

14 / 19

Convergence of 3D LERW to its scaling limit in naturalparametrization
I Recall that DN is the discretized unit ball with mesh size 1/N,
and let γN is the LERW on N −1 Z3 , from the origin andstopped at exiting DN . Recall that β ∈ (1, 5/3] is theHausdorff dimension of K.
I We writeµN := N −βPx∈γN δx

for the renormalized occupation measure of γN .

14 / 19

Convergence of 3D LERW to its scaling limit in naturalparametrization
I Recall that DN is the discretized unit ball with mesh size 1/N,
and let γN is the LERW on N −1 Z3 , from the origin andstopped at exiting DN . Recall that β ∈ (1, 5/3] is theHausdorff dimension of K.
I We writeµN := N −βPx∈γN δx

for the renormalized occupation measure of γN .

Theorem (L.-Shiraishi ’18)
As n → ∞,w(γ2n ; µ2n ) −→ (K; µ∗ ).
Moreover, µ∗ is measurable w.r.t. K.
14 / 19

Convergence of 3D LERW to its scaling limit in naturalparametrization

Theorem (L.-Shiraishi ’18)
As n → ∞,w(γ2n ; µ2n ) −→ (K; µ∗ )in the product topology of (H(D); dHaus ) and the topology of the
weak convergence on M(D). Moreover, µ∗ is measurable w.r.t. K.
HereI H(D) is the space of non-empty compact subsets of D;
I dHaus is the the Hausdorff metric on H(D);I M(D) is the space of finite measures on D.

15 / 19

Convergence of 3D LERW to its scaling limit in naturalparametrization
I The previous theorem allows as to upgrade the convergence tothe convergence of continuous curves in the uniform norm.

16 / 19

Convergence of 3D LERW to its scaling limit in naturalparametrization
I The previous theorem allows as to upgrade the convergence tothe convergence of continuous curves in the uniform norm.
I Namely, let(C(D), d∞ ) be the space of continuous curves λ : [0, tλ ] → Dequipped with the uniform normd∞ (λ1 , λ2 ) = max0≤s≤1 λ1 (stλ1 ) − λ2 (stλ2 ) + tλ1 − tλ2 ;I η (t) := γ (N β t) be the properly time-rescaled LERW as anNNelement of C(D);I

16 / 19

Convergence of 3D LERW to its scaling limit in naturalparametrization
I The previous theorem allows as to upgrade the convergence tothe convergence of continuous curves in the uniform norm.
I Namely, let(C(D), d∞ ) be the space of continuous curves λ : [0, tλ ] → Dequipped with the uniform normd∞ (λ1 , λ2 ) = max0≤s≤1 λ1 (stλ1 ) − λ2 (stλ2 ) + tλ1 − tλ2 ;I η (t) := γ (N β t) be the properly time-rescaled LERW as anNNelement of C(D);I η ∗ be the curve obtained through parametrizing K by µ∗ .

I16 / 19

Convergence of 3D LERW to its scaling limit in naturalparametrization
I The previous theorem allows as to upgrade the convergence tothe convergence of continuous curves in the uniform norm.
I Namely, let(C(D), d∞ ) be the space of continuous curves λ : [0, tλ ] → Dequipped with the uniform normd∞ (λ1 , λ2 ) = max0≤s≤1 λ1 (stλ1 ) − λ2 (stλ2 ) + tλ1 − tλ2 ;I η (t) := γ (N β t) be the properly time-rescaled LERW as anNNelement of C(D);I η ∗ be the curve obtained through parametrizing K by µ∗ .

I

Theorem (L.-Shiraishi ’18)w

As n → ∞, η2n → η ∗ with respect to the topology of (C(D), d∞ ).

RemarkAs γ2n is traversed at a constant speed, what we obtain is aconvergence in the natural parametrization. We conjecture that µ∗
can also be given through the Minkowski content of K.
16 / 19

L2 -approximation in the style of Garban-Pete-Schramm

17 / 19

Bibliography
I X. Li and D. Shiraishi.

One-point function estimates for loop-erased random walk inthree dimensions.
Preprint, available at arXiv:1807.00541, 39 pages, 2 figures.
I X. Li and D. Shiraishi.

Natural parametrization for the scaling limit of loop-erased
random walk in three dimensions. Preprint, available atarXiv:1811.11685, 74 pages, 3 figures.

Thank you for your attention!
18 / 19

-Mashikoyaki pottery made by Shoji HAMADA et al., ca. 1929.

19 / 19