One-Point Function Estimates for Loop-Erased Random Walk in Three Dimensions

One-point function estimates for Loop-erased Random Walk in three dimensions

Xinyi Li, the University of Chicago

joint work with Daisuke Shiraishi (Kyoto)

July 2018, Chengdu

1 / 15 I The model was originally introduced by Greg Lawler in 1980 as an alternative for the model of Self-avoiding walk (SAW).

I Although for the most interesting case (i.e. low dimensions), LERW and SAW belongs to diﬀerent universality classes, LERW as a model on itself attracts interests of both probabilists and statistical physicists and there is a huge literature on LERW from both mathematics and physics.

A brief introduction to LERW

I Loop-erased random walk (LERW) is the random simple path obtained by erasing all loops chronologically from a simple random walk path.

2 / 15 I Although for the most interesting case (i.e. low dimensions), LERW and SAW belongs to diﬀerent universality classes, LERW as a model on itself attracts interests of both probabilists and statistical physicists and there is a huge literature on LERW from both mathematics and physics.

A brief introduction to LERW

I Loop-erased random walk (LERW) is the random simple path obtained by erasing all loops chronologically from a simple random walk path.

I The model was originally introduced by Greg Lawler in 1980 as an alternative for the model of Self-avoiding walk (SAW).

2 / 15 A brief introduction to LERW

I Loop-erased random walk (LERW) is the random simple path obtained by erasing all loops chronologically from a simple random walk path.

I The model was originally introduced by Greg Lawler in 1980 as an alternative for the model of Self-avoiding walk (SAW).

2 / 15 I Lecture notes on LERW: Topics in loop measures and the loop-erased walk, by Greg Lawler. Probab. Surveys, 15:28-101, 2018.

A brief introduction to LERW

I LERW also has a strong connection with other models in statistical physics, e.g. the uniform spanning tree (UST), Potts model, Abelian sandpile model, O(n) model, etc...

3 / 15 A brief introduction to LERW

I LERW also has a strong connection with other models in statistical physics, e.g. the uniform spanning tree (UST), Potts model, Abelian sandpile model, O(n) model, etc...

I Lecture notes on LERW: Topics in loop measures and the loop-erased walk, by Greg Lawler. Probab. Surveys, 15:28-101, 2018.

3 / 15 and for i 1, let ≥ si := 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 In other words, chronological loop-erasure means erasing loops immediately when it is created.

Chronological loop-erasure of a path

d I Given a path λ = [λ(0), λ(1), , λ(m)] Z , we deﬁne its loop-erasure LE(λ) as follows.··· Let ⊂

s0 := max t λ(t) = λ(0) , { }

4 / 15

We write n = min i si = m . Then we deﬁne LE(λ) by { } LE(λ) = [λ(s0), λ(s1), , λ(sn)]. ··· I In other words, chronological loop-erasure means erasing loops immediately when it is created.

Chronological loop-erasure of a path

d I Given a path λ = [λ(0), λ(1), , λ(m)] Z , we deﬁne its loop-erasure LE(λ) as follows.··· Let ⊂

s0 := max t λ(t) = λ(0) , { } and for i 1, let ≥ si := max t λ(t) = λ(si−1 + 1) . { }

4 / 15 I In other words, chronological loop-erasure means erasing loops immediately when it is created.

Chronological loop-erasure of a path

d I Given a path λ = [λ(0), λ(1), , λ(m)] Z , we deﬁne its loop-erasure LE(λ) as follows.··· Let ⊂

s0 := max t λ(t) = λ(0) , { } and for i 1, let ≥ si := 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)]. ···

4 / 15 Chronological loop-erasure of a path

d I Given a path λ = [λ(0), λ(1), , λ(m)] Z , we deﬁne its loop-erasure LE(λ) as follows.··· Let ⊂

s0 := max t λ(t) = λ(0) , { } and for i 1, let ≥ si := 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 In other words, chronological loop-erasure means erasing loops immediately when it is created. 4 / 15 Picture credit: Fredrik Viklund.

5 / 15 I However, to make things precise, we have to specify the stopping time of this SRW. A very common scenario is the 3 following: let D be a ﬁnite subset of Z and let S be the SRW started from x stopped at the ﬁrst exit of D, when we call LE(S) the LERW from x stopped at exiting D.

I Other possible description of LERW: Poisson loop soup, Wilson’s algorithm for UST, etc..

LERW

I In general, we use loop-erased random walk loosely to refer to the random self-avoiding path obtained by loop-erasing from simple random walk.

6 / 15 I Other possible description of LERW: Poisson loop soup, Wilson’s algorithm for UST, etc..

LERW

I In general, we use loop-erased random walk loosely to refer to the random self-avoiding path obtained by loop-erasing from simple random walk.

I However, to make things precise, we have to specify the stopping time of this SRW. A very common scenario is the 3 following: let D be a ﬁnite subset of Z and let S be the SRW started from x stopped at the ﬁrst exit of D, when we call LE(S) the LERW from x stopped at exiting D.

6 / 15 LERW

I In general, we use loop-erased random walk loosely to refer to the random self-avoiding path obtained by loop-erasing from simple random walk.

I Other possible description of LERW: Poisson loop soup, Wilson’s algorithm for UST, etc..

6 / 15 Let D be a ﬁnite subset of 3 Z and let λ be a LERW stopped at exiting D. Write λ1 for a simple path in D from the starting point of λ and let µ be the simple random walk started from the ending point of λ1, stopped at exiting D, conditioned to avoid λ1. Then, law λ1 LE(µ) = λ conditioned on its beginning part is λ1. ⊕

Domain Markov property

I LERW is not a Markov process, however, it satisﬁes the following domain Markov property.

7 / 15 Domain Markov property

I LERW is not a Markov process, however, it satisﬁes the following domain Markov property. Let D be a ﬁnite subset of 3 Z and let λ be a LERW stopped at exiting D. Write λ1 for a simple path in D from the starting point of λ and let µ be the simple random walk started from the ending point of λ1, stopped at exiting D, conditioned to avoid λ1. Then, law λ1 LE(µ) = λ conditioned on its beginning part is λ1. ⊕

7 / 15 I The scaling limit of LERW is Brownian motion d 4. ≥ I The probability of LERW from the origin hitting a given point d x 2−d x Z (denoted by pd ) is of order x for d 5 and x∈−2(log x )−1/3 for d = 4 . | | ≥ | | | | I Intuitive explanation: in high dimensions, it is very diﬃcult for SRW to intersect itself.

I When d = 2, Lawler, Schramm and Werner showed that the scaling limit of LERW is the Schramm-Loewner evolution. Furthermore, the best one-point function estimate: px c x −3/4. (Beneˇs-Lawler-Viklund). 2 ∼ | |

LERW on Zd in diﬀerent dimensions

d I LERW on Z enjoys a Gaussian behavior if d is large.

8 / 15 I The probability of LERW from the origin hitting a given point d x 2−d x Z (denoted by pd ) is of order x for d 5 and x∈−2(log x )−1/3 for d = 4 . | | ≥ | | | | I Intuitive explanation: in high dimensions, it is very diﬃcult for SRW to intersect itself.

LERW on Zd in diﬀerent dimensions

d I LERW on Z enjoys a Gaussian behavior if d is large. I The scaling limit of LERW is Brownian motion d 4. ≥

8 / 15 I Intuitive explanation: in high dimensions, it is very diﬃcult for SRW to intersect itself.

LERW on Zd in diﬀerent dimensions

d I LERW on Z enjoys a Gaussian behavior if d is large. I The scaling limit of LERW is Brownian motion d 4. ≥ I The probability of LERW from the origin hitting a given point d x 2−d x Z (denoted by pd ) is of order x for d 5 and x∈−2(log x )−1/3 for d = 4 . | | ≥ | | | |

8 / 15 I When d = 2, Lawler, Schramm and Werner showed that the scaling limit of LERW is the Schramm-Loewner evolution. Furthermore, the best one-point function estimate: px c x −3/4. (Beneˇs-Lawler-Viklund). 2 ∼ | |

LERW on Zd in diﬀerent dimensions

8 / 15 LERW on Zd in diﬀerent dimensions

8 / 15 I Shiraishi proved in 2013 that α [ 1 , 1) such that px x −1−α. ∃ ∈ 3 3 ≈ | | I This implies that the dimension of LERW or its scaling limit is equal to 2 α. − I Numerical experiments and ﬁeld-theoretical prediction suggest that 2 α = 1.62 0.01, but there is no reason to believe that α−is any nice± number.

Previously known facts on 3D LERW

I In contrast to other dimensions, relatively little is known for LERW in three dimensions. Gady Kozma proved in 2007 the existence of the scaling limit, however, we have no nice tools like SLE to describe this limit.

9 / 15 I This implies that the dimension of LERW or its scaling limit is equal to 2 α. − I Numerical experiments and ﬁeld-theoretical prediction suggest that 2 α = 1.62 0.01, but there is no reason to believe that α−is any nice± number.

Previously known facts on 3D LERW

I Shiraishi proved in 2013 that α [ 1 , 1) such that px x −1−α. ∃ ∈ 3 3 ≈ | |

9 / 15 I Numerical experiments and ﬁeld-theoretical prediction suggest that 2 α = 1.62 0.01, but there is no reason to believe that α−is any nice± number.

Previously known facts on 3D LERW

I Shiraishi proved in 2013 that α [ 1 , 1) such that px x −1−α. ∃ ∈ 3 3 ≈ | | I This implies that the dimension of LERW or its scaling limit is equal to 2 α. −

9 / 15 Previously known facts on 3D LERW

I Shiraishi proved in 2013 that α [ 1 , 1) such that px x −1−α. ∃ ∈ 3 3 ≈ | | I This implies that the dimension of LERW or its scaling limit is equal to 2 α. − I Numerical experiments and ﬁeld-theoretical prediction suggest that 2 α = 1.62 0.01, but there is no reason to believe that α−is any nice± number.

9 / 15 I Main motivation: convergence of 3D LERW to its scaling limit in natural parametrization (work in preparation). 3 I Take a point x = 0 from the unit open ball in R . We 6 3 consider the simple random walk S on Z started at the origin and write T n for the ﬁrst time that S exits from a ball of n radius2 centered at 0. Let xn be the one of the nearest point n 3 from 2 x among Z . Finally, we set

n an,x = P xn LE(S[0, T ]) ∈

for the probability that LERW hits xn. n I Why are we considering dyadic scales (i.e. 2 ) only? Gady Kozma only provided convergence to scaling limit along dyadic scales.

Results

x I We now improve the asymptotics of p3 .

10 / 15 3 I Take a point x = 0 from the unit open ball in R . We 6 3 consider the simple random walk S on Z started at the origin and write T n for the ﬁrst time that S exits from a ball of n radius2 centered at 0. Let xn be the one of the nearest point n 3 from 2 x among Z . Finally, we set

n an,x = P xn LE(S[0, T ]) ∈

for the probability that LERW hits xn. n I Why are we considering dyadic scales (i.e. 2 ) only? Gady Kozma only provided convergence to scaling limit along dyadic scales.

Results

x I We now improve the asymptotics of p3 . I Main motivation: convergence of 3D LERW to its scaling limit in natural parametrization (work in preparation).

10 / 15 Finally, we set

n an,x = P xn LE(S[0, T ]) ∈

for the probability that LERW hits xn. n I Why are we considering dyadic scales (i.e. 2 ) only? Gady Kozma only provided convergence to scaling limit along dyadic scales.

Results

x I We now improve the asymptotics of p3 . I Main motivation: convergence of 3D LERW to its scaling limit in natural parametrization (work in preparation). 3 I Take a point x = 0 from the unit open ball in R . We 6 3 consider the simple random walk S on Z started at the origin and write T n for the ﬁrst time that S exits from a ball of n radius2 centered at 0. Let xn be the one of the nearest point n 3 from 2 x among Z .

10 / 15 n I Why are we considering dyadic scales (i.e. 2 ) only? Gady Kozma only provided convergence to scaling limit along dyadic scales.

Results

n an,x = P xn LE(S[0, T ]) ∈

for the probability that LERW hits xn.

10 / 15 Results

n an,x = P xn LE(S[0, T ]) ∈

for the probability that LERW hits xn. n I Why are we considering dyadic scales (i.e. 2 ) only? Gady Kozma only provided convergence to scaling limit along dyadic scales.

10 / 15 Results

Theorem (L.-Shiraishi 18’)

There exist universal c > 0, δ > 0 and cx > 0 depending only on + x D 0 such that n Z and x D 0 , ∈ \{ } ∀ ∈ ∈ \{ } 4 n −(1+α)nn −c −δno an,x = P xn LE(S[0, T ]) = cx 2 1 + O d 2 ∈ x

where dx = min x , 1 x . {| | − | |}

11 / 15 We are interested in 4 Es(n) = P LE(S1[0, T 1]) S2[1, T 2] = n ∩ n ∅ 1 1 the probability that LERW LE(S [0, Tn ]) and simple random 2 2 walk S [1, Tn ] do not intersect.

Results

12 / 15 Results

I As a crucial step for the main theorem, we need to estimate the following non-intersection probability of simple random walk and LERW. 1 2 3 I Let S and S be independent simple random walks on Z i i started at the origin. We write Tn for the ﬁrst time that S exits from a ball of radius n. We are interested in 4 Es(n) = P LE(S1[0, T 1]) S2[1, T 2] = n ∩ n ∅ 1 1 the probability that LERW LE(S [0, Tn ]) and simple random 2 2 walk S [1, Tn ] do not intersect.

12 / 15 This theorem immediately implies the follow asymptotics on non-interesection probabilities on general scales and lengths of LERW. Corollary 1 1 Write Mn = len LE(S [0, Tn ]) for the length of the LERW considered above. Then, it follows that for n m 1, ≥ ≥ −α 2−α Mn Es(n) n ; E(Mn) n ; is tight. n2−α

Results

Theorem (L.-Shiraishi 18’) + There exist c > 0 and δ > 0 such that for all n Z , ∈ Es(2n) = c2−αn1 + O(2−δn).

13 / 15 Results

Theorem (L.-Shiraishi 18’) + There exist c > 0 and δ > 0 such that for all n Z , ∈ Es(2n) = c2−αn1 + O(2−δn).

This theorem immediately implies the follow asymptotics on non-interesection probabilities on general scales and lengths of LERW. Corollary 1 1 Write Mn = len LE(S [0, Tn ]) for the length of the LERW considered above. Then, it follows that for n m 1, ≥ ≥ −α 2−α Mn Es(n) n ; E(Mn) n ; is tight. n2−α

13 / 15 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 (w and v in the following picture)

γn γ (t ) λn(t1,q) n 1,q

λn(t1,q) O w v O γn (1 3q)n γn(t1,q) B(2 − ) λn (1 q)n λn B(2 − ) (1 q)n B(2 − )

B(2n) B(2n) through conditioning and modifying a recent coupling result from Greg Lawler (The inﬁnite two-sided loop-erased random walk, arXiv:1802.06667).

Proof strategy for the second theorem

I Replacement of discrete objects by continuous objects: - SRW Brownian motion; - LERW−→ scaling limit provided by Kozma. −→

14 / 15 I Solution: Replace the starting points by two points far away (w and v in the following picture)

γn γ (t ) λn(t1,q) n 1,q

λn(t1,q) O w v O γn (1 3q)n γn(t1,q) B(2 − ) λn (1 q)n λn B(2 − ) (1 q)n B(2 − )

B(2n) B(2n) through conditioning and modifying a recent coupling result from Greg Lawler (The inﬁnite two-sided loop-erased random walk, arXiv:1802.06667).

Proof strategy for the second theorem

I Replacement of discrete objects by continuous objects: - SRW Brownian motion; - LERW−→ 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.

14 / 15 through conditioning and modifying a recent coupling result from Greg Lawler (The inﬁnite two-sided loop-erased random walk, arXiv:1802.06667).

Proof strategy for the second theorem

γn γ (t ) λn(t1,q) n 1,q

λn(t1,q) O w v O γn (1 3q)n γn(t1,q) B(2 − ) λn (1 q)n λn B(2 − ) (1 q)n B(2 − )

B(2n) B(2n)

14 / 15 Proof strategy for the second theorem

γn γ (t ) λn(t1,q) n 1,q

λn(t1,q) O w v O γn (1 3q)n γn(t1,q) B(2 − ) λn (1 q)n λn B(2 − ) (1 q)n B(2 − )

B(2n) B(2n) through conditioning and modifying a recent coupling result from Greg Lawler (The inﬁnite two-sided loop-erased random walk, arXiv:1802.06667). 14 / 15 Bibliography

I X. Li and D. Shiraishi. One-point function estimates for loop-erased random walk in three 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. In preparation.

Thank you for your attention!

15 / 15