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

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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). 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. 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) : f g We write n = min i si = m . Then we define LE(λ) by f g 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) ; f g 4 / 15 We write n = min i si = m . Then we define LE(λ) by f g 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) ; f g and for i 1, let ≥ si := max t λ(t) = λ(si−1 + 1) : f g 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) ; f g and for i 1, let ≥ si := max t λ(t) = λ(si−1 + 1) : f g We write n = min i si = m . Then we define LE(λ) by f g 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) ; f g and for i 1, let ≥ si := max t λ(t) = λ(si−1 + 1) : f g We write n = min i si = m . Then we define LE(λ) by f g 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 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.. 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 x2−2(log x )−1=3 for d = 4 . j j ≥ j j j j 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 ∼ j j 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 x2−2(log x )−1=3 for d = 4 . j j ≥ j j j j 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 ∼ j j 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. I When d = 2, Lawler, Schramm and Werner showed that the scaling limit of LERW is the Schramm-Loewner evolution.
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