2007/01/01 by Gady Kozma · 4 citations
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Mathematical Dynamics and Fractals #Mathematics #Scaling limit #Scaling #Universality (dynamical systems) #Random walk #Limit (mathematics) #Invariant (physics) #Statistical physics #Mathematical analysis #Pure mathematics #Mathematical physics #Geometry #Statistics #Physics #Quantum mechanics
paper · pdf · doi:10.1007/s11511-007-0018-8
openalex publication_date 2007/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
the scaling limit of loop-erased random walk in three dimensions This argument and the precise meaning of " " are contained in Lemma 5.7.To make this argument work for γ 3 , we have to use the symmetry of loop-erased random walk.Now, if γ 2 is large, then we are in the situation that was coined in [S00] a "quasiloop", namely two close points such that the path passes through both but goes a long way in-between.In two dimensions it is possible to show that LE(R) has few quasiloops, by tracing the process that creates them and seeing that it necessitates that a random walk that starts quite close to a loop-erased random walk will avoid hitting it for a long while.This, however, contradicts the discrete Beurling projection principle (see [K87]) that states that a random walk starting near any path has a high probability to intersect it.( 1 See [S00, Lemma 2.1] or [K, Lemma 18].Unfortunately this argument no longer holds in three dimensions.A random walk starting, say, at a distance n from a straight line, has a reasonable probability to intersect it only after extending to a distance of n 2 , and even after n 2 the probability to not intersecting the line only decreases logarithmically.In other words, in three dimensions not all paths are "hittable", and we have to show specifically that a loop-erased random walk is, using facts about its structure.This will be done in §4 and we shall show that the probability that a random walk starting near a loop-erased random walk will avoid hitting it decreases like a power law.This will allow us to repeat the above argumentation in three dimensions, show that there are no quasi-loops and hence that LERW is similar on our G 1 and G 2 .The proof that a loop-erased random walk is hittable is based on searching for (local) cut times.A cut time for a random walk R is a timeThe number of cut times is connected to the non-intersection exponent: by considering the parts of R up to t and from t on as two random walks, and reversing the first part, we see that it is important to estimate the probability that two random walks of length t will not intersect.This is ≈t -2ξ [L96b], where ξ is the famous non-intersection exponent of Brownian motion.See §3.2 for a description of this topic.Heuristically speaking, a set is "hittable" if its Hausdorff dimension is >1, and the set of cut times has dimension 2-ξ, so the argument terminates by the well-known fact that ξ<1 in three dimensions [BL90b].( 1 ) Kesten's theorem is stronger, and claims that the minimum probability is achieved, up to a constant, for a straight line, in which case it can be estimated directly.However we will not need this level of accuracy.1 8 , λ, G , C 8 there are two disjoint simple paths γ i starting from v i and ending at w i satisfying w i ∈B(v 1 , μ)\B v 1 , 7 8 μ and B w i , 7 8 μ ∩γ 3-i =∅.