On The Growth Exponent Of High-dimensional Loop-erased Random Walk

Let S be a simple random walk(SRW)on the lattice Zd.By chronologically erasing loops from S,we get the so-called loop-erased random walk(LERW).LERW exhibits many good properties of SRW and has close relationship with other statistical models,thus having been extensively studied.For instance,the distribution of random spanning tree constructed via LERW is the same as uniform spanning tree.Let ?n be S's first exit time of the d-dimensional ball of radius n.Consider the loop-erasure of S[0,?n]denoted by LE(S[0,?n]),we are interested in its length Mnd=len(LE(S[0,?n])).Although the distribution of Mmd is unknown,its expectation E[Mmd]has been well studied:if (?) we say the d-dimensional LERW's growth exponent is ?d.It is a classical result by Lawler that any high-dimensional(d>4)LERW's growth exponent is 2.Masson's lemma relates the distribution of points on the path of LERW with the non-intersection probability of SRW and LERW.He then gives the separation lemma between SRW and LERW by which he proves that the growth exponent of planar LERW is 5/4 thanks to the scaling limit of planar LERW.Using a similar technique,Shiraishi,utilizing ergodic theory,proves the existence of the growth exponent of LERW in three dimensions.Our first work is to extend the separation lemma between SRW and LERW to the case of high dimensions.By an induction method,we can prove more strict separation lemma and reverse separation lemma in high dimensions.As an application of the separation lemma,we give a new proof that high-dimensional LERW's exponent is 2 according to Masson's method.Based on Masson's work,Barlow et al.further give an estimation on the k-th moment of Mn2.They make use of this estimation to establish exponential tail bounds for planar LERWs,which is evidently a strong tool to further study the distribution of Mn2.Shiraishi gives a similar result in the case of three dimensions.Our second work is to study those properties in high-dimensions,which requires us to re-estimate the k-th moment of Mnd as well as prove corresponding lemmas of Mnd as in Barlow's paper,in which the critical step is based on geometric analysis and estimation on Green's function in high dimensions.Finally,we prove the following inequalities:·For any d>4,there exists a 0<c<? such that for all n?1 and ?>0,P[Mnd??E(Mnd)]?2exp(-c?).·For any d?4 and ?1/2,there exist 0<c,C<? such that for all n? 1 and??1,P(Mnd ?E(Mnd)/?)?exp(-c??).