Exponential tail bounds for loop-erased random walk in two dimensions

Abstract

Let $M_n$ be the number of steps of the loop-erasure of a simple random walk on $\mathbb{Z}^2$ from the origin to the circle of radius $n$. We relate the moments of $M_n$ to $Es(n)$, the probability that a random walk and an independent loop-erased random walk both started at the origin do not intersect up to leaving the ball of radius $n$. This allows us to show that there exists $C$ such that for all $n$ and all $k=1,2,...,\mathbf{E}[M_n^k]\leq C^kk!\mathbf{E}[M_n]^k$ and hence to establish exponential moment bounds for $M_n$. This implies that there exists $c>0$ such that for all $n$ and all $\lambda\geq0$, \[\mathbf{P}\{M_n>\lambda\mathbf{E}[M_n]\}\leq2e^{-c\lambda}.\] Using similar techniques, we then establish a second moment result for a specific conditioned random walk which enables us to prove that for any $\alpha<4/5$, there exist $C$ and $c'>0$ such that for all $n$ and $\lambda>0$, \[\mathbf{P}\{M_n<\lambda^{-1}\mathbf{E}[M_n]\}\leq Ce^{-c'\lambda ^{\alpha}}.\]Comment: Published in at http://dx.doi.org/10.1214/10-AOP539 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Universality of high-dimensional spanning forests and sandpiles

“…In particular, Barlow and Masson [14,15] proved that the UST of Z 2 has volume growth dimension 8/5 and spectral dimension 16/13 almost surely. See also [9] for more refined results.…”

Hausdorff dimension of the scaling limit of loop-erased random walk in three dimensions

“…(In fact, we will set 1 ǫEs(ǫn,n) on each b x hit by K for the density of µ ǫ , where we chose n as an arbitrary large integer so that the distance between LEW n and K is small with high probability as explained above. We also point out that for all large n, Es(ǫn, n) is of order ǫ α+o (1) for some constant α, see Theorem 2.2.3.) Finally Theorem 3.1.1 concludes that…”

“…Given a box and a simple path γ contained in the inside of the box except the end point γ(lenγ) which is lying on the boundary of the box. Following same spirits of Theorem 6.7 [1] and Theorem 8.2.6 [18], we are interested in a random walk X staring from γ(lenγ) conditioned that X[1, τ ] ∩ γ = ∅ for some stopping time τ . Estimates of such a conditioned random walk X are crucial to prove ( 4.1).…”

“…Recall that q (1) was defined as in ( 3.21) (we use the same notation here). Suppose that 0 ≤ r ≤ log 2 l − 3.…”

“…Then ( 1.7) boils down to the corresponding estimates for LERW as follows. Let Y ǫ n be the number of ǫn-cubes nb x with 1 3 ≤ |ǫx| ≤ 2 3 such that LE(S[0, τ n ]) hits nb x . Then ( 1.7) is reduced to proving that for all r > 0 there exists c r > 0 such that P Y ǫ n ≥ c r ǫ −2 Es(ǫn, n) ≥ 1 − r.…”