Convergence of loop-erased random walk to SLE(2) in the natural parametrization (10rit143)

Objectives

In one of their celebrated papers, Lawler, Schramm and Werner established the convergence of LERW to SLE(2). However, there are at least two directions in which to improve their result. The first is to get a rate of convergence of LERW to SLE(2). This problem was undertaken by Benes, Johansson and Kozdron at BIRS 08rit136. The second approach, and the goal of our project, is to prove weak convergence in a stronger topology. Lawler, Schramm and Werner proved the weak convergence of LERW to SLE(2) with respect to the supremum norm on curves modulo reparametrization. We aim to prove convergence while taking into account the parametrization of the LERW. Namely, if we let $X^n$ be the LERW from the origin to the unit circle on the lattice $(1/n)Z^2$ and $M_n$ be the number of steps of $X_n$, then one expects that $Y_n(t) = X^n(t/E[M_n])$ should converge weakly (as n tends to infinity) in the supremum norm to a suitably parametrized version of SLE(2). Although other models have been shown to scale to SLE, none of them have been proved to converge as parametrized curves.

There are two recent developments that make this problem appear tractable. The first is the identification of what the suitable parametrization for the SLE(2) curve should be. In the original definition of SLE by Schramm, the SLE curves were parametrized so that their capacity (a measure of how big the curves look in the unit disc when viewed from the origin) grew linearly. This was the best way to analyze the curves by way of the Loewner equation but is not natural when one considers the SLE curves as scaling limits of discrete models. Indeed, Beffara showed that the Hausdorff dimension of SLE(k) is d = 1 + k/8 almost surely (k < 8). This suggests that for a discrete model to converge to SLE(k) as a parametrized curve, the parametrization on the SLE(k) curve should be such that scaling the curve by a factor of r in space is equivalent to scaling by a factor of $r^d$ in time. This "natural parametrization" of SLE has recently been shown to exist by Lawler and Sheffield. It is SLE(2) in this parametrization that one expects the LERW $Y_n$ defined above to scale to. Note that for SLE(2), d = 5/4 and the fact that $E[M_n]$ grows like $n^{5/4}$ was established by Kenyon.

The second result that will be useful for this problem is a tail bound on $M_n$. Recent work by Barlow and Masson gives both upper and lower exponential tail bounds on $M_n$. This should enable suitable tightness results that will establish weak convergence in the metric defined above.

The three participants have all made contributions to this field in either SLE and the natural parametrization, the LERW, or the convergence of LERW to SLE(2). The participants are all at different universities across Canada, and therefore being able to conduct research at BIRS would be quite beneficial.