3D loop-erased random walk - Prologue

Last night, I dreamt about Gregory Lawler visiting my secondary school alma mater. I know, this is totally unhinged and I couldn’t remember much of it. Perhaps he came to remind me to publish this month-old draft.

We hope to initiate a series to discuss the progress on the three-dimensional loop-erased random walk, and mention some progress in other random geometric objects in three dimensions, whose study in two dimensions have been greatly aided by the Schramm-Loewner evolution, a representation of certain scaling limits which is not available in three dimensions. The expected waiting time for the next article is seven months.

Two-dimensional random geometry was propelled by the invention of the SLE, first emerging as a subsequential scaling limit of the loop-erased random walk in two dimensions¹. To someone outside this field, the set-up of SLEs may seem fairly arbitrary and the use of Loewner chains almost divine intervention. We will not detail the exact construction of SLE here, but rather mention what was probably the motivation for the seemingly out-of-place univalent maps. We refer the reader to Random Explorations by Gregory Lawler² for an elementary and detailed introduction to the LERW.

The LERW (which we denote by $\eta$) is defined using the loop-erasure procedure, which makes complete sense. However, it would be nice to be able to write it in a more ‘direct’ form. This is possible and this form is often referred to as the Laplacian random walk. Instead of providing transition probabilities from ${\eta }_{k-1}$ to ${\eta }_k$, as is possible for Markov chains, we are only able to give transition probabilities from $\eta [0,k-1]$ to $\eta [0,k]$, which takes into account the entire path history. Another way to phrase this is that the LERW satisfies a domain Markov property, which (roughly) states that for a LERW in a domain $D$, given the previous path $\eta [0,k-1]$, the future path $\eta [k,\infty )$ has the law of a LERW in the domain $D\setminus \eta [0,k-1]$. The LERW is a ‘potential theoretically natural’ process, arising from the space it lives in, and its evolution can be understood as cutting out slits from said space. In a more transparent manner, we mean that the transition probabilities of the LERW are determined by the conductances and the Poisson kernels of the domain. Hence, the evolution of LERW can be interpreted as the evolution of the space, whose topological property is undisturbed given that LERW does not self intersect. One may start to see how the Riemann mapping theorem comes in. It is worth mentioning that Oded Schramm’s PhD thesis was on conformal geometry, advised by William Thurston.

As we have mentioned, the Loewner evolution toolkit is not available in dimension three. This may be attributed to the lack of conformal mappings. Liouville’s theorem for conformal (not harmonic) maps states that the only conformal maps in ${\mathbb{R}}^d$, $d\ge 3$, are Möbius transformations. Such rigidity does not allow for the conformal equivalence of a domain and the same with a slit cut out. One may also attribute this failure to the transience of Brownian motion. Considering inversion, it is clear that in higher dimensions, Brownian motion is not conformally invariant³. We will continue this discussion about higher dimensional analogues after a brief digression.

The study of LERWs is intimately tied to intersections of random walks. We now state the result by Lawler⁴: for two independent simple random walks on ${\mathbb{Z}}^d$ starting at the origin, their number of intersections up to time $n$ is asymptotically given by

$$\{\begin{array}{ll}c{n}^{3/2}+O(n^{1/2}),& d=1,\\ cn+O(\ln n),& d=2,\\ c{n}^{1/2}+O(1),& d=3,\\ c\ln n+O(1),& d=4,\\ c+O(n^{(4-d)/2}),& d\ge 5.\end{array}$$

In particular, this says that the expected number of intersections is finite if and only if $d\ge 5$, and four is the critical dimension. Using this, Lawler showed⁴ that the LERW has Brownian motion as scaling limit for $d\ge 4$, with logarithmic corrections in dimension four. (Now, we have mentioned the scaling limits in all dimensions except one, in which the LERW is ill-defined, and three, the topic of this series.)

The Laplacian random walk is, from a formal perspective, a dimension-agnostic form of the LERW (but so is the original loop-erasure form). I, and at least one other person⁵, thinks that finding a continuum analogue to this process is very important to studying the scaling limit of the LERW. Yet it is unclear what this would look like. Even discounting the obvious issue of translating the transition probabilities, the continuum analogue would necessarily have to be equivalent to Brownian motion in higher dimensions, and something weird in dimension two, equivalent to SLE${}_2$. Note that, although we have a differential equation in the SLE case, the equation describes the Riemann map for the trace (hull) and not the curve. The curve will never admit a true differential-equational description as the evolution is far from local/Markovian. However, such an analogue would degenerate into a Markov process in higher dimensions. Anyway, this is too far out of my depth for me to say anything remotely useful.

With this preliminary discussion on the background of LERW, we are ready to discuss progress on its three-dimensional scaling limit.

Footnotes

1. Schramm, Oded, Scaling limits of loop-erased random walks and uniform spanning trees, Israel Journal of Mathematics 118 no. 1 (2000) 221–288, https://doi.org/10.1007/BF02803524. ↩

2. Lawler, Gregory F., Random Explorations, Student Mathematical Library, American Mathematical Society, Providence, Rhode Island, 2022, https://doi.org/10.1090/stml/098. ↩

3. Kostya_I, Answer to “Conformal invariance of Brownian motion in higher dimensions,” 2014, https://mathoverflow.net/a/178404/340556 ↩

4. Lawler, Gregory F., Intersections of Random Walks, Springer New York, New York, NY, 2013, https://doi.org/10.1007/978-1-4614-5972-9. ↩ ↩²

5. One World Probability, Daisuke Shiraishi - 27th October, 2022, https://www.youtube.com/watch?v=VzATYBAqrcM. ↩