Has this random process been studied on grid graphs?

Question

As an offshoot of a different discussion I got curious about (uniform) random spanning trees on grid graphs (torus graphs in particular, to avoid having to think about edge effects) and what their diameters would look like. Perhaps unsurprisingly, the result looks much like a random walk:

Some rough computation on graphs up to size 320 suggests that the average length of the longest path on an $n\times n$ torus is of order $n^C$ for $C\approx 1.26$; it seems distinctly larger than $5/4$, but beyond that it's difficult to discern.

I was able to find a little research on the expected diameter of random spanning trees for some graphs, particularly this article from Chung, Horn, and Lu; unfortunately, their results depend on a degree that increases with $n$ and don't seem to apply to the fixed-degree case.

This is such a natural question that it feels like it should have come up before, in a context of random walks or percolation theory or similar; does anyone know of any references or information on this particular question?

Answer 1

The scaling exponent for the diameter should be $\frac{5}{4}$.

Indeed, it follows from Wilson's algorithm that a branch of a uniform random spanning tree, say between two given vertices, is a loop-erased random walk. The typical length of LERW of size $n$ (say, between two vertices at distance $n$), is $n^{\frac{5}{4}+o(1)}$. This is an old result by Kenyon ("The Asymptotic determinant of the discrete Laplacian") which was more recently refined by Lawler and Viklund ("Convergence of loop-erased random walk in the natural parametrization"). My understanding, though, is that the $o(1)$ should lead to a logarithmic correction, since the natural SLE measure of the full radial SLE${}_2$ path is infinite.

Admittedly, and additional argument is needed to show that the diameter also scales as $n^\frac{5}{4}.$ If you are content with the bound $n^{5/4+o(1)}$, then I would guess one can extract a moderate deviation bound from the proofs in the literature, bounding the probability that the length of LERW between two given points is greater than $n^{5/4+\epsilon}$ by something super-polynomially small. The union bound will then give the result for the diameter.