### State of the Union

It’s been a while since I’ve posted! I’ve spent the last month working on trying to prove analyticity of the asymptotic velocity under the conditions that $d_0 \neq 0$, and I have run into snag after snag. We realized that while trying to bound the terms in the power series expansion mentioned previously as $\delta$ tends to $1$, my advisor noticed a striking similarity to what I was trying to accomplish and what he and coauthors had tackled in 1982. Essentially, though the terms in the series are poorly behaved, by inverting the series (which involves summing over paths in $\mathbb{Z}^d$), certain paths (paths which split into two disjoint parts) will have no contribution, and you end up with a nicely behaved sum. And it suffices to have analyticity of the inverse.

But the arguments don’t exactly line up, and without getting into the details, it’s been a tough road trying to employ their methods. I certainly need to brush up on my Fourier analysis of operators, which I’m working on doing.

Otherwise, I’ve come back to the Scaled Disorder CLT result I wrote up, and considered generalizations. Immediately, if the random perturbation of the environment decays faster than $\frac{1}{\sqrt{n}}$, where $n$ is the length of the walk, then the walk is diffusive. This is trivial given my result. I’m interested in whether or not my method will work if the initial homogenous environment, $p_0$, is not assumed to be symmetric. I don’t see why not, and I might get started on this next week.

Bricmont and Kupiainen in 1991 proved that for a random perturbation of a simple random walk in $d \geq 2$, the walk is diffusive (scales weakly to a Brownian Motion). This is one of the few results on limiting behaviour for random walks in perturbations of drift-less environments. The case where there is an underlying drift is considered heavily by Sznitman, et al. Bricmont and Kupiainen’s result doesn’t imply mine, as my result deals with scaling the disorder with the length of the walk, but it should be no surprise that what I have proven is true given their result. However, it’s possible that my martingale method might work to extend their result to perturbations of symmetric walks, instead of the stronger simplicity assumption. I’ll investigate this as well.

Also, with my advisory committee meeting coming up, I need to spend a good deal of time just reviewing the literature, and really firming up what is known and what is not.

Phew!