My Ph.D. Research

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!

Interchanging the Limit

I’ve been exploring this limit interchange for the past week or so, and the notion seemed hopeless.  The terms in the series seemed to grow to fast as the killing rate of the walk dropped to zero (survival rate to 1).  The issue really stemmed from the fact that the $k^{th}$ term looks like a sum over all possible paths of $k$ points in $Z^d$ of products of Green’s functions.  And it seemed that such a term acted like the reciprocal of the killing rate to the $k^{th}$ power.  So as this rate goes to zero, these terms blow up, and at a rate which I can’t control.  (I have one power of control, but no more.)  

However, my advisor pointed out that if instead, I consider products of derivates of Green’s functions (discrete derivative, so really just things of the form $G(z_1+e,z_2) - G(z_1,z_2)$, I can do a little better.  And due to some cancellations from the environment, such a replacement is possible. And if I go one step further and replace the Green’s functions with their discrete second derivative ( $G(z_1+e,z_2) + G(z_1-e,z_2) - 2 G(z_1,z_2)$ ), I may have the control on the terms I need.  So this is what I will be working on. 

I believe that there are very exact bounds on the discrete derivative of the heat kernel, and from here I can get estimates on the derivatives of the Green’s function.

Analyticity musings

There’s a Sznitman and Zerner result which guarantees the existence of a limiting velocity (LLN) almost surely under a certain set of conditions for our RWRE.  Sabot has a nice argument which shows that this velocity is a limiting point as $\delta$ goes to $1$ of the transition probabilities of the Kalikow Auxillary Walk taken over all points in the lattice.  In order to show analyticity, I need to understand this argument very well.  There are two limits taken, and I would like to argue that they can be interchanged, which is often a hairy situation.  If I can do this, then I can show that the Kalikow transition probability at any given point is an analytic function of $\gamma$, and this extends as $\delta$ is taken to 1, and then by this limit exchange argument, I can hopefully argue that the asymptotic velocity lies somewhere in the collection of all of these analytic functions (Convex Hull?), and is therefore analytic.  So there are a lot of things to check!  

Analyticity

The paper I’ve been reading by Sabot gives a third order expansion of the a.s. velocity of a random walk in a perturbed random environment, where the velocity is expanded as a function of the order of the perturbation, which has been called $\gamma$ in my previous blog posts.  I believe it should be fairly simple to show that this velocity is actually analytic with respect to $\gamma$ in dimension $d \geq 2$.  Basically the argument relies on this multiplication operator argument to introduce a faster killing into the green’s function, thereby solving convergence issues when the original killing time is taken to 1.  It’s a very nice trick, and I’d like to find out its origins.  

Generalizations

So I have this simple result, which says that if we scale a random perturbation of symmetric jump probabilities in $\mathbb{Z}^d$, $d \geq 3$, down to nothing with the endpoint of a random walk in the perturbed environment, then the centered and rescaled endpoint is distributed like a normal random variable, and the centering is given by the average drift of the perturbation (this is the interesting part).  

I would like to state a stronger and believable result which is this:  Suppose our random jump probabilities are given by $\omega^\gamma (z,e) = p_0(e) + \gamma \xi(z,e)$, where $\xi(z,e)$ is an i.i.d. random perturbation at each point $z$ in $\mathbb{Z}^d$. Let $\overline{d} = \mathbb{E}[\xi(0,e)]$, and let $\{X_t^{\gamma}\}$ be a random walk in the random environment $\omega^\gamma$. Suppose that we have a sequence $\{\gamma_n\}$ such that $\gamma_n \rightarrow 0$.  And further, suppose $\gamma_n \sim n^{-\alpha}$.

• If $\alpha = \frac{1}{2}$, then \[\frac{X^{\gamma_n}_n - \sqrt{n}\overline{d}}{\sqrt{n}} \Rightarrow N(0,\Sigma),\]  (This is what I’ve already shown.  Moreover, I would like to prove a pathwise invariance principle, which should be an easy corollary)
• If $\alpha > \frac{1}{2}$, then \[\frac{X^{\gamma_n}_n}{\sqrt{n}} \Rightarrow N(0,Id),\] (This should be immediate)
• If $0< \alpha < \frac{1}{2}$, then I don’t know.  I will think about this third case and get back to it.  

It’s been a while, blog.

So winter break happened, and though my work hasn’t stopped, my writing has.  And apparently so did my drive to keep up with this blog.  I believe that without the clearly defined benchmark of the paper I was working on, it’s much more difficult to measure my progress, and therefore much less enticing to write about my progress. 

But, I think now that I don’t have a clearly defined measurement of how I’m doing, this blog is more important than ever.  My entries may become more abstract and generalized, like, “I didn’t understand this section of this paper, so I will try to understand it by tomorrow.  It has something to do with Schroedinger operators.”  But hopefully such entries will keep me on track, organized, and motivated to deal with the issues that come up, rather than forgetting about them and staying confused and frustrated.  

Thanks to Andrew for reminding me that this blog exists when he noticed it on my webpage.  Ph.D. HO!

Last problem fixed, first draft done!

The problem I posted about yesterday wasn’t really a problem, just an annoyance.  I figured out the correct way of presenting the transfer from $n$ to $N$ after talking it over with Ben, and that’s that.  I wrote it up, I’ll give the paper one more proofread, and then it will be sent off!  (Not to be published, just as a benchmark in my progress)

Final issue with this first result

It’s been a while since I’ve posted.  I’ve been cleaning up the paper, which is almost entirely finished, and I’ve also been reading a paper and presenting it to my advisor, along with lecturing.  Hence the long break in posts.  But I’m a bit stuck again, so I figure writing about it might help.  

The problem I’m having in the paper is the conversion from $n$, the number of geometric walks we break our walk into (this is the variable of convergence in the Martingale argument), to $N$, the variable of the main result.  The issue is that as $N\rightarrow \infty$, I would like to choose a subsequence $n_N$ which gives convergence of 

\[\frac{X_N-\sqrt{N}\overline{d}}{\sqrt{N}}\]

in distribution to a normal, given the convergence of

\[\frac{X_{\tau_1+\cdots+\tau_n}-\gamma_n\overline{d}}{\gamma_n}.\]

And I need $n \sim N^{\frac{1}{1+\alpha}}$.  The problem is that since $n$ is growing slower than $N$, I’m not choosing a subsequence, but actually repeating terms.  So what do I set $\gamma_n$ to to equal $\sqrt{N}$.  I’m getting mixed up about these convergences, and I will write more about it later today.  

Last touches, postponed

After talking a bit with Andrew, I have convinced myself that there is no problem with the measures in the paper.  They are consistent, and it has been made clear how to define the overarching measure that is applied in the Martingale Lindeberg Feller CLT.  So as far as I see it, the only thing left to do is the following:  

After applying the Martingale CLT, I have that for almost every choice of measure, the endpoint of the RWRE (at a time which is a sum of geometric random variables) minus a sum of the expected (quenched expectation) displacements of the parts of the walk indexed by the geometric random variables converges is distributed like a Gaussian (properly normalized).  I want that for almost every environment, this sum of expected displacements, where each walk starts where the last left off, hence are random variables with respect to the starting points, converges a.s. to a deterministic drift term.  That is, if $X^1_t, \ldots, X^n_t$ is a sequence of RWRE starting where the last left off and $\tau_1,\ldots, \tau_n$ is a sequence of i.i.d. geometric random variables with failure rate $\delta$, I want that for almost every environment

\[\frac{1}{c_n}\Big(\sum_{i=1}^n\mathbb{E}[X_{\tau_1}^i - X^i_0|X^i_0 = X^{i-1}_{\tau_{i-1}}] - \frac{n\gamma\overline{d}}{1-\delta}\Big) \rightarrow 0 \]

in probability.  Here $\gamma$ is the order of the random perturbation of the environment, $c_n$ is the CLT scaling, and $\overline{d}$ is the deterministic drift, which depends only on the expected environment perturbation.  

Now, what I have to show this is that the probability (taken over environments) for one of these expectations to be too far away from the drift term $\frac{\gamma\overline{d}}{1-\delta}$ is exponentially small as $\gamma \rightarrow 0$ and as $\delta \rightarrow 1$ in proper relation to each other, both of which are happening as $n \rightarrow \infty$.  Therefore by Borel Cantelli, I have that for almost every environment, one of these expectations (with a deterministic starting point, so only random with respect to the environment) converges to the deterministic drift.  The problem is, this is for a fixed starting point $z_0$.  But I can change this probability I’m controlling to be the maximum over all starting points in a large box (which gets larger with $n$).  But then the sum of the expectations I care about above (where the starting point is random) is controlled by this maximum value, which converges to the deterministic drift for almost every environment.  How do I extend the result to be a maximum over all sites in a box?  Well, I have exponential control, and extending it naively by summing over sights in a box adds only a power law to this, so I still have great control.  That should be enough!  

I probably won’t get to this for a while, as I need to present Sabot’s paper to my advisor, so I will try to do these projects in parallel.  

Paper nearing completion

I have revamped section 1 so that I have exponential control of the probability of fluctuations for a RWRE at a geometric time.  Here’s what is left to do in the paper:  

1. Be explicit with measures and expectations, especially in the Martingale section.
2. Normalize notation throughout the paper.  Is it $\gamma$ or is it $\gamma_n$?
3. Take a maximum over sites in a box of the exponential control result in order to finish off the Martingale method.  The way it’s written now is incorrect.
4. Cite results such as Hoeffding’s inequality and the bound on the Green’s function that I use.  

These things shouldn’t take more than a week.