Given a random walk on a lattice $L$ (not necessarily centered - we allow $E[X_i] \neq 0$ for the i.i.d. increments $X_i$), let $p_t(x)$ denote the probability measure of state $x \in L$ after $t$ time increments. I am interested in the time evolution of the maximum difference of neighboring states' measures:

$$ \max_{x \in L, ~ x \sim y} |p_t(x) - p_t(y)|. $$

Here $x \sim y$ indicates sites $x$ and $y$ are adjacent, or that

$P_x(y) = P($a walker started at $x$ transitions to $y$ in a single step$) > 0$.

I've explored various texts and online notes, but they don't seem to cover questions of the pointwise nature of these distributions; typically a distribution's weak limit is found via CLT or saddle-point approximation and it's left at that.

As a first effort, it's straightforward to look at the simple random walk on $\mathbb{Z}$ (where the increments are $+1$ or $-1$ with equal probability) and use Stirling's approximation to confirm that

$$\max_{k \in \mathbb{Z}} |p_t(k+1) - p_t(k)| \approx C/t$$

for some constant $C$ (the $\approx$ indicating omission of lower order terms). One also finds that this occurs around $\lfloor k\rfloor = \sqrt{t}$, just as one would expect considering the limiting gaussian distribution, $\Phi(x) = (2\pi)^{-1/2} e^{-x^2/2}$, which has $\sup_x \Phi'(x)$ occuring around $x = 1$.

Any help in attacking the problem in higher dimensions and with non-simple probability transitions would be appreciated. I suspect a thorough answer exists in the literature; pointers to the relevant resources could also form an answer.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.