A friend of mine asked me a question in computer science which induces to the following problem in (propably) theories of random walk.
Let $d$ be a positive integer. We are given a discrete distribution $\{p_1, \ldots, p_d\}$. A particle starts walking on $\mathbb{Z}_{\geq 0}^{d}$ in the following way:
Let $(n_1(t), \ldots, n_d(t))$ be the position of the particle at time $t$, where $t$ takes discrete integer values, i.e. $t=0, 1, 2, \ldots$.
At each second, it moves one step along the direction of the $i$-th axis in the probability of $p_i$ (i.e. $(n_1(t), \ldots, n_d(t)) \mapsto (n_1(t), \ldots, n_{i-1}(t), n_{i}(t)+1, n_{i+1}(t), \ldots, n_{d}(t))$).
Then after $T$ steps (i.e. at the $T$-th second), the particle moves to the position $(n_1(t), \ldots, n_d(t))$. Then we hope to estimate the standard derviation of this sequence of $d$ numbers, i.e. $$ \Delta(t) := \dfrac{1}{d} \sqrt{\sum_{i=1}^{d} (n_i(t) - \overline{N}(t))^2}. $$ It is indeed so great if one may provide the exact formula for $\Delta(t)$. Or if it is difficult, we would like an estimate of this when $t$ is sufficiently large.
Moreover, we modifiy this with a cliff on the lattice. Let $N > 1$ be an integer. The particle still walks on the lattice in the same way as above, but when it reaches $N$ in any direction $i$, it will never go forward on that direction (i.e. even in further steps it has a possibility $p_i$ to take a step in direction $i$, it will remain unmoved in that particular direction.). Then in this case, we raise the same question: how to calculate or estimate $\Delta(t)$ in this case?
As I'm a postgraduate student in number theory and algebraic geometry, I find it hard to work on such a problem. So could anyone help us with this?
Thank you all for answering and commenting!