I am definitely not a probability guy, but I'd like to have a heuristic answer to the following question: do $n$ independently moving points in an open, connected, bounded region $R$ tend to "braid" a lot?
Here is my toy model. Firstly, if we imagine having $n$ continuous trajectories $\gamma_1, \ldots, \gamma_n : [0, T] \to R$ that does not intersect pairwise, we can define their braiding as the corresponding element of the braid group(oid) in $R$. A simple associated invariant is given by$$\textrm{#positive crossings - #negative crossings}$$which is nice because it can be computed by looking at two strands at a time.
For $n=2$, we will assume $R$ to be a cylinder, with strands having a constant velocity in the $z$-axis direction. We can further simplify the model by imagining one strand to be fixed in the center of the disk. We now consider a stochastic process $X$ on the disk and consider the second strand as being $(X(t), t)$. Given my total ignorance about such processes, I will only describe the discrete analog I think (it should be a variation of the 2D Wiener process). At time $t+ \epsilon$, $X$ will choose uniformly in which direction $v \in S^1$ to go, and then$$ X(t+ \epsilon) = X(t) + \frac{\epsilon v}{\alpha(v, \| X(t) \| )}$$Where $\alpha(v, r) > 0$ is a weight that incentivizes the process to go toward the center and prevents the process from exiting the disk. I thought about something inspired by the hyperbolic disk but I can't seem to work out a nice formula (it is independent of $v$).
The braid invariant $b(X)$ I mentioned above is a sort of winding number and can be computed in this way. Let $s_1(t), s_2(t) \in \{+,-\}$ be the sign of the projections along the $x$-axis and $y$-axis of the process respectively (not defined when such projections are zero, which happens almost never). Then $b(X)$ gains $+1$ when $X$ passes from $(+,+)$ to $(-,+)$ or from $(-,-)$ to $(+,-)$, and $-1$ when going in the opposite direction. The other two possible switches (that is $(+,+) \leftrightarrow (+,-)$ and $(-,+) \leftrightarrow (-,-)$ ) produce no change in $b(X)$.
In the end, I'd like to know the distribution of $b(X) \in \mathbb{Z}$, if such a thing has a meaning. I expect its mean value to be zero since the process is symmetric under inversion of angles. If you have a better model to suggest, it is more than welcome. Thanks!