Imagine I perform the following procedure:

[1] At time point $t_1$, I place a single point on a two-dimensional plane at the coordinate $(x, y) = (0, 0)$.

[2] At time point $t_2$, I center a circle of radius $r$ at the point placed during time $t_1$, and place another point somewhere within (or along the contour of) this circle with uniform probability across its area.

[3] For time point $t_i$, I first randomly select a previously placed point, with uniform probability for all points, and then I center a circle of radius $r$ at the point and place another point somewhere within (or along the contour of) this circle with uniform probability across its area.

GOTO [3] for $(N - 2)$ iterations.

Call $(p_1, ..., p_N) \in P$ the set of points $N$ placed on the surface via the above method.

My question is the following: What is the probability distribution for the radius and center position of the minimum circle that circumscribes (or covers) the set of $N$ points in $P$? How does this relate to the barycenter of the coordinates for the points in $P$ and the initial point at the origin?