The distance from the center to random points inside a disk is of course random, so that your question doesn't quite make sense... You can compute its distribution, or its mean for instance.

As for the decrease, it is obvious that the minimum distance from the center to points decreases with . Just give it a thought.

I show you how to find the distribution of the minimum distance from the center to points uniformly chosen in a disk.

First thing is to consider only one point uniformly distributed in a disk. I will consider a disk with center and radius . Let (i.e. is the distance from to ).

For , we have of course , and if then (because means that is inside the circle with radius , and the probability to be in a subset of the disk is proportional to its area). Using this computation, you can get the density of the distribution of (by differentiating, it is on and 0 elsewhere), and the expectation of .

Now, suppose there are independent points uniformly chosen on the same disk, and let be the minimum distance from the center to the points, i.e. where are defined like the previous , for each of the points.

Then we have , from which we can get the density of the distribution of : it is on .