There are two natural meanings for the distribution of minimum distances. One is the probability distribution for the minimum distance to one point. Another is the multiset of minimum distances to each point. They are related. This answer addresses the first sense of distribution. For a partial answer to the second, see Minimum Hamming Distance Distribution in a Random Subset of Binary Vectors+.

The cumulative distribution of the minimum of $s$ IID random variables satisfies

$$1-\operatorname{CDF}_s(d) = (1-\operatorname{CDF}_1(d))^s$$

since each side is the probability that all $s$ variables are greater than $d$.

If you assume that the symbols are uniformly random and independent, then the distance of a given string to a fixed string follows a binomial distribution. The probability that a particular string will be within a distance $d$ is the cumulative distribution function for the binomial distribution,

$$\operatorname{CDF}_1(d) = \sum_{k=0}^d {n\choose k}p^k (1-p)^{n-k}$$

where $n$ is the length of the string, and $p$ is $3/4$ or $1-\frac{1}{|\textrm{alphabet}|}$.

There isn't a closed form expression for this, but you can estimate it in many ways. Some techniques are better at estimating the cumulative distribution function near the median, and some are better for the tails. See Lower bound for sum of binomial coefficients? and the techniques in Sum of 'the first k' binomial coefficients for fixed n . For many particular values, you can use an exact computation of the distribution rather than estimating it. For example, for $s=1000, n=8,p=3/4,$ here is some code that computes the cumulative distribution function values:

```
cdf1[d_, n_, p_] := Sum[Binomial[n, k] p^k (1 - p)^(n - k), {k, 0, d}]
cdfs[d_, n_, p_, s_] := 1 - (1 - cdf1[d, n, p])^s
Table[N[cdfs[d, 8, 3/4, 1000]], {d, 0, 8}]
```

$0.01514,0.3172,0.9855,1.0000,1.0000,...$.