I am using the Kolmogorov–Smirnov two-sample test to compare distributions, and I noticed a $p$-value is frequently reported as the test statistic. How is this $p$-value determined? I know it's the probability of obtaining a result at least as large as the one obtained, but how is this $p$-value determined given this is a nonparametric test? That is, we can't assume Gaussian fluctuations in the distribution and compute the $p$-value using a $t$-test.
Thanks!