We define the frozen-square enumeration as the enumeration of $n\times
n$ ASMs under the refinement of having in a corner an s x s
square of entries that are all zeroes. We present the `guessed'
derivation of a conjectural formula for such enumeration, in terms of
a Fredholm type determinant of some definite s x s matrix. We
provide strong numerical support in favour of our conjecture. We
discuss recent proofs of our conjecture, by various authors. We also
illustrate its relevance in connection with the limit-shape phenomenon
observed in large ASMs. In particular, we show that fluctuations in
the limit shape of large ASMs are governed by the GUE Tracy-Widom
distribution.
Joint work with Andrei Pronko.