Discrete Painlevé equations in probability

Distribution functions of interest in a variety of discrete probabilistic models (like length of the longest increasing subsequence of random permutations or last passage time in directed percolation) satisfy certain recurrence relations known as discrete Painleve equations. These equations were first obtained in an algebro-geometric work on surfaces obtained by blowing up the two-dimensional projective space at 9 points. The link between probability and geometry is provided by the theory of isomonodromy transformations of linear difference equations.