Leírás
Studying analytic limit objects of discrete structures has become one of the main directions of combinatorics recently, which owes its success to the fact that quite often statements in the discrete world can be converted to analytic ones and vice versa, vastly enriching the available tools in dealing with important questions.
In my talk, I am going to discuss permutons, i.e., permutation limits, which are simply probability measures on the unit square with uniform marginals. Such a measure naturally gives rise to a distribution over the length n permutations for any n>0. We are interested in the asymptotic properties of the Shannon entropy of these distributions, which we call the pattern entropy sequence.
After discussing some simple examples, I am going to speak about an exciting connection to Kolmogorov-Sinai entropy for permutons supported by graphs of measure-preserving functions, demonstrating that given certain regularity conditions, the pattern entropy sequence has a linear growth rate.
I will also touch upon how one can prove via Baire category arguments the existence of irregular permutons, i.e., ones for which no natural normalization yields a convergent sequence.
A simple construction for random permutons comes from the random automorphism of the rooted $d$-ary tree. If time permits, I will also speak about some surprising probabilistic phenomena I encountered upon studying its pattern entropy sequence.