Medfarm Play

In analytic combinatorics, the problem of singularity analysis aims at deducing the asymptotic expansion of an array of numbers from the properties of their generating functions near its singularities. In the univariate case (where the array is a sequence), this problem is completely solved for all algebraic functions and the solution is captured by the classical transfer theorem of Flajolet and Odlyzko. In the multivariate case however, the solution to the problem remains elusive. In the past decade, Robin Pemantle and collaborators carried out a substantial amount of work which tackles the case of multivariate rational functions in the diagonal limit. In the bivariate setting, this means computing the asymptotic expansion of the coefficients $a_{m,n}$ of a rational function in the limit where $m,nto infty$ and $m/nto lambda$ for fixed $lambda >0$.

In this talk, I will present a new result which treats a different class of multivariate functions in a more general limit regime. In the bivariate case, it provides the asymptotic expansion of $a_{m,n}$ when $m,nto infty$ and $m/n^thetato lambda$ for fixed $theta, lambda >0$. This result generalizes formally Flajolet and Odlyzko's transfer theorem. In particular, it applies to all algebraic functions which are in some sense emph{$Delta$-analytic}. (It can be shown that in the multivariate setting, the class of $Delta$-analytic algebraic functions and the class of rational functions are essentially disjoint.) Applications of the result include Ising-decorated triangulations and parking processes on random trees.

Kontaktperson för denna film
Paul Thévenin, Sannolikhetsteori och kombinatorik

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