10:30-11:10
Psi-series method for random trees and quadratic convolution recurrences
By Hsien-Kuei Hwang
(Institute of Statistical Science, Academia Sinica, Taipei)
11:30-12:10
Enumeration of 3D triangular maps for creation of 3D universe
By Tetsuyuki Yukawa (Sokendai and KEK)
12:30-1:30
Lunch
1:30-2:10
Duality between population and sample in interacting particle systems on graphs
By Shuhei Mano (ISM and Sokendai)
2:30- 3:00
Free talking
3:00-3:30
Tea
(Supported by ISM Cooperative ResearchProgram 2011- ISM・CRP 1023 and JSPS Grant-in-aid for Scientific Research 23540177).
An unusual and surprising expansion of the form \[
p_n = \rho^{-n-1}\left(6n +\tfrac{18}5+
\tfrac{336}{3125} n^{-5}+\tfrac{1008}{3125}\, n^{-6}
+\text{smaller order terms}\right),
\]
as $n\to\infty$, is derived for the probability $p_n$ that two randomly chosen binary search trees are identical (in shape, hence in labels of all corresponding nodes). A quantity arising in the analysis of phylogenetic trees is also proved to have a similar asymptotic expansion. Our method of proof is new in the literature of discrete probability and the analysis of algorithms, and it is based on the logarithmic psi-series expansions for nonlinear differential equations. Such an approach is very general and applicable to many other problems involving nonlinear differential equations; many examples are discussed in this article and several attractive phenomena are discovered.