Department of Mathematics

Indian Institute of Science

Bangalore 560 012

 

SEMINAR

 

Speaker

:

 Mr. Indrajit Jana
Affiliation : Indian Institute of Science, Bangalore.

Subject Area

:

Mathematics

 

Venue

:

Department of Mathematics, Lecture Hall III

 

Time

:

11.00 a.m.-12.00 p.m.

 

Date  

:

August24, 2012 (Friday)

Title

:

"Matchings between point processes"
Abstract

:

We study several kinds of matching problems between two point processes. First we consider the set of integers $\mathbb{Z}$. We assign a color red or blue with probability 1/2 to each integer. We match each red integer to a blue integer using some algorithm and analyze the matched edge length of the integer zero. Next we go to $\mathbb{R}^{d}$. We consider matching between two different point processes and analyze a typical matched edge length $X$. There we see that the results vary significantly in different dimensions. In dimensions one and two (d=1,2), even $E[X^{d/2}]$ does not exist. On the other hand in dimensions more than two (d>2), $E[\exp(cX^{d})]$ exist, where $c$ is a constant depends on $d$ only. Next we discuss about matching problems in a finite setting, namely in a unit square. Take $n$ red points and $n$ blue points chosen independently and uniformly in a unit square. There are $n!$ many possible matchings between these red and blue points. We investigate the optimal average matched edge length and the minimum of the maximum matched edge lengths, where the minimum is over all possible $n!$ matchings. There we see that the optimal average matched edge length is like $\sqrt {\frac{\log n}{n}}$ and the min-max matched edge length is like $\frac{(\log n)^{3/4}}{\sqrt{n}}$.