Robin Pemantle (U. Penn.)

Multivariate asymptotics (why probabilists need to know math)

The single most important lesson I learned from Persi Diaconis is this. Good probability problems often lead to deep math: follow them there, learn the math, and meet the mathematicians.

Consider a generating function in several variables, for example F(x,y) = 1 / (1 - x - y - xy), which counts N-E-NE paths. We wish, as algorithmically as possible, to come up with asymptotic formulae for the coefficients a_{rs}. It turns out there is often a simple answer to this simple problem. In this case, for example, the leading term of a_{rs} is asymptotic to C (r+s)^{-1/2} x^r y^s where the point (x,y) lies on the zero set of H := 1-x-y-xy and where the logarithmic gradient of H at (x,y) is parallel to (r,s); a complete asymptotic development may in fact be obtained.

The solution leads through unexpected terrain such as complex algebraic geometry, computational commutative algebra, and stratified Morse theory. After listing some examples to which the general method can be applied, the talk will focus on the high-level description of the method.