Marc
Zimmermann
Uni Köln
Polarization of lattices: Stable cold spots and spherical designs
Abstract.
We consider the problem of finding the minimum of inhomogeneous Gaussian lattice sums: Given a lattice $L \subseteq \R^n$ and a positive constant $\alpha$, the goal is to find the minimizers of $\sum_{x \in L} e^{-\alpha \|x - z\|^2}$ over all $z \in \R^n$, which we call the cold spots of the lattice $L$, while the value of the inhomogeneous Gaussian lattice sum at such a point is called the polarization of the lattice.
By a result of B\'etermin and Petrache from 2017 it is known that for steep potential energy functions---when $\alpha$ tends to infinity---the minimizers in the limit are found at deep holes of the lattice, one might even say that "polarization converges to sphere covering".
In this talk I will discuss some expected and unexpected geometric phenomena related to the cold spots of lattices. Firstly we will discuss when a lattice can have stable cold spots, that is a point which is a minimizer for all $\alpha \geq \alpha_0$ for some finite $\alpha_0$.
It turns out that generic lattices do not have stable cold spots. For several important lattices, like the root lattices, the Coxeter-Todd lattice, and the Barnes-Wall lattice, I will discuss how to apply the linear programming bound for spherical designs to prove that the deep holes are stable cold spots.
Finally, I will discuss an example of a very famous lattice, which, somewhat unexpectedly, does not have stable cold spots.
The talk is based on joint work with C. Bachoc, O. Marzorati, P. Moustrou, and F. Vallentin.