Speaker
Description
We consider a translation invariant point process in R^d or Z^d. Let
V(N_B) be the variance in the number of points, N_B, in a ball B of
volume |B|. Generally, such as when particles with short range interactions are
distributed according to a Gibbs measure, V(N_B)/|B| >0.
There are however many interesting cases when Var(N_B)/N_B->0, as
|B|->oo. Such processes are called hyperuniform (or superhomogeneous)
This occurs when the structure function S(k), the Fourier transform of
the "full" pair correlation function, G(r)=ndelta(r)+n^2[g(r)-1], n being the
density, which is always non-negative, vanishes at k=0, S(k)=0. Just how fast
V(N_B)/|B| goes to zero depends on the way S(k) behaves as k->0.
I will discuss examples of such hyperuniform systems both old (Coulomb
systems) and recent (facilitated exclusion processes).
When S(k) vanishes in an open set M in k-space (which may or may not
include the origin) the system is maximally "rigid". Rigidity describes
the amount of information about the points in B given the configuration
of points outside B. This can be zero as in a Poisson process or "maximal" where
the exact position of the points in B are determined by the
configuration outside B.
Such systems also have other "crystaline" properties. (This is joint
work with Subhro Gosh)
