Hyperuniformity and Hyperfluctuations for Random Measures on Euclidean and Non-Euclidean spaces

Abstract

In this thesis we study fluctuations of generic random point configurations in Euclidean and symmetric curved geometries. Mathematically, such configurations are interpreted as isometrically invariant point processes, and fluctuations are recorded by the variance of the number of points in a centered ball, the number variance. Hyperuniformity and hyperfluctuation of such configurations in the sense of Stillinger-Torquato is characterized in terms of large-scale asymptotics of the number variance in relation to that of an ideal gas, and equivalently by small-scale asymptotics of the Bartlett spectral measure in the diffraction picture.

Appended to the Thesis are three papers:

In Paper I we provide lower asymptotic bounds for number variances of isometrically invariant random measures in Euclidean and hyperbolic spaces, generalizing a result by Beck. In particular, we find that geometric hyperuniformity fails for every isometrically invariant random measure on hyperbolic space. In contrast to this, we define a notion of spectral hyperuniformity which is satisfied by certain invariant random lattice configura- tions.

In Paper II we establish similar lower asymptotic bounds for number variances of automorphism invariant point processes in regular trees. The main result is that these lower bounds are not uniform for the invariant random lattice configurations defined by the fundamental groups of complete regular graphs and the Petersen graph. We also provide a criterion for when these lower bounds are uniform in terms of certain rational peaks appearing in the diffraction picture.

In Paper III we prove the existence and uniqueness of Bartlett spectral measures for invariant random measures on a large class of non-compact commutative spaces, which includes those in Papers I and II. For higher rank symmetric spaces governed by simple Lie groups, we prove that there is a power strictly less than 2 of the volume of balls that asymptotically bounds the number variance of any invariant random measure from above. Moreover, we derive Bartlett spectral measures for invariant determinantal point processes on commutative spaces and define a notion of heat kernel hyperuniformity on Euclidean and hyperbolic spaces that is equivalent to spectral hyperuniformity.