Statistical properties of determinantal point processes in high ...

SCARDICCHIO, ZACHARY, AND TORQUATO PHYSICAL REVIEW E 79, 041108 2009
M * (s)
6
5
4
3
2
1
d=1
d=2
d=3
d=4
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
s=r/λ
statistics in dimensions one to four. Our results strongly suggest
that both G P and G V are linear for sufficiently large r,
and we obtain numerical estimates of the common slope in
this limit. This behavior is to be contrasted with the equivalent
forms of G P and G V for equilibrium systems of hard
spheres and for Poisson point processes. It is known for the
former systems that both functions saturate for sufficiently
large r while G Pr=G Vr=1 for all r in the latter processes
2. The linearity of G P and G V in the determinantal case is
thus unique in the context of general point processes. We
have also shown, in accordance with 6, that in the limit as
d→ both G P and G V must saturate at unity in accordance
with the behavior of the aforementioned bounds; again, this
claim is supported by the numerical evidence we have presented
here. Also, as the dimension d grows, we observed
that the functions H P and H V become concentrated around
their maximum as do the distributions of extrema of nearestneighbor
distances M * and M*.
M * (s)
4
3
2
1
d=1
d=2
d=3
d=4
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
s=r/λ
FIG. 15. Color online Distributions of the maximum M*s; left and minimum M * s; right nearest-neighbor distances across
dimensions at unit mean nearest-neighbor separation . Results are simulated using the HKPV algorithm.
4
3.5
3
2.5
2
1.5
1
0.5
Minimum nearest-neighbor distance
Maximum nearest-neighbor distance
Unit mean nearest-neighbor distance
0
0 1 2 3 4 5
d
FIG. 16. Color online Average maxima and minima nearestneighbor
distances with standard deviations across dimensions at
unit mean nearest-neighbor separation; results are obtained using
the HKPV algorithm. Also included for reference is the unit mean
nearest-neighbor separation, which is fixed for each dimension.
041108-18
By using the HKPV algorithm to generate configurations
of points we have shown that the determinantal nature of the
n-particle probability density has a significant effect on the
Voronoi cell statistics of the Fermi-sphere point process in
two dimensions. Namely, the probability distribution of cell
sides is more peaked around n=6 hexagons than the corresponding
distribution for either the Poisson point process or
the Ginibre ensemble 38 the distribution of complex eigenvalues
of random complex matrices. The effective separation
of the points, resulting in a sharper peak in the distribution
of the number of sides of the Voronoi cells, is closely
related to the hyperuniformity of the system.
Finally, to show how the algorithm can be used for generating
determinantal processes on curved spaces, we have
presented an example of a determinantal point process on the
two-sphere.
FIG. 17. Color online Configuration of 37 points on the unit
sphere using the HKPB algorithm with the spherical harmonics as
basis functions.