Hyperuniformity on the Sphere Peter Grabner (joint work with J. - - PowerPoint PPT Presentation

Hyperuniformity on the Sphere

Peter Grabner (joint work with J. Brauchart, W. Kusner, and J. Ziefle)

Institute for Analysis and Number Theory Graz University of Technology

Optimal Point Configurations and Orthogonal Polynomials 2017

• P. Grabner

Hyperuniformity on the Sphere

Motivation: Hyperuniformity in Rd

Hyperuniformity has been introduced by Torquato and Stillinger as a measure for the distinction between order and disorder in the atomic structure of materials.

• P. Grabner

Hyperuniformity on the Sphere

Motivation: Hyperuniformity in Rd

Hyperuniformity has been introduced by Torquato and Stillinger as a measure for the distinction between order and disorder in the atomic structure of materials. Distribute N particles in a volume V ⊆ Rd according to a point process with joint density ρ(N)

V

being

• P. Grabner

Hyperuniformity on the Sphere

Motivation: Hyperuniformity in Rd

Hyperuniformity has been introduced by Torquato and Stillinger as a measure for the distinction between order and disorder in the atomic structure of materials. Distribute N particles in a volume V ⊆ Rd according to a point process with joint density ρ(N)

V

being (a) invariant under permutation of the particles

• P. Grabner

Hyperuniformity on the Sphere

Motivation: Hyperuniformity in Rd

Hyperuniformity has been introduced by Torquato and Stillinger as a measure for the distinction between order and disorder in the atomic structure of materials. Distribute N particles in a volume V ⊆ Rd according to a point process with joint density ρ(N)

V

being (a) invariant under permutation of the particles (b) invariant under Euclidean motion (for V ր Rd)

• P. Grabner

Hyperuniformity on the Sphere

Motivation: Hyperuniformity in Rd

Hyperuniformity has been introduced by Torquato and Stillinger as a measure for the distinction between order and disorder in the atomic structure of materials. Distribute N particles in a volume V ⊆ Rd according to a point process with joint density ρ(N)

V

being (a) invariant under permutation of the particles (b) invariant under Euclidean motion (for V ր Rd) Hence, a single particle is distributed with density

• V N−1 ρ(N)

V

(r1, . . . , rN) dr2 · · · drN = 1 |V | Assume N

|V | → ρ (thermodynamic limit).

This means that the distribution is asymptotically uniform with density ρ.

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Hyperuniformity on the Sphere

Hyperuniformity in Rd

Heuristic Hyperuniformity = asymptotically uniform + extra order Counting points in test sets, e.g. balls BR NR :=

N

• i=1

✶BR(Xi) , where (X1, . . . , XN) ∼ ρ(N)

V

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Hyperuniformity on the Sphere

Hyperuniformity in Rd

Heuristic Hyperuniformity = asymptotically uniform + extra order Counting points in test sets, e.g. balls BR NR :=

N

• i=1

✶BR(Xi) , where (X1, . . . , XN) ∼ ρ(N)

V

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Hyperuniformity on the Sphere

Hyperuniformity in Rd

Heuristic Hyperuniformity = asymptotically uniform + extra order Counting points in test sets, e.g. balls BR NR :=

N

• i=1

✶BR(Xi) , where (X1, . . . , XN) ∼ ρ(N)

V

The expected number of points in BR is E [NR] th. → ρ|BR|.

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Hyperuniformity on the Sphere

Hyperuniformity in Rd

The variance measures the deviation over all test sets under consideration. Example: (Xi)i i.i.d. ⇒ V[NR] th. → ρ|BR|.

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Hyperuniformity on the Sphere

Hyperuniformity in Rd

The variance measures the deviation over all test sets under consideration. Example: (Xi)i i.i.d. ⇒ V[NR] th. → ρ|BR|. Definition (ρ(N))N∈N hyperuniform ⇐ ⇒ lim

• th. V[NR] ∼ |∂BR| for large R

Remarks:

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Hyperuniformity on the Sphere

Hyperuniformity in Rd

The variance measures the deviation over all test sets under consideration. Example: (Xi)i i.i.d. ⇒ V[NR] th. → ρ|BR|. Definition (ρ(N))N∈N hyperuniform ⇐ ⇒ lim

• th. V[NR] ∼ |∂BR| for large R

Remarks: If (ρ(N))N∈N hyperuniform, i.e. Rd-term of lim

• th. V [NR]

vanishes ⇒ Rd−1-term cannot vanish.

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Hyperuniformity on the Sphere

Hyperuniformity in Rd

The variance measures the deviation over all test sets under consideration. Example: (Xi)i i.i.d. ⇒ V[NR] th. → ρ|BR|. Definition (ρ(N))N∈N hyperuniform ⇐ ⇒ lim

• th. V[NR] ∼ |∂BR| for large R

Remarks: If (ρ(N))N∈N hyperuniform, i.e. Rd-term of lim

• th. V [NR]

vanishes ⇒ Rd−1-term cannot vanish. Hyperuniformity is a long-scale property.

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Hyperuniformity on the Sphere

Hyperuniformity on the sphere

The sphere has finite volume thus the thermodynamical limit doesn’t make sense! Therefore consider distributions (ρ(N))N∈N on ❙d satisfying (a) ρ(N)(xσ1, . . . , xσN) = ρ(N)(x1, . . . , xN) for all xi ∈ ❙d, σ ∈ SN. ”particles are exchangeable”

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Hyperuniformity on the Sphere

Hyperuniformity on the sphere

The sphere has finite volume thus the thermodynamical limit doesn’t make sense! Therefore consider distributions (ρ(N))N∈N on ❙d satisfying (a) ρ(N)(xσ1, . . . , xσN) = ρ(N)(x1, . . . , xN) for all xi ∈ ❙d, σ ∈ SN. ”particles are exchangeable” (b) ρ(N)(τx1, . . . , τxN) = ρ(N)(x1, . . . , xN) for all xi ∈ Sd, τ ∈ SO(d + 1), ”isometry invariance”

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Hyperuniformity on the Sphere

Hyperuniformity on the sphere

The sphere has finite volume thus the thermodynamical limit doesn’t make sense! Therefore consider distributions (ρ(N))N∈N on ❙d satisfying (a) ρ(N)(xσ1, . . . , xσN) = ρ(N)(x1, . . . , xN) for all xi ∈ ❙d, σ ∈ SN. ”particles are exchangeable” (b) ρ(N)(τx1, . . . , τxN) = ρ(N)(x1, . . . , xN) for all xi ∈ Sd, τ ∈ SO(d + 1), ”isometry invariance”

• P. Grabner

Hyperuniformity on the Sphere

Hyperuniformity on the sphere

The sphere has finite volume thus the thermodynamical limit doesn’t make sense! Therefore consider distributions (ρ(N))N∈N on ❙d satisfying (a) ρ(N)(xσ1, . . . , xσN) = ρ(N)(x1, . . . , xN) for all xi ∈ ❙d, σ ∈ SN. ”particles are exchangeable” (b) ρ(N)(τx1, . . . , τxN) = ρ(N)(x1, . . . , xN) for all xi ∈ Sd, τ ∈ SO(d + 1), ”isometry invariance” Given any distribution averaging over permutations and isometries yields joint densities with (a) and (b).

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Hyperuniformity on the Sphere

Hyperuniformity on the sphere

Test sets BR are spherical caps, and the point counting function is Nφ :=

N

• i=1

✶C(x,φ)(Xi) , where (X1, . . . , XN) ∼ ρ(N) ✶

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Hyperuniformity on the Sphere

Hyperuniformity on the sphere

Test sets BR are spherical caps, and the point counting function is Nφ :=

N

• i=1

✶C(x,φ)(Xi) , where (X1, . . . , XN) ∼ ρ(N) The expectation remains N-dependent E [Nφ] =

N

• i=1

E[✶C(x,φ)(Xi)] = N

• C(x,φ)

ρ(N)

1

(r) dr = Nσ(C(·, φ)).

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Hyperuniformity on the Sphere

Hyperuniformity on the sphere

The variance depends on N and the pair correlation ρ(N)

2

V [Nφ] = Nσ(C(·, φ))(1 − σ(C(·, φ))) + N(N − 1)

• C(·,φ)

(ρ(N)

2

(x, y) − 1) dx dy

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Hyperuniformity on the Sphere

Hyperuniformity on the sphere

The variance depends on N and the pair correlation ρ(N)

2

V [Nφ] = Nσ(C(·, φ))(1 − σ(C(·, φ))) + N(N − 1)

• C(·,φ)

(ρ(N)

2

(x, y) − 1) dx dy Example: (Xi)i i.i.d. (i.e. ρ(N) = 1) gives E [Nφ] = Nσ(C(·, φ)) and V [NR] = Nσ(C(·, φ))(1 − σ(C(·, φ))).

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Hyperuniformity on the Sphere

Hyperuniformity on the sphere

Heuristic We want to see a similar phenomenon of a reduced order of magnitude of the variance V[Nφ] for N → ∞.

• P. Grabner

Hyperuniformity on the Sphere

Hyperuniformity on the sphere

Heuristic We want to see a similar phenomenon of a reduced order of magnitude of the variance V[Nφ] for N → ∞.

• P. Grabner

Hyperuniformity on the Sphere

Hyperuniformity on the sphere

Heuristic We want to see a similar phenomenon of a reduced order of magnitude of the variance V[Nφ] for N → ∞. Furthermore, we want to switch from random point sets to deterministically contructed point sets.

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Hyperuniformity on the Sphere

Hyperuniformity on the sphere

Heuristic We want to see a similar phenomenon of a reduced order of magnitude of the variance V[Nφ] for N → ∞. Furthermore, we want to switch from random point sets to deterministically contructed point sets. We have identified three regimes for the cap size, where we could encounter and partly characterise this phenomenon.

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Hyperuniformity on the Sphere

Definition of hyperuniformity on the sphere

Definition (Hyperuniformity) Let (XN)N∈N be a sequence of point sets on the sphere Sd. The number variance of the sequence for caps of angle φ is given by V (XN, φ) = Vx# (XN ∩ C(x, φ)) . (1) A sequence is called

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Hyperuniformity on the Sphere

Definition of hyperuniformity on the sphere

Definition (Hyperuniformity) Let (XN)N∈N be a sequence of point sets on the sphere Sd. The number variance of the sequence for caps of angle φ is given by V (XN, φ) = Vx# (XN ∩ C(x, φ)) . (1) A sequence is called hyperuniform for large caps, if V (XN, φ) = o (N) (2) for all φ ∈ (0, π

2 ) and N → ∞;

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Hyperuniformity on the Sphere

Definition continued

Definition

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Hyperuniformity on the Sphere

Definition continued

Definition hyperuniform for small caps, if V (XN, φN) = o (Nσ(C(·, φN))(1 − σ(C(·, φN)))) (3) for N → ∞ and all sequences (φN)N∈N such that

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Hyperuniformity on the Sphere

Definition continued

Definition hyperuniform for small caps, if V (XN, φN) = o (Nσ(C(·, φN))(1 − σ(C(·, φN)))) (3) for N → ∞ and all sequences (φN)N∈N such that

1

limN→∞ φN = 0

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Hyperuniformity on the Sphere

Definition continued

Definition hyperuniform for small caps, if V (XN, φN) = o (Nσ(C(·, φN))(1 − σ(C(·, φN)))) (3) for N → ∞ and all sequences (φN)N∈N such that

1

limN→∞ φN = 0

2

limN→∞ Nσ(C(·, φN)) = ∞.

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Hyperuniformity on the Sphere

Definition continued

Definition hyperuniform for small caps, if V (XN, φN) = o (Nσ(C(·, φN))(1 − σ(C(·, φN)))) (3) for N → ∞ and all sequences (φN)N∈N such that

1

limN→∞ φN = 0

2

limN→∞ Nσ(C(·, φN)) = ∞.

hyperuniform for caps at threshold order, if V (XN, φN) is bounded (4) for all sequences (φN)N∈N with limN→∞ Nσ(C(·, φN)) > 0.

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Hyperuniformity on the Sphere

Characterisation

We could show that a sequence of point sets (XN)N∈N is hyperuniform for large caps, if and only if for all n ≥ 1 lim

N→∞

1 N

N

• i,j=1

Pn(xi, xj) = 0.

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Hyperuniformity on the Sphere

Characterisation

We could show that a sequence of point sets (XN)N∈N is hyperuniform for large caps, if and only if for all n ≥ 1 lim

N→∞

1 N

N

• i,j=1

Pn(xi, xj) = 0. Notice: Uniform distribution of (XN)N∈N is equivalent to lim

N→∞

1 N2

N

• i,j=1

Pn(xi, xj) = 0

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Hyperuniformity on the Sphere

Characterisation

We could show that a sequence of point sets (XN)N∈N is hyperuniform for large caps, if and only if for all n ≥ 1 lim

N→∞

1 N

N

• i,j=1

Pn(xi, xj) = 0. Notice: Uniform distribution of (XN)N∈N is equivalent to lim

N→∞

1 N2

N

• i,j=1

Pn(xi, xj) = 0 For i.i.d. random points on the sphere 1 N

N

• i,j=1

Pn(xi, xj) distr. → χ2

Z(d,n),

which implies a law of the iterated logarithm.

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Hyperuniformity on the Sphere

Large caps (continued)

It does not suffice to assume that V (XN, φ) = o(N) for a single value of φ, since there is a countable set of exceptional φ, where the implication is not true.

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Hyperuniformity on the Sphere

Large caps (continued)

It does not suffice to assume that V (XN, φ) = o(N) for a single value of φ, since there is a countable set of exceptional φ, where the implication is not true. For such exceptional values of φ we can produce sequences of point sets, which have V (XN, φ) = o(N), but lim

N→∞

1 N

N

• i,j=1

Pn(xi, xj) = 0 for n = n0 and lim

N→∞

1 N2

N

• i,j=1

Pn0(xi, xj) = 0, which implies that (XN)N∈N is not even uniformly distributed.

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Hyperuniformity on the Sphere

Small caps

If (XN)N∈N is hyperuniform for small caps, then for all n ≥ 1 the sequence 1 N

N

• i,j=1

Pn(xi, xj) is bounded, which again implies uniform distribution.

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Hyperuniformity on the Sphere

Small caps

If (XN)N∈N is hyperuniform for small caps, then for all n ≥ 1 the sequence 1 N

N

• i,j=1

Pn(xi, xj) is bounded, which again implies uniform distribution. Spherical designs of optimal order are hyperuniform for small caps.

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Hyperuniformity on the Sphere

Determinantal point process in ❙2

A well-studied determinantal point process is given by the distribution of the zeros of random polynomials. ❙

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Hyperuniformity on the Sphere

Determinantal point process in ❙2

A well-studied determinantal point process is given by the distribution of the zeros of random polynomials. Applying stereographic projection g : C × R → C, (z, x) →

z 1−x

transforms the probability density of this process to ρ(N)(p1, . . . , pN) := g∗˜ ρ(N)(p1, . . . , pN) = const.

• i<j

pi − pj2

R3,

with resp. to the normalized Lebesgue measure σ on ❙2.

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Hyperuniformity on the Sphere

Hyperuniformity of determinantal point processes

• n ❙2

Lemma (Alishahi, Zamani ’15) If Nσ(C) → ∞, when N → ∞ and φN → 0. Then for all ǫ > 0: V [#(XN ∩ C)] =

• Nσ(C) + o(log(Nσ(C))1/2+ǫ).
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Hyperuniformity on the Sphere

Hyperuniformity of determinantal point processes

• n ❙2

Lemma (Alishahi, Zamani ’15) If Nσ(C) → ∞, when N → ∞ and φN → 0. Then for all ǫ > 0: V [#(XN ∩ C)] =

• Nσ(C) + o(log(Nσ(C))1/2+ǫ).
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Hyperuniformity on the Sphere

Hyperuniformity of determinantal point processes

• n ❙2

Lemma (Alishahi, Zamani ’15) If Nσ(C) → ∞, when N → ∞ and φN → 0. Then for all ǫ > 0: V [#(XN ∩ C)] =

• Nσ(C) + o(log(Nσ(C))1/2+ǫ).

This implies that point sets sampled from a determinantal point process are hyperuniform for small caps.

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Hyperuniformity on the Sphere

Determinantal point process in ❙2

Figure: 10000 sampled points from an i.i.d. process and a DPP , resp.

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Hyperuniformity on the Sphere

Higher dimensional spheres

In a recent paper by C. Beltr´ an, J. Marzo and J. Ortega-Cerd` a determinantal point processes on ❙d are constructed for certain values of N, which exhibit a similar behaviour as for the process on ❙2.

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Hyperuniformity on the Sphere

Higher dimensional spheres

In a recent paper by C. Beltr´ an, J. Marzo and J. Ortega-Cerd` a determinantal point processes on ❙d are constructed for certain values of N, which exhibit a similar behaviour as for the process on ❙2. They study discrepancy

• f the sample points.
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Hyperuniformity on the Sphere

Higher dimensional spheres

In a recent paper by C. Beltr´ an, J. Marzo and J. Ortega-Cerd` a determinantal point processes on ❙d are constructed for certain values of N, which exhibit a similar behaviour as for the process on ❙2. They study discrepancy Riesz energy

• f the sample points.
• P. Grabner

Hyperuniformity on the Sphere

Higher dimensional spheres

In a recent paper by C. Beltr´ an, J. Marzo and J. Ortega-Cerd` a determinantal point processes on ❙d are constructed for certain values of N, which exhibit a similar behaviour as for the process on ❙2. They study discrepancy Riesz energy separation

• f the sample points.
• P. Grabner

Hyperuniformity on the Sphere

Higher dimensional spheres

In a recent paper by C. Beltr´ an, J. Marzo and J. Ortega-Cerd` a determinantal point processes on ❙d are constructed for certain values of N, which exhibit a similar behaviour as for the process on ❙2. They study discrepancy Riesz energy separation

• f the sample points.

It can be expected that hyperuniformity for small caps can be

• btained also for this process.
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Hyperuniformity on the Sphere

Concluding remarks

The limit S(n) = lim

N→∞

1 N

N

• i,j=1

Pn(xi, xj) that occurred in the characterisation of hyperuniformity is an analogue of the structure factor S(k) = lim

V →Rd

1 |V |

• x,y∈V

e2πik,x−y studied in the context of classical hyperuniformity.

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Hyperuniformity on the Sphere

Concluding remarks

The limit S(n) = lim

N→∞

1 N

N

• i,j=1

Pn(xi, xj) that occurred in the characterisation of hyperuniformity is an analogue of the structure factor S(k) = lim

V →Rd

1 |V |

• x,y∈V

e2πik,x−y studied in the context of classical hyperuniformity. The variance V(XN ∩ C(x, φ)) is a localised version of the classical L2-discrepancy.

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Hyperuniformity on the Sphere

Open questions

Find relations with other measures of uniformity: discrepancy, error of integration, energy. . .

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Hyperuniformity on the Sphere

Open questions

Find relations with other measures of uniformity: discrepancy, error of integration, energy. . . Establish hyperuniformity for known deterministic constructions of point sets on the sphere.

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Hyperuniformity on the Sphere

Open questions

Find relations with other measures of uniformity: discrepancy, error of integration, energy. . . Establish hyperuniformity for known deterministic constructions of point sets on the sphere. Find explicit deterministic constructions for point sets achieving the best possible discrepancy bound (or even a bound better than N− 1

2 )

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Hyperuniformity on the Sphere