The spherical ensemble and uniform distribution of points on the sphere

The spherical ensemble is a well-studied determinantal process with a fixed number of points on the sphere. The points of this process correspond to the generalized eigenvalues of two appropriately chosen random matrices, mapped to the surface of the sphere by stereographic projection. This model can be considered as a spherical analogue for other random matrix models on the unit circle and complex plane such as the circular unitary ensemble or the Ginibre ensemble, and is one of the most natural constructions of a (statistically) rotation invariant point process with repelling property on the sphere. In this paper we study the spherical ensemble and its local repelling property by investigating the minimum spacing between the points and the area of the largest empty cap. Moreover, we consider this process as a way of distributing points uniformly on the sphere. To this aim, we study two"metrics"to measure the uniformity of an arrangement of points on the sphere. For each of these metrics (discrepancy and Riesz energies) we obtain some bounds and investigate the asymptotic behavior when the number of points tends to infinity. It is remarkable that though the model is random, because of the repelling property of the points, the behavior can be proved to be as good as the best known constructions (for discrepancy) or even better than the best known constructions (for Riesz energies).


1.1.
Background. The aim of this paper is to study the statistical properties of a natural point process on the sphere where the points exhibit repulsive behavior. This point process was introduced in [23] and is known as spherical ensemble; see [21] and [22]. The model was studied earlier in [12] and [15], but without observing the connection to random matrices. It was shown in [12,15] that there exists a connection between this model and the one-component plasma on the sphere for a special value of temperature. See [17] for further discussion.
Let A n and B n be independent n × n random matrices with independent and identically distributed standard complex Gaussian entries, and let {λ 1 , λ 2 , . . . , λ n } denotes the set of eigenvalues of A −1 n B n . We can consider these eigenvalues as a (simple) random point process on complex plane C. The point process {λ 1 , . . . , λ n } can be described using the k-point correlation functions ρ (n) k : C k → R ≥0 , 1 ≤ k ≤ n, defined in such a way 1 that C k F (z 1 , . . . , z k )ρ (n) k (z 1 , . . . , z k ) dµ(z 1 ) . . . dµ(z k ) (1.1) Moreover from the repulsive nature of determinantal point processes we expect that the points of process X (n) are typically more evenly distributed than n independently chosen uniform points on the sphere. This repelling property is the common feature of many models in random matrix theory and has been comprehensively studied in some special models such as the Gaussian unitary or the circular unitary ensembles. For example, the distribution or the minimum or the maximum of the gaps between consecutive eigenvalues have been computed and compared to simpler models as a way to measure and understand the repulsive structure. One of the goals of this paper is to do the same computations for the spherical model. We specially focus on the minimum distance between the points, the area of the largest empty cap, the hole probability and the limiting distribution of the nearest neighbors distances as natural two-dimensional extensions of the so called metrics studied in one dimensional models.
On the other hand we exploit this model and its properties to the classic problem of distributing points on the sphere. The problem of distributing a given number of points on the surface of a sphere "uniformly", is a challenging and old problem. Contrary to the one dimensional case (i.e. distributing points on a circle) where the most uniform arrangement clearly exists and is attained when the points are equidistributed, it seems that there is no arrangement on the sphere that can be considered as completely uniform, and the answer for the best arrangement depends on what criteria do we use to quantify the uniformity of an arrangement. Among the mostly used criteria are those related to the electrostatic potential energy and its generalizations where one tries to distribute the points in a way that minimizes some energy function. Another common metric is the discrepancy that measures the maximum deviance between the number of points and the expected number, in some class of regions in the underlying space (sphere, in our case). Both the energy and the discrepancy optimization problems, i.e. finding the most optimum arrangement or even obtaining some relatively sharp upper and lower bounds are open and challenging problems for the sphere. We study these metrics (discrepancy and Riesz energies) to measure the uniformity of points of X (n) . For each of these metrics we obtain some bounds and investigate the asymptotic behavior when the number of points tends to infinity. It is remarkable that though the model is random, because of the repelling property of the points, the behavior can be proved to be as good as the best known constructions (for discrepancy) or even better than the best known constructions (for Riesz energies).
The main results of the paper are stated in the next subsection, together with definition and some properties of the metrics discussed above.

Main results.
Discrepancy. The geometrically most natural measure for the uniformity of the distribution of an n-point set on S 2 is the spherical cap discrepancy. Let P n = {x 1 , . . . , x n } be an n-point set on S 2 . The spherical cap discrepancy of P n is defined as where A is the set of all spherical caps on S 2 . A spherical cap is defined as the intersection of the sphere and a half-space. In [4], it was shown that there is constant c > 0, independent of n, such that for any n-point set P n on S 2 we have On the other hand, using probabilistic methods it has been shown (see [5]) that for any n ≥ 1, there exists n-point set P n on S 2 such that D(P n ) ≤ Cn 1/4 log n.
The following theorem shows that the point process X (n) has small spherical cap discrepancy.
For every M > 0 independent of n, one has D({P 1 , . . . , P n }) = O n 1/4 log n with probability 1 − 1 n M . Note that for independent uniform points on sphere, the discrepancy is of order √ n (up to a logarithmic factor).
The key to the proof of Theorem 1.1 is an estimate on the variance of the number of points of X (n) on a spherical cap. The asymptotic expansion of this variance and the proof of Theorem 1.1 will be presented in Section 2.
Largest empty cap. Given P n = {x 1 , . . . , x n } ⊂ S 2 , define the covering radius of P n as the infimum of the numbers t > 0 such that every point of S 2 is within distance t of some x j . If we let τ be the covering radius of P n , then the area of the largest spherical cap which does not contain any point of P n in its interior is equal to πτ 2 (note that, for fixed q, the area of the spherical cap {p ∈ S 2 : |p − q| ≤ r} is πr 2 ). We will be interested in studying the asymptotic behavior of the area of the largest empty cap for the spherical ensemble.
Let M n be the area of largest empty cap for random point process X (n) . In the following theorem, we derive first-order asymptotic for M n .
The proof of this theorem is given in Section 3. For the proof we need asymptotics of the hole probability, the probability that there are no points of X (n) in a given spherical cap. The desired asymptotics of the hole probability will be given in Lemma 3.1. Then we will prove Theorem 1.2 using a method similar to the proof of Theorem 1.3 in [3]. Notice that for independent uniform points on S 2 , the area of largest empty cap is of order log n n .
At the end of Section 3 we study the nearest neighbour statistics of spherical ensemble and show a connection between the local behavior of this model and the Ginibre ensemble.
Riesz and logarithmic energy. In Section 4, we compute the expectations of the logarithmic energy and Riesz s-energy of the random point process X (n) on S 2 . The discrete logarithmic energy of n points x 1 , . . . , x n on S 2 is given by Also we define E log (n) := min{E log (x 1 , . . . , x n ); x 1 , . . . , x n ∈ S 2 }. The n-tuples that minimize this energy are usually called elliptic Fekete Points. Define C n by E log (n) = 1 2 − log 2 n 2 − 1 2 n log n + C n n.
In [27] it was shown that C n satisfies the following estimates For a given s, the Riesz s-energy of n points x 1 , . . . , x n on S 2 are defined as Also, we consider the optimal n-point Riesz s-energy, The important special case s = 1 corresponds to electrostatic potential energy of electrons on S 2 that repel each other with a force given by Coulomb's law. We remark that this problem is only interesting for s > −2. It is known that for the potential-theoretic regime, −2 < s < 2, we have See e.g. [10]. Consider the difference E s (n) − 2 1−s 2−s n 2 . Wagner ([33] lower bound for 0 < s < 2 and upper bound for −2 < s < 0, and [34] upper bound for 0 < s < 2 and lower bound for −2 < s < 0) proved that where c 1 and c 2 are positive constants depending only on s. In [27], an alternative method which is based on partitioning S 2 into regions of equal area and small diameter is used to prove the upper bound in the case 0 < s < 2 (and lower bound in the case −2 < s < 0). Let ε > 0 be arbitrary, this method gives for n ≥ n 0 (ε, s). We show that the better bounds than (1.4) and (1.5) can be obtained by considering the expectation of Riesz s-energy of point process X (n) . It is conjectured (see [10], Conjecture 3) that the asymptotic expansion of the optimal Riesz s-energy for −2 < s < 4, s = 2 has the form

Also, in [10] (see Proposition 3 and its remark therein), it is shown that
Considering the expectation of Riesz s-energy of point process X (n) , we are able to omit the log log n term in this estimate. (See Conjecture 5 in [10] for the asymptotic expansion of the optimal Riesz 2-energy.) We remark that the first term of the asymptotics of E s (n) for s > 2 is not known. In the following theorem, we give the expectations of the logarithmic energy and Riesz s-energy of the random point process X (n) on S 2 . Theorem 1.3. For the point process X (n) on S 2 , n ≥ 2, we have i) (Logarithmic energy) ii) (Riesz s-energy: s < 4 and s = 2) iii) (Riesz s-energy: s = 2) As a corollary to Theorem 1.3, we obtain the following bounds for optimal Riesz s-energy.
Suppose that x 1 , . . . , x n are chosen randomly and independently on the sphere, with the uniform distribution. One can easily show that Minimum spacing. Define the minimum spacing by We are interested in the asymptotic behavior of m n as n tends to infinity. For 0 < t < 2, set (1.14) G t,n := to be the number of non-ordered pairs of distinct points at Euclidean distance at most t apart. Our result concerning distribution of G t,n is the following.
Theorem 1.5. Let G t,n defined by (1.14) and assume that t = x n 3/4 then G t,n converges in distribution to the Poisson random variable with mean x 4 64 . The proof of this theorem is given in Section 5. As a consequence, since P(G t,n = 0) = P(m n > t), Theorem 1.5 clearly implies that Suppose that x 1 , . . . , x n are chosen randomly and independently on the sphere, with the uniform distribution. It was shown that (see Theorem 2 of [11])

Discrepancy
Let D be a spherical cap on the sphere S 2 . Define the number of points of X (n) in D. Clearly, the expected value of N D is equal to n|D| 4π . In order to prove Theorem 1.1 we need to control the variance of N D . The following lemma gives the asymptotic behavior of the variance.
Then for any ε > 0 where |D| denotes the area of D and D c = S 2 − D.
Proof. The distribution of N D is invariant under isometries of the sphere, so we may assume without loss of generality that g(D) is a disk centred at the origin with radius r. From [20], Theorem 26, the set {|λ k | 2 : 1 ≤ k ≤ n} has the same distribution as set where the random variables Q k are jointly independent and Q k for 0 ≤ k ≤ n − 1 has density n n−1 k q k (1 + q) n+1 , q ≥ 0 (Note that K(z, w) = n−1 k=0 n−1 k (zw) k and µ has density φ(|z|) where φ(x) = n π(1+x 2 ) n−1 ). Thus Q k has the beta prime distribution with parameters k + 1 and n − k, i.e., Q k 1+Q k has the beta distribution with parameters as before.
Thus we have And so by the independence of Q k , 0 ≤ k ≤ n − 1, we deduce that where the random variables η k are independent and distributed as Bernoulli variables with P(η k = 1) = P(Q k < r 2 ). By the properties of incomplete beta function (see e.g. [36]) we have Now suppose B 1 , . . . , B n are i.i.d. Bernoulli random variables with parameter α := r 2 1+r 2 . Let S n := B 1 + · · · + B n . One can write the right-hand side of above equation as P(S n > k). Therefore, we obtain for every 0 ≤ k ≤ n − 1 We also note that |D| = 4πr 2 1 + r 2 = 4πα. Now from (2.3) and (2.4) it follows that for every spherical cap D, where S n is a binomial random variable with parameters n and |D| 4π .
, by symmetry, we may assume that α ≤ 1/2. Using the Bernstein's inequality we see that for any t > 0, one has

(see for example Lemma 2 in [5]). We can use this to show that
The assumption of the lemma implies that σ → +∞ as n → +∞ and thus we have Also, by comparing the integral of exp −x 2 4 with its Riemann sum, we conclude that Thus, from (2.7), (2.8) and above estimate, we obtain Moreover, from the Berry-Esséen theorem we see that for some absolute constant C, one has (see [6]) where Φ is the cumulative distribution function of the standard normal distribution. From (2.5), and by using the above two estimates, we conclude that

By considering the Riemann sum of
(The difference in the left-hand side of above equation is less than the total variation of the integrand, possibly plus a constant.) Using the standard bound for any x > 0, we have Combining all these facts, we deduce that

On the other hand, from integration by parts, we have
and (2.1) follows.
Remark. Clearly, the restriction of point process X (n) to a spherical cap D is also a determinantal point process with kernel the integral operator in L 2 (g(D)) obtained by considering this kernel. Then the distribution of N D is the sum of independent Bernoulli random variables, whose expectations are the eigenvalues of K g(D) are equal to P(S n > k), 0 ≤ k ≤ n − 1, where S n is a binomial random variable with parameters n and |D| 4π . From Lemma 2.1 and the general central limit theorem for determinantal point processes [29] (due to Costin and Lebowitz [13] in case of the sine kernel) we have the following theorem.
Next we prove Theorem 1.1.
Proof of Theorem 1.1. We use the notation from the proof of Lemma 2.1. As we have seen before

Similar to inequality (2.7), we have
Also, we see So we deduce that there exist absolute constants c 1 , c 2 such that for any spherical cap D, and since σ ≤ √ n 2 , we have for some absolute constant c 3 , We know that for any spherical cap D, the random variable N D has the same distribution as k η k where η k are independent Bernoulli random variables (see equation (2.3)). From Bernstein's inequality we get a concentration estimate for N D :  log P(S n ≤ k).
By the exponential Chebyshev inequality, we have for any k ≤ nα where I(x) = sup t∈R (tx − log(E e tB 1 )) is the Legendre transform of the cumulant generating function of B 1 . One easily computes that On the other hand, from Stirling's formula we have for 0 < k ≤ nα and some absolute constant c 0 . Therefore, for all 0 < k ≤ nα, we have (3.4) log P(S n ≤ k) ≥ −nI(k/n) − 1 2 log(nα) + log c 0 .

So we conclude that for sufficiently large constant c
uniformly in c ′ /n < α < 1 − c ′ /n.

largest empty cap.
Proof of Theorem 1.2. We follow the proof of Theorem 1.3 in [3]. Let ε > 0 be arbitrary and set X n := n 8π √ log n M n we easily see the inequality

Hence, it suffices to show that the last two terms go to zero as n tends to infinity. Integrating by parts, we get
s(u − 1) s−1 P(X n > u) du + ǫ s P(X n > 1 + ǫ).
Next, we give an upper bound for P(X n > u). We can choose on S 2 at most 4n points q 1 , . . . , q m so that every spherical cap with area 4π/n contains at least one of these points. (We can find m ≤ 4n spherical caps with area π/n such that there exists no spherical cap with this area that does not intersect these m spherical caps. Let q 1 , . . . , q m be the center of these m spherical caps. One can check that q 1 , . . . , q m have desired properties.) Let Y i , 1 ≤ i ≤ m, be the area of the largest empty cap centred at q i . Note that there exists some q j within distance 2/ √ n of the center of the largest empty cap. From this, we can conclude that for any u ≥ 1 + ε and sufficiently large n, Thus, for sufficiently large n (3.9) P(X n > u) ≤ 4n∆ n 2 √ log n n (1 + ε/2) .
The first integral on the right hand side goes to zero as n → +∞, thanks to (3.10).

Using a similar argument as in (3.8) and the crude upper bound
∆ n (α) ≤ P(S n = 0) = (1 − α) n we conclude that for any fixed C > 0, one has P(X n > u) = o(n −C ) uniformly for u ≥ log n. Thus the second integral also goes to zero as n → +∞.
To prove Theorem 1.2, it suffices to show that P(X n < 1 − ε) converges to zero.  (1−ǫ)8π √ log n n . If F n be the number of such caps free of eigenvalues then from previous lemma and Chebyshev's inequality one has (1) , which implies that P(X n < 1 − ǫ) → 0 as desired.

Nearest neighbour statistics. Consider the random point process X
the minimum distance from P j to the remaining points. We define, as in [9], the nearest neighbour spacing measure µ(P 1 , . . . , P n ) on [0, +∞) by Let We want to show that, Theorem 3.3. As n → +∞, One can easily check that for independent uniform points on sphere, this measure converges to e −x dx as n tends to infinity. See Figure 1.
First, we compute the expectation of Y n . Let∆ n (α) denote the probability that the spherical cap D with area 4πα centred at P j has no other points of {P 1 , . . . , P n } in its interior. With this definition we obtain (3.14) E Y n = n(1 −∆ n (x/n)).
We first show that where S n is a binomial random variable with parameters n and α. For α ′ < α define ∆ n (α, α ′ ) := P N D\D ′ = 0 where |D| = 4πα, |D ′ | = 4πα ′ and D, D ′ have the same center. ∆ n (α, α ′ ) − ∆ n (α) is the probability that there are no points of X (n) in the D\D ′ , but there is at least one point (of X (n) ) in the D ′ . Now, conditioning on the {N D ′ ≥ 1} and letting α ′ → 0 (keeping only the first-order terms), one obtains

Similar to (2.2), we have
where Q k , 0 ≤ k ≤ n − 1, are as in Lemma 2.1. A straightforward computation yields

Thus, inserting this into (3.16), we obtain (3.15).
Next, we show that For this, we will use identity (3.15) and Poisson approximation. From a result in [25] we have uniformly for all k. From (2.6) we see that there exists M = O(log n) such that for every k ≥ M we have where, for the second inequality, note that from Taylor's theorem and then use Stirling's formula. Therefore, using the inequality

In [26] (See Theorem 3.5 therein) it has been shown that for every 1-Lipschitz function
.

Using the Borel-Cantelli Lemma, this gives (3.13) as desired.
Remark. The infinite Ginibre ensemble is a translation invariant determinantal point process on the complex plane C with kernel K(z, w) = e zw with respect to the Gaussian measure 1 π e −|z| 2 dz. It can be viewed as the local limit of the law of eigenvalues of random matrices from the complex Ginibre ensemble (see [21] for more details). From (3.17) we deduce that lim n→+∞ ∆ n (x/n) = exp(−x)E ∞ (x).

Riesz and logarithmic energy
The purpose of this section is to establish Theorem 1.3 and Corollary 1.4. We start with computing the correlation functions of X (n) on S 2 . Letρ (n) k (p 1 , . . . , p k ), k ≥ 1, be the correlation functions of the point process X (n) with respect to the surface area measure dν(p). Set g(p) = z and g(q) = w. Since dν(p) = 4 (1+|z| 2 ) 2 dz we conclude that Also, one can easily verify that Recall that K (n) (z, w) = (1 + zw) n−1 . A short computation then shows that
Making the change of variable u = r 2 , we see that .

In the last line, we have used the beta function identity
By using induction on n, one can show that for s = 2 Combining this with (4.5), we obtain ( It remains to prove (1.8). By differentiation right hand side of (4.4) with respect to s at s = 0, we conclude that
Also, For 0 < β < 1 and x > 0, we have (See [35]) Hence, for 0 < s < 2, Γ(n + 1 − s/2) Γ(n) ≤ n 1−s/2 . Therefore, using similar argument as above, we get (1.12). If we set β = −s/2 and x = 1 in (4.7), we then have for −2 < s < 0, Γ(1 − s/2) < 1, therefore Remark. For s ≥ 4, the expected value of Riesz s-energy for X (n) is infinite. Compare with the points that are chosen randomly and independently on the sphere, with the uniform distribution. in this case, the expected value of Riesz s-energy is infinite for s ≥ 2. Remark. Another interesting random point process on S 2 is the roots of random polynomials via the stereographic projection. Let f (z) = n j=0 a j z j where the coefficients a j are independent complex Gaussian random variables with mean 0 and variance n j . Let z 1 , . . . , z n are the complex zeros of f (z). In [14] the expectation of the logarithmic energy for point process {g −1 (z 1 ), . . . , g −1 (z n )} was computed. It was shown that E E log (g −1 (z 1 ), . . . , g −1 (z n )) = 1 2 − log 2 n 2 − 1 2 n log n − 1 2 − log 2 n.

Minimum spacing
Point pair statistics. Recall the definition (1.14) of the function G t,n . We first compute the expectation of G t,n . The argument is similar to what was done in the proof of Theorem 1.3 in Section 4. Let F (p, q) = 1 {|p−q|≤t} . From (4.3) and Fubini's theorem we have If we set t = o( 1 √ n ) then we see that and for t = x n 3/4 , where x > 0 is fixed, we get The above equation shows that the correct scaling for the minimum spacing is n −3/4 . To prove Theorem 1.5, similar to [3], we will use a modification of the method from Soshnikov. This method has been used in [3] and [30] (see also [28]) to solve similar problems in one-dimensional case. We will modify this method that works in our case. Moreover this modification makes the proof simpler. The following two lemmas will be used frequently in the proof. The next lemma will be used to control the k-point correlation function in terms of the lower order correlation functions. For the proof see [19]. Fix c > 0 and let t = c n 3/4 . Define for p ∈ S 2 C (n) (p) = {q ∈ S 2 : |p − q| ≤ t} Consider the random point process We define a new point processX (n) . First consider all pairs (P i , P j ) such that n 3/4 |P i − P j | < x and Then from each pair select independently with probability 1 2 one of the two items, and consider all this points asX (n) . (Compare this with the modified processes that have been used in [3,30] in similar cases.) Then let Z n =X (n) (S 2 ).
Proof. We show that G t,n − Z n L 1 −→ 0. First note that we have To see this, observe that G t,n − Z n is equal to number of pairs (P i , P j ) such that 0 < |P i − P j | < t and there exist some point P k where 0 < |P i − P k | < t or 0 < |P j − P k | < t. By considering the triples (P i , P j , P k ) or (P j , P i , P k ) we see that Thus, taking expectation gives (5.4). If q 1 , q 2 ∈ C (n) (p) then |q 1 − q 2 | = O(n −3/4 ) and using Hadamard-Fischer inequality and (5.3) we obtain ρ (n) 3 (p, q 1 , q 2 ) ≤ρ   For n large enough we make assume that for i = j |p i − p j | > 4t. . . .