Eigenvalues of truncated unitary matrices: disk counting statistics

Let $T$ be an $n\times n$ truncation of an $(n+\alpha)\times (n+\alpha)$ Haar distributed unitary matrix. We consider the disk counting statistics of the eigenvalues of $T$. We prove that as $n\to + \infty$ with $\alpha$ fixed, the associated moment generating function enjoys asymptotics of the form \begin{align*} \exp \big( C_{1} n + C_{2} + o(1) \big), \end{align*} where the constants $C_{1}$ and $C_{2}$ are given in terms of the incomplete Gamma function. Our proof uses the uniform asymptotics of the incomplete Beta function.


Introduction
Let n ∈ N >0 , α > 0, and consider the joint probability measure where Z n is the normalization constant.Note that the z j 's are constrained to lie in the unit disk D := {z ∈ C : |z| ≤ 1}.A main motivation for studying this point process stems from its connection with random matrices: it is shown in [19] that for α ∈ N >0 , (1.1) is the law of the eigenvalues of an n × n truncated unitary matrix T , i.e.T is the upper-left n × n submatrix of a Haar distributed unitary matrix of size (n + α) × (n + α).By rewriting (1.1) in the form we infer that for general α > 0 (not necessarily α ∈ N >0 ), (1.1) is also the law of a Coulomb gas with n particles at inverse temperature β = 2 associated with the potential Q [11].
We emphasize that (1.1) is a probability measure only for α > 0. If α = 0, the above matrix T is an n × n Haar distributed unitary matrix, for which the n eigenvalues z 1 , . . ., z n lie exactly on the unit circle according to the probability measure proportional to As is well-known, the equilibrium measure associated with (1.2) is the uniform measure on the unit circle.
This work focuses on the point process (1.1) as n → +∞ with α > 0 fixed.In this regime, Q(z) = O(n −1 ) for any fixed z ∈ D, and the associated equilibrium measure µ is defined as the unique measure minimizing the following energy functional among all Borel probability measures ν supported on D. This problem is a so-called classical (or unweighted) electrostatics problem, and as such the support of µ must be the boundary of D [16] (and this, despite the fact that the point process (1.1) is two-dimensional).Because the density of (1.1) is invariant under rotation, we conclude that µ is the uniform measure on the unit circle ∂D.
In the language of random matrix theory, ∂D is a "hard wall" of (1.1).For two-dimensional Coulomb gases, it is a standard fact that along a hard wall the equilibrium measure is singular with respect to the two-dimensional Lebesgue measure; moreover, a non-zero percentage of the points are expected to accumulate (as n → +∞) in a very small interface of width 1/n around the hard wall, see e.g.[16,17,3].Following [3], we call this small interface "the hard edge regime".A particular feature of (1.1) is that the associated equilibrium measure µ is purely singular (i.e.µ has no absolutely continuous component and ∂D dµ = 1).Hence, for large n and α fixed, most of the points z 1 , . . ., z n of (1.1) are expected to lie in a 1/n-neighborhood of ∂D, see also Figure 1.
There already exists a fairly rich literature on truncated unitary matrices.For example, the convergence of the distribution of the maximal modulus max j |z j | to the Weibull distribution has been studied in [13,14,17], several characterizations in terms of Painlevé transcendents for expectations of powers of the characteristic polynomial are established in [9], and results on the eigenvectors can be found in [10].Also, besides (1.1), other two-dimensional point processes whose points are distributed within a narrow interface (or "band") have been considered, see e.g.[12,2,7]; however, the point processes considered in these works only feature "soft edges", and are thus very different from (1.1).
Let N(y) := #{z j : |z j | < y} be the random variable that counts the number of points of (1.1) in the disk centered at 0 of radius y.The goal of this paper is to understand the large n behavior of the multivariate moment generating function We consider the hard edge regime, i.e. the radii r 1 , . . ., r m are merging near 1 at the critical speed 1 − r j n −1 (we also allow r m = 1), see also Figure 2.More precisely, we define Note that if t m = 0, then r m = 1 and trivially N(r m ) = n with probability one.In Theorem 1.2 below, we prove that and we give explicit expressions for the constants C 1 and C 2 in terms of the following functions where s a−1 e −s ds is the Gamma function, and Q is the normalized incomplete Gamma function: Note that H α is also well-defined at x = 0, while G α is well-defined at x = 0 only for α ≥ 1.
The statement of our main theorem involves ln H α , and the function H α also appears in the denominator of (1.6).The following lemma implies that ln H α and G α are well-defined and real-valued for x ∈ (0, 1], u = (u 1 , . . ., u m ) ∈ R m , and t 1 > . . .> t m ≥ 0. In this paper, ln always denotes the principal branch of the logarithm.

Comparison with other works on counting statistics.
There has been a lot of interest recently on counting statistics of two-dimensional point processes.We will not attempt to survey this literature here, but refer the interested reader to [5] and the introduction of [1].Our main goal in this subsection is to compare Theorem 1.2 with the works [8,3].
In [8], the following Mittag-Leffler ensemble is considered: where b > 0 and a > −1 are parameters of the model.The associated equilibrium measure µ ML is supported on the disk {|z| ≤ b − 1 2b } and given by µ Note that µ ML is absolutely continuous with respect to the Lebesgue measure d 2 z (this contrasts with the equilibrium measure µ of (1.1), which is purely singular).Let m ∈ N >0 , r ∈ (0, b − 1 2b ), s 1 , . . ., s m ∈ R, a > −1 and b > 0 be fixed parameters such that s 1 < . . .< s m .The main result of [8] is the large n asymptotics of the m-point moment generating function of the disk counting statistics of (1.18) when the radii are merging either in the bulk, i.e. r = r 1 + for all ∈ {1, . . ., m}, or at the soft edge, i.e.
for all ∈ {1, . . ., m}.In both cases, it is shown in [8] that and the constants D 1 , . . ., D 4 are determined explicitly.The constant D 1 is particularly simple; for example, in the bulk regime it is given by The constant D 2 is more complicated and given by u j dx, for the bulk regime, for the soft edge regime, where χ (−∞,0) (x) = 1 if x < 0 and χ (−∞,0) (x) = 0 otherwise, s := (s 1 , . . ., s m ), and We find it curious that the above function has the same structure as the function H α in (1.5); namely, there are both of the form where X (x) := Q(α, t x) in the present paper and X (x) := erfc(x − s )/2 in [8].Note also that • H α already appears in the leading constant C 1 , while H ML appears in D 2 , • t 1 , . . ., t m are dilation parameters of X , in the sense that they appear in the multiplicative form "t x" in Q(α, t x), while s 1 , . . ., s m are translation parameters of X , in the sense that they appear in the additive form "x − s " in erfc(x − s )/2.
Let 0 < ρ < b − 1 2b be fixed.The following point process was considered in [3]: The only (but important) difference between the point processes (1.18) and (1.20) is that in (1.20) the points are constrained to lie in the disk {|z| ≤ ρ}.Because ρ < b − 1 2b , the circle {|z| = ρ} is a hard wall of (1.20) and it is shown in [3] that the associated equilibrium measure µ ML h is given by where z = re iθ , r > 0, θ ∈ (−π, π] and c ρ : where This function is also in the form (1.19), with . It is also interesting to note the presence of the term E 2 ln n in (1.22), while in (1.9) there is no term proportional to ln n.We believe the reason for this is that µ ML h has a non-trivial component µ ML reg which is absolutely continuous with respect to d 2 z, while the equilibrium measure µ of (1.1) is purely singular.This belief is supported by the following fact: when ρ → 0, the measure µ ML h becomes purely singular (because c ρ → 1), and E 2 → 0 (as can be easily checked from [3,Theorem 1.3

]).
A transition regime between the hard edge and the bulk was also considered in [3].This regime is called "the semi-hard edge regime" and corresponds to the case when the radii are at a distance of order 1/ √ n from the hard edge.More precisely, for r = ρ 1 + where The function H ML sh is also in the form (1.19), with X (x) = erfc(x−s ) erfc(x) .The above discussion is summarized in Figure 4.

Point process
Regime Figure 4: Summary.r ) , and define

Preliminaries
By rewriting 1≤j<k≤n |z k − z j | 2 as the product of two Vandermonde determinants, and then using standard algebraic manipulations, we get Since w is rotation-invariant, only the diagonal elements in (2.2) are non-zero, and thus Substituting (2.1), we then find It will be convenient for us to rewrite ω (defined in (2.1)) as follows: where r m+1 := +∞.Using (2.5) in (2.4) yields the following expression for ln E n : ) where B(j, α) is the Beta function (2.9) Many properties of these functions are stated e.g. in [15, Sections 5.12 and 8.17].It is also convenient for us to consider the normalized incomplete Beta function, which is given by so that F n,j, = I(r 2 , j, α).
Hence, to analyze the right-hand side of (2.6), we need the asymptotics of I(v, j, α) when (v−1) n −1 and simultaneously j ∈ {1, . . ., n} and α fixed.The large n behavior of I(r 2 , j, α) depends crucially on whether j remains bounded or not as n → +∞.We will therefore split the sum (2.6) in two parts as follows where and is a new parameter such that N n/M → +∞ as n → +∞.(A more naive choice for M would be M = n/M where M is large but fixed, but this choice does not yield a good control over certain error terms in the proof.The precise reason as to why we choose M n is technical and will become apparent at the end of Section 3.) The following lemma establishes an exact identity that will be useful to handle the sum S 0 , i.e. to obtain the large n asymptotics of F n,j, when j is "not very large".
To analyze S 1 , we will use the uniform asymptotics of the incomplete Beta function (this is the main novelty of the proof, as earlier works such as [8,3] on the Mittag-Leffler ensemble rely instead on the uniform asymptotics of the incomplete gamma function).The following lemma is due to Temme [18,Section 11.3.3.1](this result can also be found in e.g.[15, Section 8.18(ii)]).Lemma 2.2 (Temme [18]).Let N ∈ N >0 .As j → +∞ with α > 0 fixed, uniformly for v in compact subsets of (0, 1].The coefficients F k = F k (v, j, α) are defined by with the initial assignments where Q is defined in (1.7) and the coefficients d k = d k (v, α) are defined through the generating function In particular, Remark 2.3.(Determinants with circular root-type singularities.)Note from (2.2) that E n can be seen as a ratio of two determinants.The determinant on the numerator involves w, and this weight has a root-type singularity along the unit circle (i.e.along the hard edge).Other determinants with circular root-type singularities have been considered in [4]; however, the singularities in [4] lie in the bulk, and the asymptotics of the corresponding determinants involve the so-called associated Hermite polynomials (this contrasts drastically with the asymptotics of E n , which are given in Theorem 1.2).Remark 2.4.(Partition function.)Asymptotic expansions of partition functions of two-dimensional point processes are a classical topic of interest, see e.g.[5,Section 5.3].For rotation-invariant (and determinantal) ensembles with soft edges, precise formulas up to and including the term of order 1 have been obtained in the recent work [6].The class of ensembles considered in [6] includes (1.1) when α is proportional to n, see [6,Section 4.2].As mentioned earlier, for α fixed, the ensemble (1.1) has a hard edge and is therefore not considered in [6].As a minor aside, we compute here the partition function of (1.1) with α fixed using a similar formula as (2.3).As in (2.3) (but with w(u) replaced by (1 − |u| 2 ) α−1 ), we get where for the last identity we have used the functional equation for the Barnes G-function to write Using the expansion (see [15,Eq. 5.17.5]) we then get 3 Proof of Theorem 1.2 As mentioned in Section 2, it is convenient to split the sum (2.6) into two parts: where and ) −1 n 2 2+α .Define also Ω := e u1+•••+um .We first obtain the large n asymptotics of S 0 using Lemma 2.1.Lemma 3.1.Let x 1 , . . ., x m ∈ R be fixed.There exists δ > 0 such that Proof.Let K := n/M − 1. Recalling that F n,j, = I(r 2 , j, α) and r = (1 − t n ) 1/2 , and using Lemma 2.1, we infer that as n → +∞ uniformly for j ∈ {1, . . ., K} and ∈ {1, . . ., m}.Using (see e.g.[15, formula 5.11 we infer that B(j, α) = O(j −α ) as j → +∞ with α fixed, so that the error term in (3.3) can be replaced by O((j/n) α+1 ).Hence, since 1 The O-term after the first equality is clearly independent of u 1 , . . ., u m , and therefore the O-term after the second equality is uniform for u for any fixed δ > 0. We can (and do) choose δ > 0 sufficiently small such that Ω remains bounded away from (−∞, 0] for The above O-term can also be written as O( n M 2+α + n M 1+2α ), and thus By (2.8) and the functional relation Γ(z + 1) = zΓ(z), This last sum can be rewritten as −k and then changing indices of summations, we obtain We now use the so-called "parallel summation formula" Substituting the above in (3.6) yields This formula can easily be expanded as K = n M − 1 → +∞ using (3.4):We now turn to the analysis of S 1 .We will rely on Lemma 2.2, as well as on the following Riemann sum approximation lemma (whose proof is omitted).as n → +∞ uniformly for ∈ {1, . . ., m} and j ∈ { n M , . . ., n}.Moreover, using (2.11), (2.12), (2.13) and (3.4), we get F n,j, = Q(α, t j/n) + (j/n) α j e −t j/n t α Γ(α)

Figure 2 :
Figure 2: Left: two circles (in black) merging near the unit circle (in red).Right: a zoom is taken around i.For both pictures, α = 10 and n = 500.