On the characteristic polynomial of the eigenvalue moduli of random normal matrices

We study the characteristic polynomial $p_{n}(x)=\prod_{j=1}^{n}(|z_{j}|-x)$ where the $z_{j}$ are drawn from the Mittag-Leffler ensemble, i.e. a two-dimensional determinantal point process which generalizes the Ginibre point process. We obtain precise large $n$ asymptotics for the moment generating function $\mathbb{E}[e^{\frac{u}{\pi} \, \mathrm{Im} \ln p_{n}(r)}e^{a \, \mathrm{Re} \ln p_{n}(r)}]$, in the case where $r$ is in the bulk, $u \in \mathbb{R}$ and $a \in \mathbb{N}$. This expectation involves an $n \times n$ determinant whose weight is supported on the whole complex plane, is rotation-invariant, and has both jump- and root-type singularities along the circle centered at $0$ of radius $r$. This"circular"root-type singularity differs from earlier works on Fisher-Hartwig singularities, and surprisingly yields a new kind of ingredient in the asymptotics, the so-called associated Hermite polynomials.


Introduction and statement of results
The Mittag-Leffler ensemble with parameters b > 0 and α > −1 is the joint probability distribution where Z n is the normalization constant.This determinantal point process can be realized as the eigenvalues of a random normal matrix M with distribution proportional to | det(M )| 2α e −n tr((MM * ) b ) dM [51].The special case (b, α) = (1, 0) also corresponds to the eigenvalue distribution of a Ginibre matrix [39], i.e. an n × n matrix with independent complex Gaussian entries with mean 0 and variance 1 n .Consider the characteristic polynomial p n (x) = ) is fixed means that we focus on "the bulk regime".By definition, the expectation (1.2) is equal to D n /Z n , where , (1.3) and the weight w is given by w(z) := |z| 2α e −n|z| 2b ω(|z|), ω(x) := |x − r| a e u , if x < r, 1, if x ≥ r. (1.4) Hence our results can also be seen as large n asymptotics for n × n determinants whose weight is supported on C, rotation-invariant, has both jump-and root-type singularities along the circle centered at 0 of radius r (which we will call "circular" jump-and root-type singularities), and a "pointwise" root-type singularity at 0.
Over the past 50 years or so a lot of works have been done on structured determinants with singularities, and we briefly pause here to review the literature.In their pioneering work [34], Fisher and Hartwig made a conjecture for the asymptotics of large Toeplitz determinants when the weight is supported on the unit circle and has root-and jump-type singularities-such singularities are now called Fisher-Hartwig singularities.Many authors have contributed in proving this conjecture for certain parameter ranges, among which Lenard [50], Widom [62], Basor [6], Böttcher and Silbermann [12], and Ehrhardt [30].A counterexample to the Fisher-Hartwig conjecture was found by Basor and Tracy in [9], and the corrected conjecture was solved for general values of the parameters by Deift, Its and Krasovsky [28].The study of these singular determinants was motivated mainly from questions arising in the Ising model and impenetrable bosons, see [8,29] for more historical background.In recent years, these determinants have also attracted considerable attention in the random matrix community.One reason for that is the well-known work [41] of Keating and Snaith, where numerical evidences were found of links between the characteristic polynomials of unitary and Hermitian random matrices and the zeros of the Riemann zeta function on the critical line.Expectations of powers of the absolute value of the characteristic polynomial of the Gaussian Unitary Ensemble, which are Hankel determinants with a Gaussian weight on R and root-type singularities, were investigated in [13] and their asymptotics were obtained by Garoni [38] for integer values of the parameters and by Krasovsky [42] for the general case.This result was then generalized by Berestycki, Webb and Wong [10] for one-cut regular ensembles.In a different direction, Its and Krasovsky in [40] obtained asymptotics of Hankel determinants with a jump-type singularity and a Gaussian weight.Such determinants provide information about the imaginary part of the log-characteristic polynomial of the Gaussian Unitary Ensemble.The results [40,10] have been generalized in [16] for general Fisher-Hartwig singularities and one-cut regular ensembles.The case of one-cut regular ensembles with hard edges was then treated in [20], and the multi-cut case in [21].Strong results on Toeplitz determinants with merging Fisher-Hartwig singularities are also available in the literature [24,32]; these results have been useful to prove a conjecture of [37] on "the moments of the moments" of the characteristic polynomial of random unitary matrices, and [24] has also been used by Webb in [60] to establish a connection between random matrix theory and Gaussian multiplicative chaos.There exists also a vast literature on other structured determinants with Fisher-Hartwig singularities, see e.g.[25,26] for Fredholm determinants, [7,35,23] for Toeplitz+Hankel determinants, and [19] for a biorthogonal generalization of Hankel determinants.
The literature on determinants associated with a singular weight supported on C is more limited.For a = 0, (1.2) is the moment generating function of the disk counting function E e u π Im ln pn(r) = E e u #{zj :|zj|<r} , (1.5) and in this case w is discontinuous along a circle but has no "circular" root-type singularity.Counting statistics of two-dimensional point processes have attracted a lot of interest in recent years [45,15,43,44,33,31,56,57,17,1,22].The first two terms in the large n asymptotics of (1.5) were derived in [15,44] 1 .More precise asymptotics, including the third term of order 1, were obtained in [33] for the Ginibre ensemble (i.e.(b, α) = (1, 0)) 2 , and the asymptotics of (1.5) including the fourth term of order n − 1 2 were then obtained in [17] for general b > 0 and α > −1.Several works on determinants with "pointwise" root-type singularities in dimension two are also available in the literature.In [4,5,46,11,47,48], the orthogonal polynomials for the planar Gaussian weight perturbed with a finite number of "pointwise" root-type singularities have been studied, see also [2] where microscopic properties of the associated point process have been analyzed.Building on [4,46], Webb and Wong in [61] obtained a precise asymptotic formula for E[e a Re ln qn (r) ] where a ∈ C, Re a > −2, r < 1 and q n is the characteristic polynomial of a Ginibre matrix, i.e. q n (x) = n j=1 (z j − x) and the z j are drawn from (1.1) with (b, α) = (1, 0).This expectation involves a determinant with a single "pointwise" root-type singularity in the bulk.The case where r is close to 1, which corresponds to the edge regime, was then investigated by Deaño and Simm in [27].Determinants with two merging planar "pointwise" root-type singularities were also considered in [27], and the asymptotics were found to involve some Painlevé transcendents.
Determinants with "circular" root-type singularities have not been considered before to our knowledge.A main difficulty in the analysis of "pointwise" root-type singularities in dimension two stems from the fact that they break the rotation-invariance of the weight (unless of course if they are located at 0).The "circular" root-type singularities preserve the rotation-invariance of the weight, which makes them simpler to analyze in this respect, but they also pose a series of new challenges, which we discuss at the end of this section, and which we have been able to overcome only for integer values of a. Interestingly, these "circular" root-type singularities also produce some associated Hermite polynomials (see below for the definition) in the asymptotics.This came as a surprise to us and appears to be completely new.There are of course many exact formulas in random matrix theory which involve the Hermite polynomials, but we are not aware of an earlier work where these polynomials show up explicitly in the asymptotics of large determinants (let alone the associated Hermite polynomials).For comparison, "pointwise" root-type singularities typically produce other kinds of ingredients in the asymptotics, such as Barnes' G-function (as was discovered by Basor [6] in dimension one and by Webb and Wong [61] in dimension two), and "circular" jump-type singularities involve the error function.We also mention that ensembles with "circular" root-type singularities have been studied in [64,55], and ensembles with "elliptic" root-type singularities in [52].In [64,52,55], the singularities are located at the hard edge and the focus was on the leading order behavior of the kernel; in particular, the (associated) Hermite polynomials do not show up in these works.
The ν-th associated Hermite polynomials {He and satisfy the orthogonality relations [3] 1 [15] considers counting statistics on more general domains (not only centered disks) for a class of determinantal processes on a Kähler manifold (which includes Ginibre), and [44] considers general "one-cut" rotation-invariant potentials (including Mittag-Leffler).
2 Some generalizations of the Ginibre point process (different from the Mittag-Leffler ensemble) and some hyperbolic models have also been considered in [33].
where D −ν is the parabolic cylinder function.These polynomials are explicitly given by He (ν) see [63, eq. (4.12)].For ν = 0, they reduce to the standard Hermite polynomials 3 , i.e.He (0) k (x) = He k (x) for all k ∈ N. We refer to [59] for basic properties of general associated polynomials, and to [63] for a focus on the Hermite case.
Only the polynomials {He k , He k : k = 0, 1, . ..}, corresponding to ν = 0 and ν = 1, will appear in our asymptotic formula.Our results can be presented in a unified way if we formally define He k , He (1) k for the first few negative k as follows:

He
(1) These definitions are consistent with the recurrence (1.6).For general a ∈ N, we define (1.12) where the brackets in (1.12) emphasize that for the first values of a, one needs to use (1.8), namely To be concrete, the first polynomials are given by p 0,a (x) = 1, x, x 2 + 1, x 3 + 3x, x 4 + 6x 2 + 3 , q 0,a (x) = 0, 1, x, x 2 + 2, x 3 + 5x , (1.14) 3 There are two commonly used Hermite polynomials in the literature, denoted He k and H k , and which are related by For us it is more convenient to work with He k .
In the above, the 5 polynomials inside the brackets correspond, from left to right, to a = 0, 1, 2, 3, 4.
The large n asymptotics of E e u π Im ln pn(r) e a Re ln pn(r) are naturally described in terms of the two functions where y ∈ R, u ∈ R, a ∈ N, and erfc is the complementary error function In the statement of our theorem, G 0 appears inside a logarithm and in a denominator.It turns out that G 0 (y; u, a) > 0 for all y ∈ R, u ∈ R and a ∈ N.This fact is not obvious so we defer the proof to Section 4, see Lemma 4.9. where dy, dµ(y) = 2b 2 y 2b−1 dy, and χ (−∞,0) (y) = 0 for y ≥ 0 and χ (−∞,0) (y) = 1 for y < 0.
Remark 1.2.For a = 0, the next term in (1.20) is of order n − 1 2 and was obtained explicitly in [17].For a ≥ 1, numerical simulations suggest that the O-term is in fact of order n − 1 2 + n − 1 2b .This also suggests that our estimate in (1.20) for the O-term is optimal for b > 4 and a ≥ 1.

Outline of the proof of Theorem 1.1.
Let E n := E e u π Im ln pn(r) e a Re ln pn (r) .Our starting point is the following exact formula where γ(ã, z) is the incomplete gamma function Formula (1.21) can be derived using the facts that (1.1) is determinantal and that w is rotationinvariant, see Lemma 2.1 below.For fixed j and a ≥ 1, it is easy to see that the summand in (1.21) contains a term proportional to n − 1 2b in its large n asymptotics.This already explains why our estimate for the error term in (1.20) contains n − 1 2b .
To obtain precise asymptotics for E n , up to and including the term C 3 of order 1, we must take into account each of the n terms in the sum (1.21) (they all contribute).As can be seen from (1.21), this means that we need precise uniform asymptotics for γ(ã, z) as both z → +∞ and ã → +∞ at various different relative speeds.Fortunately, these asymptotics are available in the literature [58].Following the approach of [36] (which was further developed in [17,18]), we will split the sum (1.21) in several parts, where S ℓ , ℓ = 0, 1, 2, 3 are given in (2.5)-(2.8).There is a critical transition in the large ã asymptotics of γ(ã, z) when z → +∞ is such that λ = z ã ≈ 1.The sum S 2 is the hardest one and precisely corresponds to this critical transition; it requires a "local analysis" involving the j-terms in (1.21) for which bnr 2b j ≈ 1.
We found that, quite surprisingly, "circular" root-type singularities are significantly more involved to analyze than "circular" jump-type singularities.Let us highlight some of the reasons for that: • The "global analysis" needed for S 0 , S 1 and S 3 requires some precise Riemann sum approximations for functions with singularities.For comparison, the analogue of S 0 , S 1 and S 3 in [17] in the case of pure "circular" jump-type singularities are straightforward to analyze, because the corresponding Riemann sum approximations only involve constant functions.
• Huge cancellations occur in the "local analysis" of S 2 .In fact, to obtain C 3 , we need to expand up to the (a + 2)-th order the summand of the k-sum in (1.21).This is because, curiously, the first a terms in the expansion cancel perfectly after summing over k.To treat the general case a ∈ N, this means that we need to expand various quantities to all orders.An important technical obstacle is that the coefficients in these various expansions are not always readily available in an explicit form; sometimes they can only be found recursively and involve heavy combinatorics, see e.g.Lemma 4.5 and the all-order expansion of γ in Lemma A.2.The analysis of S 2 is in fact the only part in the proof where solving the problem for general a ∈ N is clearly harder than solving the problem for a finite number of values of a, say a ∈ {0, 1, 2, 3, 4}.This is also the only place in the proof where the (associated) Hermite polynomials arise, see Lemmas 4.6 and 4.7.
Remark 1.3.For non-integer values of a, formula (1.21) does not hold and the connection with the incomplete gamma function is lost (and therefore the strong results from [58] cannot be used anymore).This is the main reason as to why we decided to restrict ourselves to a ∈ N in this work.For a / ∈ N, the exact expression for E n involves hypergeometric functions that generalize the incomplete gamma function.Also, because of the well-known relation [53, eq 12.7.2]), it is tempting to conjecture that for the general case a ∈ (−1, +∞) the large n asymptotics of E n involve the parabolic cylinder function.That would be very interesting to figure that out in detail and we intend to come back to this problem in a future publication.
Outline of the paper.In Section 2, we prove (1.21), define the sums S j , j = 0, 1, 2, 3, and establish many useful lemmas.In Section 3, we obtain the large n asymptotics of S 0 , S 3 and S 1 .The large n asymptotics of S 2 are then obtained in Section 4. We finish the proof of Theorem 1.1 in Section 5.

Preliminaries
This section contains the proof of (1.21) and the definitions of S 0 , . . ., S 3 .We also establish here various preliminary lemmas that will be used in Sections 3 and 4.
Proof.The partition function Z n of the Mittag-Leffler ensemble is known to be see e.g.[17, eq. (1.23)].Since E n = D n /Z n , it only remains to find a simplified exact expression for D n .Since w is rotation-invariant, C z j z k w(z)d 2 z = 0 for j = k, and therefore, by (1.3), we have Since a ∈ N, and thus , nr 2b ) . (2. 2) The claim now follows directly from (2.1), (2.2) and Through the paper, c and C denote positive constants which may change within a computation, and ln always denotes the principal branch of the logarithm.
Let M ′ be a large integer independent of n, let ǫ > 0 be a small constant independent of n, and let We choose ǫ small enough so that with ) ) ) ) . (2.8) In Sections 3 and 4, we will analyze these sums in order of increasing difficulty: first S 0 , then S 3 , then S 1 , and finally S 2 .The sum S 0 is straightforward to analyze, but S 1 , S 2 and S 3 are more involved and require some preparation.This preparation is carried out in the next subsection.
The next lemma establishes yet another representation of g ℓ .
The sums S 1 , S 2 and S 3 naturally involve the functions ( The next lemma collects some properties of γ ℓ .

2.19)
The claim is now a straightforward consequence of Lemma 2.3. where , and B (k) ℓ (x) is the generalized Bernoulli 4 polynomial of degree ℓ defined through the generating function Remark 2.7.The degree 2ℓ polynomial p 2ℓ satisfies p 2ℓ (0) = 0.This is consistent with the fact that for k = 0 the left-hand side of (2.20) is 1.

B
(1) ℓ (x) is the classical Bernoulli polynomial of degree ℓ, and For the large n analysis of S 3 , S 1 , S 2 , we will need to approximate various large sums involving functions with singularities; for this we will also rely on the following lemma from [18].
where, for a given function g continuous on [min{ an n , A}, max{ bn n , B}], 3 Global analysis: large n asymptotics of S 0 , S 3 and S 1 As mentioned earlier, we will analyze the sums (2.5)-(2.8) in order of increasing difficulty: first S 0 , then S 3 , then S 1 , and finally S 2 .In this section we focus on S 0 , S 3 and S 2 .We defer the analysis of S 2 to the next section.
Lemma 3.1.As n → +∞, Proof.Using (2.5) and Lemma A.1, we obtain Since M ′ is fixed, only the (k = 0)-terms contribute to order 1 in the large n asymptotics of S 0 ; the other terms are O(n − 1 2b ).
Recall that S 1 , S 3 are given by (2.6) and (2.8).Following the approach of [17,18], we define and for j = 1, . . ., n and k = 0, 1, . . ., a, we also define Proof.Using (2.8) and Lemma A.2 (ii) with a and λ replaced by a j,k and λ j,k respectively, where j ∈ {j + + 1, . . ., n} and k ∈ {0, . . ., a}, we obtain 3 hold for some positive constants {c j , c ′ j } 3 j=1 , for all n sufficiently large, for all j ∈ {j + + 1, . . ., n} and for all k ∈ {0, . . ., a}.Thus To complete the proof of this lemma, we need the following weaker version of (2.24): as n → +∞ and simultaneously j ∈ {j + + 1, . . ., n}.Note from (2.3) that j/n lies in (br 2b , 1] and remains bounded away from br 2b as n → +∞ and simultaneously j ∈ {j + + 1, . . ., n}; in particular γ 0 (j/n) remains bounded away from 0. Hence, by substituting the above expansion in (3.6) and using (2.16) with ℓ = 0, 1, 2, we obtain after a computation that ) where we have also used that ln(1 as n → +∞.We then obtain the claim for S 3 after a computation using the simplification The large n asymptotics of S 1 are harder to obtain than those of S 3 .The main reason for it is that S 1 involves small j's, and for such j's the quantities γ ℓ (j/n) have a singular behavior, see (2.17).This is also the reason why the error term in Lemma 3.3 is more complicated than in Lemma 3.2.
Proof.Lemma A.2 (i) implies that for any ǫ ′ > 0 there exist and where η is given by (A.1).Let us take ǫ ′ = ǫ 2 and choose M ′ so large that as n → +∞.From a direct analysis of (3.4), we infer that a j,k η 2 j,k decreases as j increases from M ′ + 1 to j − − 1, and a j,k η 2 j,k decreases also as k increases from 0 to a. Therefore for a small enough c > 0. Thus Substituting (2.25) in (3.10) and using (2.16) with ℓ = 0, 1, 2, we obtain as n → +∞, where f 1 is given by (3.7).Note that as n → +∞.Furthermore, by a direct analysis of f 0 and f 1 , as n → +∞.The claim follows after substituting the above expansions in (3.11).

Large n asymptotics of S 2
It remains to obtain the large n asymptotics of S 2 , which was defined in (2.7).For this, let us split S 2 in three pieces, for v = 1, 2, 3, λ j is given by (3.3), and Equivalently, the above sums can be rewritten using where Note that the sums S (1) 2 and S (3) 2 each contain a number of elements proportional to n, while S (2) 2 contains roughly M √ n elements.

Global analysis: large n asymptotics of S
(1) 2 and S (3) 2 We first treat 2 .These sums are delicate to analyze because they involve the asymptotics of γ(a, z) in the regime a → +∞, z → +∞, when λ = z a is close to 1 but not very close (more precisely, , the O-term above is small as n → +∞.

Local analysis: large n asymptotics of S
(2) 2 Our next goal is to obtain the large n asymptotics of S (2) 2 .This is the most technical part of the proof of Theorem 1.1.As mentioned in the introduction, a major obstacle in the asymptotic analysis of S (2) 2 is that, in order to treat the general case a ∈ N, we need to expand various quantities to all orders.Lemma 4.5 below provides a general scheme to compute the coefficients appearing in these expansions in a recursive way.These coefficients are not all readily available in explicit forms, however only a few of those will really matter for us.Lemmas 4.6 and 4.7 establish some non-trivial identities between those relevant coefficients and the (associated) Hermite polynomials.The large n asymptotics of S ) 3.4) that λ j,k is close to λ j for large n, and from (4.26)-(4.27) that M j,k is close to M j for large n.For convenience, for j ∈ {g − , . . ., g + }, we also define where a j,k and η j,k are given in (3.4).
Proof.It follows from (4.36) that a−2s is of the form a ℓ=0 cℓ (k/b) a , where c0 , . . ., ca ∈ C are independent of b.Thanks to the prefactor (2b) a , the above expression is thus independent of b.Replacing b by 1  2 yields (2b) a r ab (−1) a a!
Remark 4.8.The degree of the polynomial in the right-hand side of (4.52) is given by In particular, q a+1,a,a+1 = q (4) a+1,a,a+2 = 0. Proof.Let us first rewrite the sums on the left-hand sides of (4.51) and (4.52) in terms of the coefficients {q ℓ,m,p , q s,s−1,p x p were already simplified in (4.46) and (4.47).Also, by (4.37), where [(Ξ formal j,k ) s ] ℓ is the coefficient of the term of order n − ℓ 2 in the asymptotic series (Ξ formal j,k ) s .Using (4.30), we infer that Also, a direct computation using (4.28) shows that  For the first sum, we use (4.37) to get A direct computation using (4.30) shows that and by (4.28) and (4.57), Substituting the above in (4.61), for ℓ ≥ 0 we get This means that for ℓ ≥ 1, the only term that contributes in (4.63) corresponds to m = s = ℓ and q = 0. Thus, for any ℓ ≥ 0, we have .
A direct computation shows that e 0 = − 1 3r b .Using also (4.56), we get  The polynomial s a can actually be considerably simplified.Indeed, since

.72)
Let us now simplify the sum in (4.66).First, substituting the definitions (4.67) and (1.9), we rewrite it as where To

.75)
For A 1 , we use a−1 s−2 = a s−1 − a−1 s−1 together with (4.74) and (4.75), and find Similarly, by (4.59) and (4.74), Simplifying A 3 is a simpler task as it only relies on (4.59), namely  In the large n asymptotics of S 2 , which are obtained in Lemma 4.11 below, the function G 0 (y; u, a), defined in (1.17), will appear inside a logarithm and in a denominator.The next lemma ensures that this function is positive for relevant values of the parameters.Lemma 4.9.The function G 0 (y; u, a) given in (1.17) is positive for all y ∈ R, u ∈ R, and a ∈ N. where Recall that for all j ∈ {j :    , which finishes the proof of Theorem 1.1.

A Uniform asymptotics of the incomplete gamma function
In this section, we collect some known asymptotic formulas for γ(ã, z) that are useful for us.

3 will
get perfectly canceled by other terms in the large n asymptotics of S

. 62 ) 2 ,
where He −2 (x) ≡ He −1 (x) ≡ 0, and for the second line we have used the three-term recurrence relation(1.6)  of He ℓ .Now we turn to the problem of simplifying the second sum in (4.60).Let us write as n → +∞ for some {e ℓ } +∞ ℓ=0 ⊂ C. By (4