## Point processes and the infinite symmetric group. Part III: Fermion point processes (1998)

Citations: | 40 - 20 self |

### BibTeX

@TECHREPORT{Olshanski98pointprocesses,

author = {Grigori Olshanski},

title = {Point processes and the infinite symmetric group. Part III: Fermion point processes},

institution = {},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. We study a 2-parametric family of probability measures on an infinite– dimensional simplex (the Thoma simplex). These measures originate in harmonic analysis on the infinite symmetric group (S. Kerov, G. Olshanski and A. Vershik, Comptes Rendus Acad. Sci. Paris I 316 (1993), 773-778). Our approach is to interprete them as probability distributions on a space of point configurations, i.e., as certain point stochastic processes, and to find the correlation functions of these processes. In the present paper we relate the correlation functions to the solutions of certain multidimensional moment problems. Then we calculate the first correlation function which leads to a conclusion about the support of the initial measures. In the appendix, we discuss a parallel but more elementary theory related to the well–known Poisson–Dirichlet distribution. The higher correlation functions are explicitly calculated in the subsequent paper (A. Borodin). In the third part (A. Borodin and G. Olshanski) we discuss some applications and relationships with the random matrix theory. The goal of our work is to understand new phenomena in noncommutative harmonic analysis which arise when the irreducible representations depend on countably many continuous parameters.

### Citations

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Citation Context ...final version of the paper. In particular, one of the devices of [B] allowed me to simplify the derivation of Theorem 5.2. §1. Coherent systems of distribitions on the Young graph Symmetric functions =-=[M]-=-. Let Λ denote the algebra of symmetric functions over the base field R. Formally, Λ may be defined as R[p1, p2, . . .], the algebra of polynomials over infinitely many indeterminates p1, p2, . . ., c... |

320 |
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Citation Context ...of the type (3.7). Such processes were considered by several authors, see [Me], [DVJ]. Furthermore, determinantal formulas for correlation functions appear in some models of mathematical physics, see =-=[KBI]-=-. For a more detailed discussion of these processes see [BO]. It turns out that the main result of Section 2.4 (Theorem 2.4.1) also has a nice reformulation in terms of the lifted processes. It expres... |

180 |
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Citation Context ...on functions via the matrix Whittaker kernel is our main result. Determinantal form for the correlation functions appears in many problems of random matrix theory and mathematical physics, see, e.g., =-=[Dy]-=-, [Me2], [KBI]. In most situations the kernels are symmetric or hermitian (see, however, [[B2]). But ] the 1 0 matrix Whittaker kernel turns out to be ‘J-symmetric’ where J = , see 0 −1 Remark 2.9 bel... |

176 |
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Citation Context ...ing sections we shall see that the lifting substantially simplifies the formulas for the correlation functions of our processes Pzz ′. Moreover, it also works for the Poisson-Dirichlet processes, see =-=[Ki]-=- for definitions. The following claim also follows from section 9.4 in [Ki]. Proposition 3.1.2. The lifting of the Poisson-Dirichlet process PD(t) is the Poisson process on (0, +∞) with density te −x ... |

137 |
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Citation Context ...consider the point process in the restricted phase space R+ ⊂ Ĩ) then all correlation functions are given by the determinants of the type (3.7). Such processes were considered by several authors, see =-=[Me]-=-, [DVJ]. Furthermore, determinantal formulas for correlation functions appear in some models of mathematical physics, see [KBI]. For a more detailed discussion of these processes see [BO]. It turns ou... |

127 |
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Citation Context ...lity measures P on Ω as random measures on [−1, 1]. The next construction looks rather natural from the point of view of the theory of random measures (see, e.g., [DVJ]) or the exchangeability theory =-=[A]-=-. We take the infinite product ν (ω) ∞ = ν (ω) × ν (ω) × . . ., ω ∈ Ω, which is a probability measure on the infinite–dimensional cube [−1, 1] ∞ = [−1, 1] × [−1, 1] × . . ., and we average ν (ω) ∞ wit... |

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Citation Context ... series F [n] B (a, b ; c | y) = ∑ m1,...,mn≥0 (a1)m1(b1)m1 . . .(an)mn (bn)mn (c)m1+...mn m1! . . .mn! y m1 1 . . .y mn n where the series is absolutely convergent for |y1| < 1, . . ., |yn| < 1, see =-=[AK]-=-, [Ex]. When n = 1, this is Gauss’ hypergeometric function, and when n = 2, this is Appell’s hypergeometric function F3. Note also that the function FB remains invariant when the couples (a1, b1), . .... |

110 | The spectrum edge of random matrix ensembles. Nucl. Phys. B402, 709–728 - Forrester - 1993 |

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Multiple Hypergeometric Functions and Applications
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Citation Context ...s F [n] B (a, b ; c | y) = ∑ m1,...,mn≥0 (a1)m1(b1)m1 . . .(an)mn (bn)mn (c)m1+...mn m1! . . .mn! y m1 1 . . .y mn n where the series is absolutely convergent for |y1| < 1, . . ., |yn| < 1, see [AK], =-=[Ex]-=-. When n = 1, this is Gauss’ hypergeometric function, and when n = 2, this is Appell’s hypergeometric function F3. Note also that the function FB remains invariant when the couples (a1, b1), . . ., (a... |

70 | A New Approach to the Representation Theory of the Symmetric Groups - Vershik |

63 | Harmonic analysis on the infinite symmetric group
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Citation Context ... the subsequent paper [B] by Alexei Borodin. In the third paper [BO] we discuss certain applications. The point processes in question originated from harmonic analysis on the infinite symmetric group =-=[KOV]-=-: they govern decomposition of the so-called generalized regular representations. I shall briefly discuss the link with representation theory, as this is the main motivation of the work. I believe tha... |

61 |
Level spacing distributions and the Airy kernel
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Citation Context ...m are expressed through Painlevé transcendents, 12see [TW3, Sections III and V]. Various kernels (14.1) restricted to suitable intervals commute with Sturm–Liouville operators (see [G], [Me1, §5.3], =-=[TW1]-=-, [TW2]). The same is true for the Whittaker kernel restricted to [τ, +∞), see [Part III, Section 6]. The Whittaker kernel degenerates to the Laguerre kernel of order N and parameter α > −1 if we form... |

56 |
The representation theory of the symmetric group, Encyclopedia of Mathematics and its applications
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Citation Context ...′(dω) where (p|q) is considered as the Frobenius notation for a Young diagram λ; dimλ is the dimension of the complex irreducible representation of the symmetric group Sn corresponding to λ (see [M], =-=[JK]-=-); and ˜sλ stands for the so-called extended Schur function, see [O], [KOO] for details. These moments lie in the base of all our computations. Now we shall introduce the point process, see [DVJ] for ... |

56 |
The coincidence approach to stochastic point processes
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Citation Context ...processes were already considered in the 1st edition (1967) of Mehta’s book on random matrices [Me] and some earlier papers, the first (to our knowledge) general discussion appeared in Macchi’s paper =-=[Ma1]-=-. In her works and in the book [DVJ] these processes are called the fermion point processes, and we shall adopt this terminology. Let us list some general properties of the fermion processes (see [Ma1... |

50 | The boundary of Young graph with Jack edge multiplicities
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Citation Context ...ram λ; dimλ is the dimension of the complex irreducible representation of the symmetric group Sn corresponding to λ (see [M], [JK]); and ˜sλ stands for the so-called extended Schur function, see [O], =-=[KOO]-=- for details. These moments lie in the base of all our computations. Now we shall introduce the point process, see [DVJ] for general information about point processes. Let us denote by I the punctured... |

46 |
Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables, Ellis Horwood Limited
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Citation Context ...s. The results of this section represent a generalization of the well known formulas for the analytic continuation of the Gauss hypergeometric function and Appell hypergeometric function F3, see [E], =-=[Mar]-=-, [Ex1], [Ex2]. We start with Mellin-Barnes type integrals. Proposition 4.2.1. If ai, bi ̸= 0, −1, −2, . . . for all i = 1, . . . , m then F [m] B (a, b; c|y) = × Γ(c) m∏ Γ(ai)Γ(bi) i=1 1 (2πi) m ∫ +i... |

41 |
Asymptotic theory of the characters of a symmetric group
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Citation Context ...t of all, via the correspondence M ↔ χ, (1.7) implies that the characters are parametrized by the points ω ∈ Ω (we shall write them as χ (ω) ). 3 This result is known as Thoma’s theorem [T]; see also =-=[VK]-=-. The extreme coherent system M (ω) ↔ χ (ω) is given by the formula M (ω) (λ) = dim λ · ˜sλ(ω), which is equivalent to Thoma’s formula [T] χ (ω) (ρ) = ˜pρ1 (ω)˜pρ2 (ω) . . . . 3 Thus, the Thoma simple... |

41 |
A method of integration over matrix variables
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(Show Context)
Citation Context ... with such symmetry in determinantal formulas for correlation functions seems to be new. At the end of the paper we consider the systems of eigenvalues of two random coupled matrices studied in [IZ], =-=[Me1]-=-, [MS], [EM], [MN], [Ey]. As was recently proved in [EM], the correlation functions of such systems are also given by determinantal formulas. We show that this result and our considerations have commo... |

37 |
An introduction to the theory of point processes, Springer Series in Statistics
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Citation Context .... This is a direct consequence of the definition of the Thoma measure and that of ˜pn. □ Let Prob[−1, 1] denote the set of probability Borel measure on [−1, 1]; this set has a natural Borel structure =-=[DVJ]-=-. The map Ω → Prob[−1, 1], ω ↦→ ν (ω) , 11is Borel–measurable (this is a routine exercise). Therefore, equiping Ω with a probability measure P, we obtain a random measure on [−1, 1]. Since the above ... |

37 |
Correlation functions of random matrix ensembles related to classical orthogonal polynomials
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- 1991
(Show Context)
Citation Context ...t space L2 (R+, dx) on the linear span of the functions xi+µ x − e 2 , where i = 0, . . ., N −1; this is exactly the kernel associated with the “N-point Laguerre polynomial ensemble”, see [FK], [Br], =-=[NW1]-=-. Finally, note that the restriction µ < 0, which comes from the assumption N−1 < z, z ′ < N, is inessential, because there exists a natural “degenerate series” of the measures Pzz ′ with the paramete... |

35 | Introduction to random matrices. In: Geometric and quantum aspects of integrable systems (Scheveningen - Tracy, Widom - 1992 |

27 |
The K-functor (Grothendieck group) of the infinite symmetric group
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(Show Context)
Citation Context ...sian Foundation for Basic Research under grant 98-01-00303 (G. O) and by the Russian Program for Support of Scientific Schools under grant 96-15-96060 (A. B. and G. O.)called the Thoma simplex [VK], =-=[KV]-=-. Note that Ω is compact in the topology of pointwise convergence. Our aim is to understand these measures. Our results show that the measures Pzz ′ are close to stochastic point processes arising in ... |

26 |
Momentum distribution in the ground state of the one-dimensional system of impenetrable bosons
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Citation Context ... K+− K++ ] Determinantal form for the correlation functions (like (7.2)) appears in different problems of random matrix theory and mathematical physics, see, e.g., [Dy], [Me1], [Ma1], [Ma2], [TW1–3], =-=[L]-=-, [KBI]. In most situations the kernel K is symmetric or Hermitian (see, however, [B]). Appearance of J–symmetric kernels seems to be new. 78. The L–kernel [Part V, §2]. Consider the operator K in th... |

25 |
A characterization of supersymmetric polynomials
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Citation Context ... γt 1 + βit ↦→ e 1 − αit . i≥1 This is a generalization of the well–known “super” realization of Λ; indeed, setting γ = 0 converts the above expressions to “supersymmetric” functions in α and −β, see =-=[S]-=- and [M, §I.3, Ex. 23]. The Thoma simplex [VK, KV, KOO]. We shall abbreviate α = (α1, α2, . . .), β = (β1, β2, . . .). Let Ω be the set of the triples ω = (α, β, γ) such that α1 ≥ α2 ≥ · · · ≥ 0, β1 ≥... |

23 |
Matrices coupled in a chain
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(Show Context)
Citation Context ...mmetry in determinantal formulas for correlation functions seems to be new. At the end of the paper we consider the systems of eigenvalues of two random coupled matrices studied in [IZ], [Me1], [MS], =-=[EM]-=-, [MN], [Ey]. As was recently proved in [EM], the correlation functions of such systems are also given by determinantal formulas. We show that this result and our considerations have common combinator... |

22 |
Limit measures arising in the asymptotic theory of symmetric groups. i. Theory of Probability and its applications
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(Show Context)
Citation Context ...the final step 5 is much easier, because of a simpler structure of the correlation functions. We get in this way that for PD(t), (xj) 1/j tends to e −t , the result originally obtained (for t = 1) in =-=[VS]-=- by a quite different way. We can also use lifting, as suggested on step 4, which provides a quick reduction to the law of large numbers for the Poisson process, see [Ki], section 4.2. Thus, both for ... |

20 |
Exponential ensembles for Random matrices
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(Show Context)
Citation Context ...Hilbert space L2 (R+, dx) on the linear span of the functions xi+µ x − e 2 , where i = 0, . . ., N −1; this is exactly the kernel associated with the “N-point Laguerre polynomial ensemble”, see [FK], =-=[Br]-=-, [NW1]. Finally, note that the restriction µ < 0, which comes from the assumption N−1 < z, z ′ < N, is inessential, because there exists a natural “degenerate series” of the measures Pzz ′ with the p... |

20 |
Sur la loi Limite de l’éspacement des valeurs propres d’une matrice aléatoire
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(Show Context)
Citation Context ...ernels of this form are expressed through Painlevé transcendents, 12see [TW3, Sections III and V]. Various kernels (14.1) restricted to suitable intervals commute with Sturm–Liouville operators (see =-=[G]-=-, [Me1, §5.3], [TW1], [TW2]). The same is true for the Whittaker kernel restricted to [τ, +∞), see [Part III, Section 6]. The Whittaker kernel degenerates to the Laguerre kernel of order N and paramet... |

18 |
Die unzerlegbaren, positive-definiten Klassenfunktionen der abzählbar unendlichen symmetrischen Gruppe
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(Show Context)
Citation Context ...e the form Ωi = {ω ∈ Ω ∣ ∣ αi = αi+1}, Ω−i = {ω ∈ Ω ∣ ∣ βi = βi+1}, i ≥ 1, (1.5) Ω0 = {ω ∈ Ω ∣ ∣ γ = 0}. (1.6) The simplex Ω is called the Thoma simplex in connection with the pioneering Thoma’s work =-=[T]-=-. We equip Ω with the weakest topology in which the coordinates αi and βi (but not γ) are continuous functions. In this topology, Ω is a metrizable compact space, and the face Ω0 is a dense subset. We... |

18 |
Two coupled matrices: eigenvalue correlations and spacing functions
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- 1994
(Show Context)
Citation Context ...uch symmetry in determinantal formulas for correlation functions seems to be new. At the end of the paper we consider the systems of eigenvalues of two random coupled matrices studied in [IZ], [Me1], =-=[MS]-=-, [EM], [MN], [Ey]. As was recently proved in [EM], the correlation functions of such systems are also given by determinantal formulas. We show that this result and our considerations have common comb... |

17 | Unitary Representations of Infinite-Dimensional Pairs (G, K) and the Formalism of R. Howe - Ol’shanskii - 1990 |

17 |
Painlevé functions of the third kind
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Citation Context ...stand whether the Macdonald kernel is somehow related to random matrix ensembles. I am grateful to Alexei Borodin for numerous discussions and to Craig A. Tracy for drawing my attention to the papers =-=[MTW]-=-, [T], [TW4]. 1. A model In this section we fix a finite set X which will serve as a “state space”. We shall deal with kernels K(x, y), L(x, y) on X × X which will also be considered as matrices of or... |

15 | Thoma’s theorem and representations of the infinite bisymmetric group, Funct - Okounkov - 1994 |

14 | Combinatorial examples in the theory of AF-algebras - Kerov - 1989 |

13 | On the distribution of allele frequencies in a diffusion model - Griffiths - 1979 |

10 |
The sampling theory of selectively neutral alleles
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Citation Context ... Poisson process on R with constant density 1. Proof. Recall that the correlation functions of PD(t) are given by Watterson’s formula ρn(x1, . . ., xn) = tn (1 − x1 − · · · − xn) t−1 + x1 . . .xn see =-=[W]-=- and Part I, Corollary 7.4. In particular, the first correlation function is , ρ1(x) = t(1 − x)t−1 x . These correlation functions fit into the above scheme with C = t and all the functions fn identic... |

9 |
A.: Asymptotics of a τ-function arising in the two-dimensional Ising model
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- 1991
(Show Context)
Citation Context ...hether the Macdonald kernel is somehow related to random matrix ensembles. I am grateful to Alexei Borodin for numerous discussions and to Craig A. Tracy for drawing my attention to the papers [MTW], =-=[T]-=-, [TW4]. 1. A model In this section we fix a finite set X which will serve as a “state space”. We shall deal with kernels K(x, y), L(x, y) on X × X which will also be considered as matrices of order |... |

7 | Eigenvalue distribution of random matrices at the spectrum edge - Nagao, Wadati - 1993 |

7 | Matrices coupled in a chain: II. Spacing functions
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(Show Context)
Citation Context ... in determinantal formulas for correlation functions seems to be new. At the end of the paper we consider the systems of eigenvalues of two random coupled matrices studied in [IZ], [Me1], [MS], [EM], =-=[MN]-=-, [Ey]. As was recently proved in [EM], the correlation functions of such systems are also given by determinantal formulas. We show that this result and our considerations have common combinatorial ba... |

7 | Biorthogonal ensembles
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(Show Context)
Citation Context ...ifferent problems of random matrix theory and mathematical physics, see, e.g., [Dy], [Me1], [Ma1], [Ma2], [TW1–3], [L], [KBI]. In most situations the kernel K is symmetric or Hermitian (see, however, =-=[B]-=-). Appearance of J–symmetric kernels seems to be new. 78. The L–kernel [Part V, §2]. Consider the operator K in the Hilbert space L 2 (R ∗ , du) ≃ L 2 (R+, dx) ⊕ L 2 (R+, dx) (8.1) given by the kerne... |

6 |
Handbook of hypergeometric integrals. Theory, applications, tables, computer programs,” Mathematics and its Applications
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(Show Context)
Citation Context ... of this section represent a generalization of the well known formulas for the analytic continuation of the Gauss hypergeometric function and Appell hypergeometric function F3, see [E], [Mar], [Ex1], =-=[Ex2]-=-. We start with Mellin-Barnes type integrals. Proposition 4.2.1. If ai, bi ̸= 0, −1, −2, . . . for all i = 1, . . . , m then F [m] B (a, b; c|y) = × Γ(c) m∏ Γ(ai)Γ(bi) i=1 1 (2πi) m ∫ +i∞ −i∞ · · · +i... |

6 | Detection and “emission” processes of quantum particles in a “chaotic state - Benard, Macchi - 1973 |

6 | Nonuniversal correlations for random matrix ensembles, J.Math.Phys. 34 - Nagao - 1993 |

5 |
spacing distributions and the Bessel kernel
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Citation Context ...xpressed through Painlevé transcendents, 12see [TW3, Sections III and V]. Various kernels (14.1) restricted to suitable intervals commute with Sturm–Liouville operators (see [G], [Me1, §5.3], [TW1], =-=[TW2]-=-). The same is true for the Whittaker kernel restricted to [τ, +∞), see [Part III, Section 6]. The Whittaker kernel degenerates to the Laguerre kernel of order N and parameter α > −1 if we formally su... |

5 |
A class of integral transforms
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Citation Context ...K++ with varying µ form a commutative family (Corollary 2.5). Consider the following functions on R+: We have fa,m(x) = 1 x Wa,im(x), m > 0. (3.3) D(a)fa,m = (a 2 + 1 4 + m2 )fa,m. (3.4) According to =-=[W]-=-, the functions fa,m with a fixed and m ranging over R+ form a continual basis in L2 (R+) diagonalizing D(a). Moreover, an explicit Plancherel formula holds: where (f, g) L 2 (R+) = (fa,m, fa.m) := ∫ ... |

4 |
Waveforms processing
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Citation Context ...t) are known as Ewens partition structures [Ki1, Ki2]. According to Kingman’s theorem, they determine certain probability measures on ∆. The latter are called the Poisson–Dirichlet distributions, see =-=[Ki2]-=-, and denoted as PD(t). So, the link between M (t) and PD(t) is as follows: M (t) ∫ (λ) = ∆ k ˜mλ(α)(PD(t))(dα). Given a probability measure P on ∆, we may regard it as a measure on Ω. Hence, the defi... |

4 | Unitary representations of (G, K)-pairs connected with the infinite symmetric group S(∞), Algebra i Analiz 1 - Olshanskii - 1989 |

4 |
Multiplicative distributions on Young graph, in: Representation Theory, Dynamical Systems, Combinatorial and Algorithmic Methods II
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Citation Context ...omplementary series was observed in 1995 by Borodin. The z-systems can be characterized as the only coherent systems of distributions on the Young graph satisfying a “multiplicativity condition”, see =-=[R]-=-. In the present paper we do not deal with a “degenerate series” of coherent systems which arises when one of the parameters z, z ′ is integral. The ”degenerate” coherent systems live, in essence, on ... |

4 | New family of unitary random matrices - Muttalib, Chen, et al. - 1993 |