Harmonic analysis for rank-1 Randomised Horn Problems

Zhang, Jiyuan; Kieburg, Mario; Forrester, Peter J.

doi:10.48550/arxiv.1911.11316

Cited by 5 publications

(13 citation statements)

References 36 publications

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“…This operation is equivalent to put an additional β 2 term in the exponent of each determinant in (15), except for the last one.…”

Section: Heckman-opdam and The Spherical Integralmentioning

confidence: 99%

Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform

Mergny,

Potters

2020

Preprint

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In this note, we study the asymptotic of spherical integrals, which are analytical extension in index of the normalized Schur polynomials for β = 2 , and of Jack symmetric polynomials otherwise. Such integrals are the multiplicative counterparts of the Harish-Chandra-Itzykson-Zuber (HCIZ) integrals, whose asymptotic are given by the so-called R-transform when one of the matrix is of rank one. We argue by a saddle-point analysis that a similar result holds for all β > 0 in the multiplicative case, where the asymptotic is governed by the logarithm of the S-transform. As a consequence of this result one can calculate the asymptotic behavior of complete homogeneous symmetric polynomials.

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“…This operation is equivalent to put an additional β 2 term in the exponent of each determinant in (15), except for the last one.…”

Section: Heckman-opdam and The Spherical Integralmentioning

confidence: 99%

Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform

Mergny,

Potters

2020

Preprint

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“…(103). However, the case with the exponential function in the second Vandermonde, appeared in the randomized multiplicative Horn problem [51], in the DPMK equation for transport in semiconductors [52] and in the multiplicative analogue of Dyson Brownian Motion [53].…”

Section: Relations To Other Modelsmentioning

confidence: 99%

Stability of large complex systems with heterogeneous relaxation dynamics

Mergny,

Majumdar

2021

Preprint

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We study the probability of stability of a large complex system of size N within the framework of a generalized May model, which assumes a linear dynamics of each population size ni (with respect to its equilibrium value):The ai > 0's are the intrinsic decay rates, Jij is a real symmetric (N × N ) Gaussian random matrix and √ T measures the strength of pairwise interaction between different species. Unlike in May's original homogeneous model, each species has now an intrinsic damping ai that may differ from one another. As the interaction strength T increases, the system undergoes a phase transition from a stable phase to an unstable phase at a critical value T = Tc. We reinterpret the probability of stability in terms of the hitting time of the level b = 0 of an associated Dyson Brownian Motion (DBM), starting at the initial position ai and evolving in 'time' T . In the large N → ∞ limit, using this DBM picture, we are able to completely characterize Tc for arbitrary density µ(a) of the ai's. For a specific flat configuration ai = 1 + σ i−1 N , we obtain an explicit parametric solution for the limiting (as N → ∞) spectral density for arbitrary T and σ. For finite but large N , we also compute the large deviation properties of the probability of stability on the stable side T < Tc using a Coulomb gas representation.

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“…the eigenvalue distributions are Dirac deltas, Horn's problem generalises to a sum of randomized adjoint orbits of O(n) and U(n), see [13,34,7,11]. We would like to draw also attention to some recent works discussing a multiplicative version of Horn's problem [11,33]. For general eigenvalues, a general unitarily invariant ensemble, the Pólya ensemble, on Herm(n) have been identified in [28,12].…”

Section: Introductionmentioning

confidence: 99%

“…For general eigenvalues, a general unitarily invariant ensemble, the Pólya ensemble, on Herm(n) have been identified in [28,12]. For this particular problem it has been proven, analogous to elementary probability theory, that harmonic analysis on matrix spaces is an essential tool for studies of sums of random matrices, e.g., see [28,12,33].…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, one only needs to replace f diag by a product of weight functions. In recent works [12,22,23,24,25,33], harmonic analysis for random matrices has been extended to sums of other matrix spaces as well as to products of random matrices with the help of other Harish-Chandra-Itzykson-Zuber-type integrals [17,21] like the Berezin-Karpelevich integral [2,16] or the Gelfand-Naȋmark integral [15]. Interestingly, the corresponding Pólya ensembles that arise out of these computations have a distribution that have a similar structure as the right side of (1.2).…”

Section: Introductionmentioning

confidence: 99%

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Derivative principles for invariant ensembles

Kieburg¹,

Zhang²

2020

Preprint

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A big class of ensembles of random matrices are those which are invariant under group actions so that the eigenvectors become statistically independent of the eigenvalues. In the present work we show that this is not the sole consequence but that the joint probability distribution of the eigenvalues can be expressed in terms of a differential operator acting on the distribution of some other matrix quantities which are sometimes easier accessible. Those quantities might be the diagonal matrix entries as it is the case for Hermitian matrices or some pseudo-diagonal entries for other matrix spaces. These representations are called derivative principles. We show them for the additive spaces of the Hermitian, Hermitian antisymmetric, Hermitian anti-self-dual, and complex rectangular matrices as well as for the two multiplicative matrix spaces of the positive definite Hermitian matrices and of the unitary matrices. In all six cases we prove a uniqueness of the derivative principle as it is not trivial to see that the derivative operator has a trivial kernel on the selected spaces of the identified matrix quantities.

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Harmonic analysis for rank-1 Randomised Horn Problems

Cited by 5 publications

References 36 publications

Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform

Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform

Stability of large complex systems with heterogeneous relaxation dynamics

Derivative principles for invariant ensembles

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