Ergodic Decomposition for Inverse Wishart Measures on Infinite Positive-Definite Matrices

Assiotis, Theodoros

doi:10.3842/sigma.2019.067

Cited by 10 publications

(29 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Many classical results from probability and ergodic theory fall into this abstract framework, the simplest example being de Finetti's theorem. Here, we are interested in the infinite dimensional unitary group U(∞) = lim → U(N) 4…”

Section: The Origins Of the Random Variable X(s)mentioning

confidence: 99%

“…N on H(N) which are also known in random matrix theory as the Cauchy ensemble and under the Cayley transform as the circular Jacobi ensemble on U(N). We refer the reader to the extensive literature for properties of these [40], [50], [9], [56], [17], [12], [11] and closely related measures [54], [14], [15], [16], [4], and connections ranging from Painlevé equations [31] to stochastic processes [2], [3].…”

Section: Acting On the Space Of Infinitementioning

confidence: 99%

“…Thus, our results from Section 1.3 on the random variable X(s) are of independent interest since in particular the parameter γ 1 is arguably the hardest parameter to study, not only in the problem of ergodic decomposition of M (s) on Ω, but in essentially all the other allied problems on possibly different spaces we have alluded to [10], [48], [37], [14], [15], [16], [4].…”

Section: Acting On the Space Of Infinitementioning

confidence: 99%

See 2 more Smart Citations

Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts

Assiotis

Keating

2020

Random Matrices: Theory Appl.

Self Cite

View full text Add to dashboard Cite

The aim of this note is to give a combinatorial and non-computational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity. This is achieved by relating these quantities to a lattice point count problem. As a byproduct, we obtain a new expression for the leading order coefficient in the asymptotic as a volume of a certain region involving continuous Gelfand-Tsetlin patterns with constraints.

show abstract

Section: The Origins Of the Random Variable X(s)mentioning

confidence: 99%

Section: Acting On the Space Of Infinitementioning

confidence: 99%

Section: Acting On the Space Of Infinitementioning

confidence: 99%

See 1 more Smart Citation

Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts

Assiotis

Keating

2020

Random Matrices: Theory Appl.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Remark 5.2. It follows from the results in [2], see also Section 3 in [3], that Y 2 (ν) is equal to the sum of the (random) inverse points of the Bessel determinantal point process, see [18]. Moreover, for general β it follows from the results of [34] that Y β (ν) is the trace of the random integral operator in (1.4) in [34] (after multiplication by β/2 and the correspondence of parameters ν = β(a + 1)/2 − 1).…”

Section: Proof That G (τ)mentioning

confidence: 81%

“…and the joint distribution N of the sequence x (i) N i=1 satisfies: (2) , dx (1) , where ν N is the distribution of the top row x (N) in the sequence. A consistent random family of infinite interlacing arrays with parameter β is a family of random infinite sequences x (i) ∞ i=1 such that the above holds for all N ≥ 1.…”

Section: A Moments Convergence Results For Rows Of Random Interlacing...mentioning

confidence: 99%

Convergence and an explicit formula for the joint moments of the Circular Jacobi $β$-Ensemble characteristic polynomial

Assiotis,

Gunes,

Soor

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The problem of convergence of the joint moments, which depend on two parameters s and h, of the characteristic polynomial of a random Haar-distributed unitary matrix and its derivative, as the matrix size goes to infinity, has been studied for two decades, beginning with the thesis of Hughes [23]. Recently, Forrester [19] considered the analogous problem for the Circular β-Ensemble (CβE) characteristic polynomial, proved convergence and obtained an explicit combinatorial formula for the limit for integer s and complex h. In this paper we consider this problem for a generalisation of the CβE, the Circular Jacobi β-ensemble (CJβE δ ), depending on an additional complex parameter δ and we prove convergence of the joint moments for general positive real exponents s and h. We give a representation for the limit in terms of the moments of a family of real random variables of independent interest. This is done by making use of some general results on consistent probability measures on interlacing arrays. Using these techniques, we also extend Forrester's explicit formula to the case of real s and δ and integer h. Finally, we prove an analogous result for the moments of the logarithmic derivative of the characteristic polynomial of the Laguerre β-ensemble.

show abstract

Convergence and an Explicit Formula for the Joint Moments of the Circular Jacobi $$\beta $$-Ensemble Characteristic Polynomial

Assiotis

Gunes

Soor

2022

Math Phys Anal Geom

View full text Add to dashboard Cite

The problem of convergence of the joint moments, which depend on two parameters s and h, of the characteristic polynomial of a random Haar-distributed unitary matrix and its derivative, as the matrix size goes to infinity, has been studied for two decades, beginning with the thesis of Hughes (On the characteristic polynomial of a random unitary matrix and the Riemann zeta function, PhD Thesis, University of Bristol, 2001). Recently, Forrester (Joint moments of a characteristic polynomial and its derivative for the circular $$\beta $$ β -ensemble, arXiv:2012.08618, 2020) considered the analogous problem for the Circular $$\beta $$ β -Ensemble (C$$\beta $$ β E) characteristic polynomial, proved convergence and obtained an explicit combinatorial formula for the limit for integer s and complex h. In this paper we consider this problem for a generalisation of the C$$\beta $$ β E, the Circular Jacobi $$\beta $$ β -ensemble (CJ$$\beta \text {E}_\delta $$ β E δ ), depending on an additional complex parameter $$\delta $$ δ and we prove convergence of the joint moments for general positive real exponents s and h. We give a representation for the limit in terms of the moments of a family of real random variables of independent interest. This is done by making use of some general results on consistent probability measures on interlacing arrays. Using these techniques, we also extend Forrester’s explicit formula to the case of real s and $$\delta $$ δ and integer h. Finally, we prove an analogous result for the moments of the logarithmic derivative of the characteristic polynomial of the Laguerre $$\beta $$ β -ensemble.

show abstract

Ergodic Decomposition for Inverse Wishart Measures on Infinite Positive-Definite Matrices

Cited by 10 publications

References 34 publications

Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts

Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts

Convergence and an explicit formula for the joint moments of the Circular Jacobi $β$-Ensemble characteristic polynomial

Convergence and an Explicit Formula for the Joint Moments of the Circular Jacobi $$\beta $$-Ensemble Characteristic Polynomial

Contact Info

Product

Resources

About