# Universal Behavior of the Corners of Orbital Beta Processes

@article{Cuenca2019UniversalBO, title={Universal Behavior of the Corners of Orbital Beta Processes}, author={Cesar Cuenca}, journal={International Mathematics Research Notices}, year={2019} }

There is a unique unitarily-invariant ensemble of $N\times N$ Hermitian matrices with a fixed set of real eigenvalues $a_1> \dots > a_N$. The joint eigenvalue distribution of the $(N-1)$ top-left principal submatrices of a random matrix from this ensemble is called the orbital unitary process. There are analogous matrix ensembles of symmetric and quaternionic Hermitian matrices that lead to the orbital orthogonal and symplectic processes, respectively. By extrapolation, on the dimension of…

## 11 Citations

The Boundary of the Orbital Beta Process

- MathematicsMoscow Mathematical Journal
- 2021

The unitarily invariant probability measures on infinite Hermitian matrices have been classified by Pickrell, and by Olshanski and Vershik. This classification is equivalent to determining the…

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With operators on formal series in $x_i$, $1\leq i\leq N$, which are symmetric in $N-1$ of the $x_i$, probability measures can be studied through Bessel generating functions. These operators are used…

Projections of Orbital Measures and Quantum Marginal Problems

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This paper studies projections of uniform random elements of (co)adjoint orbits of compact Lie groups. Such projections generalize several widely studied ensembles in random matrix theory, including…

PR ] 2 0 A ug 2 02 0 THE BOUNDARY OF THE ORBITAL BETA PROCESS

- Mathematics
- 2020

The unitarily invariant probability measures on infinite Hermitian matrices have been classified by Pickrell [35] and by Olshanski and Vershik [33]. This classification is equivalent to determining…

Random entire functions from random polynomials with real zeros

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We point out a simple criterion for convergence of polynomials to a concrete entire function in the Laguerre-Pólya (LP) class (of all functions arising as uniform limits of polynomials with only real…

Turning Point Processes in Plane Partitions with Periodic Weights of Arbitrary Period

- Computer ScienceRepresentation Theory, Mathematical Physics, and Integrable Systems
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It is shown that near the vertical boundary the system develops up to as many turning points as the period of the weights, and that these turning points are separated by vertical facets which can have arbitrary rational slope.

Operators on Bessel Generating Functions With General Coefficients UROP+ Final Paper, Summer 2021

- Mathematics
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With operators on formal series in xi, 1 ≤ i ≤ N , which are symmetric in N−1 of the xi, Bessel generating funtions can be studied. These operators are used with the Dunkl transform on Bessel…

On the Moments of the Partition Function of the C$$\beta $$E Field

- MathematicsJournal of Statistical Physics
- 2022

We obtain a combinatorial formula for the positive integer moments of the partition function of the $$C\beta E_{N}$$
C
β
E
N
field, or equivalently the moments of the moments of the…

Matrix addition and the Dunkl transform at high temperature

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- 2021

We develop a framework for establishing the Law of Large Numbers for the eigenvalues in the random matrix ensembles as the size of the matrix goes to inﬁnity simultaneously with the beta (inverse…

PR ] 2 0 N ov 2 02 0 ON THE MOMENTS OF THE PARTITION FUNCTION OF THE C β E FIELD

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## References

SHOWING 1-10 OF 79 REFERENCES

Crystallization of random matrix orbits

- Mathematics
- 2017

Three operations on eigenvalues of real/complex/quaternion (corresponding to $\beta=1,2,4$) matrices, obtained from cutting out principal corners, adding, and multiplying matrices can be extrapolated…

ERGODIC UNITARILY INVARIANT MEASURES ON THE SPACE OF INFINITE HERMITIAN MATRICES

- Mathematics
- 1996

Let $H$ be the space of all Hermitian matrices of infinite order and $U(\infty)$ be the inductive limit of the chain $U(1)\subset U(2)\subset...$ of compact unitary groups. The group $U(\infty)$…

Random matrices with prescribed eigenvalues and expectation values for random quantum states

- MathematicsTransactions of the American Mathematical Society
- 2020

Given a collection
λ
_
=
{
λ
1
\underline {\lambda }=\{\lambda _1
,
…
\dots
,
λ
n
}
\lambda _n\}
of real numbers, there is a canonical…

Beta ensembles, stochastic Airy spectrum, and a diffusion

- Mathematics
- 2011

We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schroedinger operator -d^2/dx^2 + x +…

Eigenvalues of GUE Minors

- Mathematics
- 2006

Consider an infinite random matrix $H=(h_{ij})_{0 < i,j}$ picked from the Gaussian Unitary Ensemble (GUE). Denote its main minors by $H_i=(h_{rs})_{1\leq r,s\leq i}$ and let the $j$:th largest…

Interpretations of some parameter dependent generalizations of classical matrix ensembles

- Mathematics
- 2002

Abstract.Two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions (PDFs). As the parameter is varied, one interpolates between…

Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues

- Mathematics
- 2016

We prove a new CLT for the difference of linear eigenvalue statistics of a Wigner random matrix $H$ and its minor $\hat H$ and find that the fluctuation is much smaller than the fluctuations of the…

Fluctuations for linear eigenvalue statistics of sample covariance random matrices

- Mathematics
- 2018

We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix $\widetilde{W}$ and its minor $W$. We find that the fluctuation of this difference is…

Six-vertex models and the GUE-corners process

- Mathematics
- 2016

In this paper we consider a class of probability distributions on the six-vertex model from statistical mechanics, which originate from the higher spin vertex models of…

Macdonald processes

- Mathematics
- 2014

Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters $$q,t \in…