scholarly journals Limits for circular Jacobi beta-ensembles

2017 ◽  
Vol 215 ◽  
pp. 40-67 ◽  
Author(s):  
Dang-Zheng Liu
Keyword(s):  
Beta Ensembles

scholarly journals The bead process for beta ensembles

2021 ◽  
Author(s):  
Joseph Najnudel ◽  
Bálint Virág
Keyword(s):  
Point Processes ◽  
Scaling Limit ◽  
Companion Paper ◽  
Point Counting ◽  
Infinite Dimensional ◽  
Transition Mechanism ◽  
Beta Ensemble ◽  
Beta Ensembles ◽  
Orthogonal Ensembles ◽  
Gaussian Beta Ensembles

AbstractThe bead process introduced by Boutillier is a countable interlacing of the $${\text {Sine}}_2$$ Sine 2 point processes. We construct the bead process for general $${\text {Sine}}_{\beta }$$ Sine β processes as an infinite dimensional Markov chain whose transition mechanism is explicitly described. We show that this process is the microscopic scaling limit in the bulk of the Hermite $$\beta $$ β corner process introduced by Gorin and Shkolnikov, generalizing the process of the minors of the Gaussian Unitary and Orthogonal Ensembles. In order to prove our results, we use bounds on the variance of the point counting of the circular and the Gaussian beta ensembles, proven in a companion paper (Najnudel and Virág in Some estimates on the point counting of the Circular and the Gaussian Beta Ensemble, 2019).

scholarly journals Moments of traces of circular beta-ensembles

2015 ◽  
Vol 43 (6) ◽  
pp. 3279-3336 ◽  
Author(s):  
Tiefeng Jiang ◽  
Sho Matsumoto
Keyword(s):  
Beta Ensembles

scholarly journals Matrix models for beta ensembles

2002 ◽  
Vol 43 (11) ◽  
pp. 5830-5847 ◽  
Author(s):  
Ioana Dumitriu ◽  
Alan Edelman
Keyword(s):  
Matrix Models ◽  
Beta Ensembles

scholarly journals Universality of general $\beta$ -ensembles

2014 ◽  
Vol 163 (6) ◽  
pp. 1127-1190 ◽  
Author(s):  
Paul Bourgade ◽  
László Erdős ◽  
Horng-Tzer Yau
Keyword(s):  
Beta Ensembles

scholarly journals Edge Universality of Beta Ensembles

2014 ◽  
Vol 332 (1) ◽  
pp. 261-353 ◽  
Author(s):  
Paul Bourgade ◽  
László Erdös ◽  
Horng-Tzer Yau
Keyword(s):  
Beta Ensembles

scholarly journals Gap universality of generalized Wigner and $\beta$-ensembles

2015 ◽  
Vol 17 (8) ◽  
pp. 1927-2036 ◽  
Author(s):  
László Erdős ◽  
Horng-Tzer Yau
Keyword(s):  
Beta Ensembles

Beta ensembles

2018 ◽  
Author(s):  
Edouard Brezin ◽  
Sinobu Hikami
Keyword(s):  
Random Matrix ◽  
Scaling Limit ◽  
Planck Equation ◽  
Body System ◽  
Many Body ◽  
Boltzmann Factor ◽  
Random Matrix Ensembles ◽  
Gaussian Ensemble ◽  
Matrix Ensembles ◽  
Beta Ensembles

This article deals with beta ensembles. Classical random matrix ensembles contain a parameter β, taking on the values 1, 2, and 4. This parameter, which relates to the underlying symmetry, appears as a repulsion sβ between neighbouring eigenvalues for small s. β may be regarded as a continuous positive parameter on the basis of different viewpoints of the eigenvalue probability density function for the classical random matrix ensembles - as the Boltzmann factor for a log-gas or the squared ground state wave function of a quantum many-body system. The article first considers log-gas systems before discussing the Fokker-Planck equation and the Calogero-Sutherland system. It then describes the random matrix realization of the β-generalization of the circular ensemble and concludes with an analysis of stochastic differential equations resulting from the case of the bulk scaling limit of the β-generalization of the Gaussian ensemble.

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