Abstract
The bead process introduced by Boutillier is a countable interlacing of the {text {Sine}}_2 point processes. We construct the bead process for general {text {Sine}}_{beta } 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 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).
Highlights
In Boutillier [6], a remarkable family of point processes on R × Z, called bead processes, and indexed by a parameter γ ∈ (−1, 1), has been defined
The limiting process is defined as an infinite-dimensional Markov chain, the transition from one line to the being explicitly described. This transition can be viewed as a generalization of the limit, when the dimension n goes to infinity, of the random reflection walk on the unitary group U (n)
For β = 2, this process is the bead process itself, and it is the limit of the eigenvalues of the Gaussian Unitary Ensemble (GUE) minors when the dimension goes to infinity
Summary
In Boutillier [6], a remarkable family of point processes on R × Z, called bead processes, and indexed by a parameter γ ∈ (−1, 1), has been defined. The limiting process is defined as an infinite-dimensional Markov chain, the transition from one line to the being explicitly described This transition can be viewed as a generalization of the limit, when the dimension n goes to infinity, of the random reflection walk on the unitary group U (n). 8, we show, under some technical conditions, a property of continuity of the Markov chain with respect to the initial point measure and the weights From this result, and from a bound, proven in a companion paper [15] on the variance of the number of points of the Gaussian beta ensemble in intervals, we deduce in Sect. For other values of β, we get a similar result of convergence for the renormalized points of the Hermite β corner defined in [10]
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