Gaussian asymptotics of discrete $\beta$-ensembles
Abstract
We introduce and study stochastic $N$-particle ensembles which are discretizations for general-$\beta$ log-gases of random matrix theory. The examples include random tilings, families of non-intersecting paths, $(z,w)$-measures, etc. We prove that under technical assumptions on general analytic potential, the global fluctuations for such ensembles are asymptotically Gaussian as $N\to\infty$. The covariance is universal and coincides with its counterpart in random matrix theory. Our main tool is an appropriate discrete version of the Schwinger-Dyson (or loop) equations, which originates in the work of Nekrasov and his collaborators.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2015
- DOI:
- 10.48550/arXiv.1505.03760
- arXiv:
- arXiv:1505.03760
- Bibcode:
- 2015arXiv150503760B
- Keywords:
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- Mathematics - Probability;
- Mathematical Physics;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems
- E-Print:
- 54 pages, 4 figures. v2: misprint in Theorem 7.1 corrected