Continuum limits of random matrices and the Brownian carousel
By Benedek Valkó, B'alint Vir'ag. 2007
223 citations recorded by the source corpus.
We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general β siblings converge to Sine β, a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane.The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We show that the gap probability of Sine β is continuous in the gap size and β, and compute its asymptotics for large gaps. Moreover, the stochastic differential equation version of the Brownian carousel exhibits a phase transition at β=2.
Subfields: Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/04fb8906d7048535b4b85d32b3141fe80a859f53
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In Mathematical Physics
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Citations in the corpus
- Beta ensembles, stochastic Airy spectrum, and a diffusion (2006, background): “The second and third papers rely in part on the results here, and based on [30] those authors have gone on to prove a refined form of Dyson’s conjectures for the bulk eigenvalue spacing for all β > 0 [31].”
- Universality of Wigner random matrices: a survey of recent results (2010, methodology): “A different method, the Brownian carousel representation [103], has been used to generalize the tail distribution of large eigenvalue gaps (“Wigner surmise”) for Gaussian β-ensembles [104].”
- Coulomb Gases and Ginzburg - Landau Vortices (2014, background): “as at inverse temperature , see [DE]. This and other methods allow again to compute a lot explicitly, and to derive that the microscopic laws of the eigenvalues are those of a so called sine-process [VV]. Generally speaking, much is known for log gases in one dimension, for any value of and a wide c”
- A Dynamical Approach to Random Matrix Theory (2017, background): “We remark that the gap distribution in the Gaussian case for any β > 0 can be described by an interesting stochastic process [135].”
- Edge Universality of Beta Ensembles (2013, methodology): “Using this connection, Ramı́rez, Rider and Virág were able to characterize the edge distributions for all β [48]; a characterization of the bulk statistics of the Gaussian beta ensembles was obtained in [58].”
- Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals (2013, background): “probability literature, such questions are now understood in dimension 2 for the particular determinantal case ˇ D 2 [2, 3] and in dimension 1 with the logarithmic interaction [11, 12, 78].”
- Large deviation principle for empirical fields of Log and Riesz gases (2015, methodology): “map (x 1;:::;x N) 7! P N i=1 Nx i. The following is an immediate consequence of [VV09, Theorem1]: (4.13) For any x2( 2;2), for any >0, we have NxP N;=)Sine (x): The convergence =)is proven in [VV09] \in law with respect to vague topology for the counting measure of the point process" which”
- Universality of local spectral statistics of random matrices (2011, methodology): “For all β > 0, the local statistics for the Gaussian case V (x) = x(2)/2 can, however, be characterized by the “Brownian carousel” [57, 74] which was derived from a tridiagonal matrix representation [19] of Gaussian random matrices.”
- Eigenvalue statistics for CMV matrices: From Poisson to clock via random matrix ensembles (2009, background): “Actually, that is a lie since ψ is merely the relative Prüfer phase; nevertheless, because of the way we choose our boundary condition, it is true that the eigenvalues of C(n) are related to the random increasing function ψn−1 in the sense of Definition 1.10.”