### Abstract

We define a "renormalized energy" as an explicit functional on arbitrary point configurations of constant average density in the plane and on the real line. The definition is inspired by ideas of Sandier and Serfaty (From the Ginzburg-Landau model to vortex lattice problems, 2012; 1D log-gases and the renormalized energy, 2013). Roughly speaking, it is obtained by subtracting two leading terms from the Coulomb potential on a growing number of charges. The functional is expected to be a good measure of disorder of a configuration of points. We give certain formulas for its expectation for general stationary random point processes. For the random matrix β-sine processes on the real line (β = 1,2,4), and Ginibre point process and zeros of Gaussian analytic functions process in the plane, we compute the expectation explicitly. Moreover, we prove that for these processes the variance of the renormalized energy vanishes, which shows concentration near the expected value. We also prove that the β = 2 sine process minimizes the renormalized energy in the class of determinantal point processes with translation invariant correlation kernels.

Original language | English (US) |
---|---|

Pages (from-to) | 199-244 |

Number of pages | 46 |

Journal | Communications in Mathematical Physics |

Volume | 320 |

Issue number | 1 |

DOIs | |

State | Published - 2013 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*320*(1), 199-244. https://doi.org/10.1007/s00220-013-1716-z

**Renormalized Energy Concentration in Random Matrices.** / Borodin, Alexei; Serfaty, Sylvia.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 320, no. 1, pp. 199-244. https://doi.org/10.1007/s00220-013-1716-z

}

TY - JOUR

T1 - Renormalized Energy Concentration in Random Matrices

AU - Borodin, Alexei

AU - Serfaty, Sylvia

PY - 2013

Y1 - 2013

N2 - We define a "renormalized energy" as an explicit functional on arbitrary point configurations of constant average density in the plane and on the real line. The definition is inspired by ideas of Sandier and Serfaty (From the Ginzburg-Landau model to vortex lattice problems, 2012; 1D log-gases and the renormalized energy, 2013). Roughly speaking, it is obtained by subtracting two leading terms from the Coulomb potential on a growing number of charges. The functional is expected to be a good measure of disorder of a configuration of points. We give certain formulas for its expectation for general stationary random point processes. For the random matrix β-sine processes on the real line (β = 1,2,4), and Ginibre point process and zeros of Gaussian analytic functions process in the plane, we compute the expectation explicitly. Moreover, we prove that for these processes the variance of the renormalized energy vanishes, which shows concentration near the expected value. We also prove that the β = 2 sine process minimizes the renormalized energy in the class of determinantal point processes with translation invariant correlation kernels.

AB - We define a "renormalized energy" as an explicit functional on arbitrary point configurations of constant average density in the plane and on the real line. The definition is inspired by ideas of Sandier and Serfaty (From the Ginzburg-Landau model to vortex lattice problems, 2012; 1D log-gases and the renormalized energy, 2013). Roughly speaking, it is obtained by subtracting two leading terms from the Coulomb potential on a growing number of charges. The functional is expected to be a good measure of disorder of a configuration of points. We give certain formulas for its expectation for general stationary random point processes. For the random matrix β-sine processes on the real line (β = 1,2,4), and Ginibre point process and zeros of Gaussian analytic functions process in the plane, we compute the expectation explicitly. Moreover, we prove that for these processes the variance of the renormalized energy vanishes, which shows concentration near the expected value. We also prove that the β = 2 sine process minimizes the renormalized energy in the class of determinantal point processes with translation invariant correlation kernels.

UR - http://www.scopus.com/inward/record.url?scp=84876931518&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84876931518&partnerID=8YFLogxK

U2 - 10.1007/s00220-013-1716-z

DO - 10.1007/s00220-013-1716-z

M3 - Article

VL - 320

SP - 199

EP - 244

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -