### Abstract

We prove a local central limit theorem (LCLT) for the number of points N(J) in a region J in (Formula presented.) specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of N(J) tends to infinity as (Formula presented.). This extends a previous result giving a weaker central limit theorem for these systems. Our result relies on the fact that the Lee–Yang zeros of the generating function for (Formula presented.)—the probabilities of there being exactly k points in J—all lie on the negative real z-axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble. A LCLT is also established for the probability density function of the k-th largest eigenvalue at the soft edge, and of the spacing between k-th neighbors in the bulk.

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

Pages (from-to) | 60-69 |

Number of pages | 10 |

Journal | Journal of Statistical Physics |

Volume | 157 |

Issue number | 1 |

DOIs | |

State | Published - Oct 1 2014 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Keywords

- Central limit theorem
- Lee–Yang zeros
- Local central limit theorem
- Random matrices

### Cite this

*Journal of Statistical Physics*,

*157*(1), 60-69. https://doi.org/10.1007/s10955-014-1071-2

}

*Journal of Statistical Physics*, vol. 157, no. 1, pp. 60-69. https://doi.org/10.1007/s10955-014-1071-2

**Local Central Limit Theorem for Determinantal Point Processes.** / Forrester, Peter J.; Lebowitz, Joel L.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Local Central Limit Theorem for Determinantal Point Processes

AU - Forrester, Peter J.

AU - Lebowitz, Joel L.

PY - 2014/10/1

Y1 - 2014/10/1

N2 - We prove a local central limit theorem (LCLT) for the number of points N(J) in a region J in (Formula presented.) specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of N(J) tends to infinity as (Formula presented.). This extends a previous result giving a weaker central limit theorem for these systems. Our result relies on the fact that the Lee–Yang zeros of the generating function for (Formula presented.)—the probabilities of there being exactly k points in J—all lie on the negative real z-axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble. A LCLT is also established for the probability density function of the k-th largest eigenvalue at the soft edge, and of the spacing between k-th neighbors in the bulk.

AB - We prove a local central limit theorem (LCLT) for the number of points N(J) in a region J in (Formula presented.) specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of N(J) tends to infinity as (Formula presented.). This extends a previous result giving a weaker central limit theorem for these systems. Our result relies on the fact that the Lee–Yang zeros of the generating function for (Formula presented.)—the probabilities of there being exactly k points in J—all lie on the negative real z-axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble. A LCLT is also established for the probability density function of the k-th largest eigenvalue at the soft edge, and of the spacing between k-th neighbors in the bulk.

KW - Central limit theorem

KW - Lee–Yang zeros

KW - Local central limit theorem

KW - Random matrices

UR - http://www.scopus.com/inward/record.url?scp=84913595774&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84913595774&partnerID=8YFLogxK

U2 - 10.1007/s10955-014-1071-2

DO - 10.1007/s10955-014-1071-2

M3 - Article

AN - SCOPUS:84913595774

VL - 157

SP - 60

EP - 69

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1

ER -