# FY2011 Annual Report

## Mathematical and Theroretical Physics Unit

**Professor Shinobu Hikami**

### Abstract

The Mathematical and Theoretical Physics Unit is studying a statistical field theory. In this year, random matrix theory (RMT) was investigated, for the universal behavior near the edge of the spectrum in the external source. The dynamical correlation function of the eigenvalues of the random matrix was studied near the edge spectrum through the duality theorem and from the topological invariants. The applications to the two dimensional conformal field theory (CFT) and the supersymmetric theories are pursued.

### 1. Staff

**Theoretical Group**

Dr. Reuven Pnini, Researhcer

Dr. Ayumi Kikkawa, Researcher

**Research Administrator/Secretary**

Ms. Kaori Yamashiro

### 2. Collaborations

- Theme: Random matrix theory with an external source
- Type of collaboration: joint research.
- Researchers:

Professor Edouard Brezin, Ecole Normale Superieure

Professor Shinobu Hikami, The University of Tokyo

### 3. Activities and Findings

The universal behavior near the edge spectrum in a random matrix theory is described by the determinant of Airy kernel for the n-point correlation function. The time dependent behavior of a random matrix model is reduced to two-matrix model, and in the scaling region near the edge, the model is reduced to Airy matrix model with a logarithmic potential.

The Airy matrix model is a generating function for the intersection numbers of the moduli space in the Riemann surface with marked points. In the case of the presence of the logarithmic potential term, the modifications of the intersection numbers and the topological invariants are obtained and the deviation from KdV equation was obtained.

### 4. Publications

#### 4.1 Journals

- Brezin, E. & Hikami, S. On an Airy matrix model with a logarithmic potential.
*Journal of Physics A: Mathematical and Theoretical***45**, 045203 (2012).

#### 4.2 Books and other one-time publications

- Brezin, E., Hikami, S. “Characteristic Polynomials” in Chap.18, The Oxford Handbook of Random Matrix Theory, Edited by Baik, J. et. al., Oxford Univ. Press, 2012. January.

#### 4.3 Oral and Poster Presentations

- Hikami, S., On an Airy matrix model with a logarithmic potential, LPT ENS, Paris, 2011 August.
- Hikami,S.,On an Airymatrix model with a logarithmic potential, Workshop “Spectrum of random operator and related topics”, Kyoto University, 2011,December.
- Hikami, S., Random matrix theory for complex systems, The University of Tokyo, 2012, January.

### 5. Intellectual Property Rights and Other Specific Achievements

Nothing to report

### 6. Meetings and Events

#### 6.1 Seminar

- Date: December 21, 2011.
- Venue: OIST Campus Lab1
- Speakers: Prof. Susumu Komiyama (The University of Tokyo) “Photon-detectors in the terahertz region”, Prof. Kenji Ikushima (The University of Agriculture and Technology) “Quantum-dot applications to terahertz physics”, Dr. Yusuke kajihara (The University of Tokyo) “Near-field microscopy of THz thermal radiation”.

#### 6.2 Workshop

Organizer of workshop: “Matrix models and related topics” Rikkyo University, 2012 February

### 7. Other

Nothing to report.