Japanese version is here.

Recent works

**Random matrix theory and its applications to physics**Random matrix theory is the theory of matrices with random number elements. It was invented by mathematical statisticians at the beginning of the 20th century and after World War II introduced to nuclear physics. Now it is transversally applied to many areas, including analytic theory of numbers, combinatorics, elementary particle physics, solid state physics, ecology and financial engineering. Its deep structure and meanings are currently being revealed one after another.

I study random matrices from the viewpoints of both the fundamental theory and various applications. Related research topics are the followings.

**Entropy maximization method in random matrix theory****Sypersymmetry, quaternion, special functions in mathematical physics and eigenvalue correlations of random matrices****Conductance distribution of mesoscopic systems and random matrices****Brownian motion models of matrices and collective dynamics of energy levels****Random matrix model as an effective theory of QCD (Quantum chromodynamics)****Vicious random walkers, growth process and discretizations of random matrices**

**Semiclassical theory of spectral statistics**It is known that the spectral statistics of quantum systems reflect the features of the underlying classical dynamics. In particular, when the corresponding classical system is chaotic, universal spectral correlations predicted by random matrix theory are observed. In attempt to elucidate the origin of the universality, the following topics are studied.

**Analysis of algebraic structures which enable the evaluation of periodic orbit sums****Correspondence between the Brownian motions of classical parameters and the the matrix Brownian motions****Semiclassical analysis of the spectral statistics of quantum graphs (quantum mechanics on graphs)**

**Semiclassical electric conduction in antidot superlattices**Antidot superlattices are regular or irregular potential arrays fabricated on semiconductor substrates. The lattice spacing can be made less than 1 micrometer so that the resultant electronic system is in the boundary region between quantum and classical theories. Related research topics are the followings.

**The effect of unstable diffusive orbits on the magnetoresistance oscillation****Negative Hall resistance and the chaotic autocorrelation of the electron orbits****Quantization of classical dynamical systems with mixed phase spaces**

Java applet for a single cell antidot system (You can apply a "magnetic field"!)