Dimensional Reduction by Conformal Bootstrap

The dimensional reductions in the branched polymer and the random field Ising model (RFIM) are discussed by a conformal bootstrap method. The small size minors are applied for the evaluations of the scale dimensions of these two models and the results are compared to D'=D-2 dimensional Yang-Lee edge singularity and to pure D'=D-2 dimensional Ising model, respectively. For the former case, the dimensional reduction is shown to be valid for $3 \le D \le 8$, and for the latter case, the deviation from the dimensional reduction can be seen below five dimensions.Comment: 23 page, 13 figure

Punctures and p-spin curves from matrix models III. Dl type and logarithmic potential

The intersection numbers for p spin curves of the moduli space M(g,n) are considered for D type by a matrix model. The asymptotic behavior of the large genus g limit and large p limit are derived. The remarkable features of the cases of p= 1/2, - 1/2, -2, -3 are examined in the Laurent expansion for multiple correlation functions. The strong coupling expansions for the negative p cases are considered.Comment: 40 page

Spectral Form Factor for Time-dependent Matrix model

The quantum chaos is related to a Gaussian random matrix model, which shows a dip-ramp-plateau behavior in the spectral form factor for the large size $N$. The spectral form factor of time dependent Gaussian random matrix model shows also dip-ramp-plateau behavior with a rounding behavior instead of a kink near Heisenberg time. This model is converted to two matrix model, made of $M_1$ and $M_2$. The numerical evaluation for finite $N$ and analytic expression in the large $N$ are compared for the spectral form factor.Comment: 38 pages,16 figure

Instanton and Superconductivity in Supersymmetric CP(N-1) Model

The two dimensional supersymmetric CP(N-1) model has a striking similarity to the N=2 supersymmetric gauge theory in four dimensions. The BPS mass formula and the curve of the marginal stability (CMS), which exist in the four dimensional gauge theory, appears in this two dimensional CP(N-1) model. These two quntities are derived by a one-dimensional n-vector spin model in the large n limit for the N=2 case. This mapping is further investigated at the critical point. An application of the study of the BPS mass formula is proposed to the phenomena of the spin and charge separations in the Higgs phase.Comment: 6 page

Exponent of n-Ising matter fields coupled to 2d gravity

n-Ising spins on a random surface represented by a matrix model is studied as a model of the 2D gravity coupled to matter field with the central charge c > 1. The magnetic field is introduced to discuss the scaling exponent $\Delta$, and the value of this magnetic field exponent is estimated by the series expansion.Comment: 9page

Logarithmic moments of characteristic polynomials of random matrices

In a recent article we have discussed the connections between averages of powers of Riemann's $\zeta$-function on the critical line, and averages of characteristic polynomials of random matrices. The result for random matrices was shown to be universal, i.e. independent of the specific probability distribution, and the results were derived for arbitrary moments. This allows one to extend the previous results to logarithmic moments, for which we derive the explicit universal expressions in random matrix theory. We then compare these results to various results and conjectures for $\zeta$-functions, and the correspondence is again striking.Comment: 10 pages, late

Perturbative analysis of an n-Ising model on a random surface

Two dimensional quantum gravity coupled to a conformally invariant matter field of central charge c=n/2, is represented, in a discretized version, by n independent Ising spins per cell of the triangulations of a random surface. The matrix integral representation of this model leads to a diagrammatic expansion at large orders, when the Ising coupling constant is tuned to criticality, one extracts the values of the string susceptibility exponent. We extend our previous calculation to order eight for genus zero and investigate now also the genus one case in order to check the possibility of having a well-defined double scaling limit even c>1.Comment: 9p