Reply to Comment on Dirac spectral sum rules for QCD in three dimensions

I reply to the comment by Dr S. Nishigaki (hep-th/0007042) to my papers Phys. Rev. D61 (2000) 056005 and Phys. Rev. D62 (2000) 016005.Comment: 2 pages, LaTeX, no figure

Singularities of the Partition Function for the Ising Model Coupled to 2d Quantum Gravity

We study the zeros in the complex plane of the partition function for the Ising model coupled to 2d quantum gravity for complex magnetic field and real temperature, and for complex temperature and real magnetic field, respectively. We compute the zeros by using the exact solution coming from a two matrix model and by Monte Carlo simulations of Ising spins on dynamical triangulations. We present evidence that the zeros form simple one-dimensional curves in the complex plane, and that the critical behaviour of the system is governed by the scaling of the distribution of the singularities near the critical point. Despite the small size of the systems studied, we can obtain a reasonable estimate of the (known) critical exponents.Comment: 22 pages, LaTeX2e, 10 figures, added discussion on antiferromagnetic transition and reference

Comment on Dirac spectral sum rules for QCD_3

Recently Magnea hep-th/9907096 , hep-th/9912207 [Phys.Rev.D61, 056005 (2000); Phys.Rev.D62, 016005 (2000)] claimed to have computed the first sum rules for Dirac operators in 3D gauge theories from 0D non-linear sigma models. I point out that these computations are incorrect, and that they contradict with the exact results for the spectral densities unambiguously derived from random matrix theory by Nagao and myself.Comment: REVTeX 3.1, 2 pages, no figure. (v2) redundant part removed, conclusion unchange

The Width of the Colour Flux Tube

We discuss and rederive in a general way the logarithmic growth of the mean squared width of the colour flux tube as a function of the interquark separation. Recent data on 3D $Z_2$ gauge theory, combined with high precision data on the interface physics of the 3D Ising model fit nicely this behaviour over a range of more than two orders of magnitude.Comment: 3 pages, contribution to the Lattice '94 conference, uuencoded compressed ps-fil

Fluid Interfaces in the 3D Ising Model as a Dilute Gas of Handles

We study the topology of fluid interfaces in the 3D Ising model in the rough phase. It turns out that such interfaces are accurately described as dilute gases of microscopic handles, and the stiffness of the interface increases with the genus. The number of configurations of genus $g$ follows a Poisson-like distribution. The probability per unit area for creating a handle is well fitted in a wide range of the inverse temperature $\beta$ near the roughening point by an exponentially decreasing function of $\beta$. The procedure of summing over all topologies results in an effective interface whose squared width scales logarithmically with the lattice size.Comment: 15 pages, Latex and 10 ps figs (uuencoded file) DFTT 27/9

Spin-spin correlation functions of spin systems coupled to 2-d quantum gravity for 0<c<10 < c < 1

We perform Monte Carlo simulations of 2-d dynamically triangulated surfaces coupled to Ising and three--states Potts model matter. By measuring spin-spin correlation functions as a function of the geodesic distance we provide substantial evidence for a diverging correlation length at $\beta_c$. The corresponding scaling exponents are directly related to the KPZ exponents of the matter fields as conjectured in [4] (NPB454(1995)313).Comment: Talk presented at LATTICE96(gravity

Complex zeros of the 2d Ising model on dynamical random lattices

We study the zeros in the complex plane of the partition function for the Ising model coupled to $2d$ quantum gravity for complex magnetic field and for complex temperature. We compute the zeros by using the exact solution coming from a two matrix model and by Monte Carlo simulations of Ising spins on dynamical triangulations. We present evidence that the zeros form simple one-dimensional patterns in the complex plane, and that the critical behaviour of the system is governed by the scaling of the distribution of singularities near the critical point.Comment: 3 pages, 8 figures, latex2e, uses espcrc2.sty. Contribution to Lattice '97, Edinburgh, July 1997, to appear on Nucl. Phys. B (Proc. Suppl.

Universality of random matrices in the microscopic limit and the Dirac operator spectrum

We prove the universality of correlation functions of chiral unitary and unitary ensembles of random matrices in the microscopic limit. The essence of the proof consists in reducing the three-term recursion relation for the relevant orthogonal polynomials into a Bessel equation governing the local asymptotics around the origin. The possible physical interpretation as the universality of the soft spectrum of the Dirac operator is briefly discussed

Three-dimensional QCD in the adjoint representation and random matrix theory

In this paper we complete the derivations of finite volume partition functions for QCD using random matrix theories by calculating the effective low-energy partition function for three-dimensional QCD in the adjoint representation from a random matrix theory with the same global symmetries. As expected, this case corresponds to Dyson index $\beta =4$, that is, the Dirac operator can be written in terms of real quaternions. After discussing the issue of defining Majorana fermions in Euclidean space, the actual matrix model calculation turns out to be simple. We find that the symmetry breaking pattern is $O(2N_f) \to O(N_f) \times O(N_f)$, as expected from the correspondence between symmetric (super)spaces and random matrix universality classes found by Zirnbauer. We also derive the first Leutwyler--Smilga sum rule.Comment: LaTeX, 19 pages. Minor corrections, added comments, to appear on Phys. Rev.