Comment on ``Casimir force in compact non-commutative extra dimensions and radius stabilization''

We call attention to a series of mistakes in a paper by S. Nam [JHEP 10 (2000) 044, hep-th/0008083].Comment: 6 pages, LaTeX, uses JHEP.cl

Fluctuating Commutative Geometry

We use the framework of noncommutative geometry to define a discrete model for fluctuating geometry. Instead of considering ordinary geometry and its metric fluctuations, we consider generalized geometries where topology and dimension can also fluctuate. The model describes the geometry of spaces with a countable number $n$ of points. The spectral principle of Connes and Chamseddine is used to define dynamics.We show that this simple model has two phases. The expectation value , the average number of points in the universe, is finite in one phase and diverges in the other. Moreover, the dimension $\delta$ is a dynamical observable in our model, and plays the role of an order parameter. The computation of is discussed and an upper bound is found, $< 2$. We also address another discrete model defined on a fixed $d=1$ dimension, where topology fluctuates. We comment on a possible spontaneous localization of topology.Comment: 7 pages. Talk at the conference "Spacetime and Fundamental Interactions: Quantum Aspects" (Vietri sul Mare, Italy, 26-31 May 2003), in honour of A. P. Balachandran's 65th birthda

Anisotropic Lifshitz Point at O(ϵL2)O(\epsilon_L^2)

We present the critical exponents $\nu_{L2}$, $\eta_{L2}$ and $\gamma_{L}$ for an $m$-axial Lifshitz point at second order in an $\epsilon_{L}$ expansion. We introduced a constraint involving the loop momenta along the $m$-dimensional subspace in order to perform two- and three-loop integrals. The results are valid in the range $0 \leq m < d$. The case $m=0$ corresponds to the usual Ising-like critical behavior.Comment: 10 pages, Revte

Yang-Lee Zeros of the Ising model on Random Graphs of Non Planar Topology

We obtain in a closed form the 1/N^2 contribution to the free energy of the two Hermitian N\times N random matrix model with non symmetric quartic potential. From this result, we calculate numerically the Yang-Lee zeros of the 2D Ising model on dynamical random graphs with the topology of a torus up to n=16 vertices. They are found to be located on the unit circle on the complex fugacity plane. In order to include contributions of even higher topologies we calculated analytically the nonperturbative (sum over all genus) partition function of the model Z_n = \sum_{h=0}^{\infty} \frac{Z_n^{(h)}}{N^{2h}} for the special cases of N=1,2 and graphs with n\le 20 vertices. Once again the Yang-Lee zeros are shown numerically to lie on the unit circle on the complex fugacity plane. Our results thus generalize previous numerical results on random graphs by going beyond the planar approximation and strongly indicate that there might be a generalization of the Lee-Yang circle theorem for dynamical random graphs.Comment: 19 pages, 7 figures ,1 reference and a note added ,To Appear in Nucl.Phys

Yang-Lee Zeros of the Two- and Three-State Potts Model Defined on ϕ3\phi^3 Feynman Diagrams

We present both analytic and numerical results on the position of the partition function zeros on the complex magnetic field plane of the $q=2$ (Ising) and $q=3$ states Potts model defined on $\phi^3$ Feynman diagrams (thin random graphs). Our analytic results are based on the ideas of destructive interference of coexisting phases and low temperature expansions. For the case of the Ising model an argument based on a symmetry of the saddle point equations leads us to a nonperturbative proof that the Yang-Lee zeros are located on the unit circle, although no circle theorem is known in this case of random graphs. For the $q=3$ states Potts model our perturbative results indicate that the Yang-Lee zeros lie outside the unit circle. Both analytic results are confirmed by finite lattice numerical calculations.Comment: 16 pages, 2 figures. Third version: the title was slightly changed. To be published in Physical Review

Fluctuating Dimension in a Discrete Model for Quantum Gravity Based on the Spectral Principle

The spectral principle of Connes and Chamseddine is used as a starting point to define a discrete model for Euclidean quantum gravity. Instead of summing over ordinary geometries, we consider the sum over generalized geometries where topology, metric and dimension can fluctuate. The model describes the geometry of spaces with a countable number $n$ of points, and is related to the Gaussian unitary ensemble of Hermitian matrices. We show that this simple model has two phases. The expectation value $$, the average number of points in the universe, is finite in one phase and diverges in the other. We compute the critical point as well as the critical exponent of$$. Moreover, the space-time dimension $\delta$ is a dynamical observable in our model, and plays the role of an order parameter. The computation of $$ is discussed and an upper bound is found, $< 2$.Comment: 10 pages, no figures. Third version: This new version emphasizes the spectral principle rather than the spectral action. Title has been changed accordingly. We also reformulated the computation of the dimension, and added a new reference. To appear in Physical Review Letter