E. Montaldi

LARGE N PHASE TRANSITIONS IN LOW DIMENSIONS

Quantum models with fields in matrix representations of classical groups exhibit a high order phase transition in the limit of large order of the group. In zero dimension of space-time the Green functions of the newly found branch are those of a Gaussian model; in one dimension, the physical mass vanishes at the critical point.

Large rectangular random matrices

Models with a multiplet of field variables arranged into rectangular matrices, in the limit of infinite dimensions of the matrices, are studied. In zero‐dimensional space (where the problem is a combinatorial one) a closed solution is given that improves the one previously known. In arbitrary space dimension a symmetry is described that connects rectangular models with vector models.

Matrix models and graph colouring

Abstract We study an edge-colouring problem on random planar graphs which is one of the simplest vertex models that may be analyzed by standard methods of large N matrix models. The main features of the saddle point solution and its critical behaviour are described. At the critical value of the coupling gcr the eigen value density u(λ)M is found to vanish at the border of the support as |λ−a| 2 3 .

Multicritical points in matrix models

The simplest matrix model which exhibits multicritical points is carefully analysed. The authors reproduce results of potential interest for the non-perturbative theory of strings in the region where the orthogonal polynomials were correctly used. However, the analysis holds for the whole parameter space.

A matrix model for random surfaces with dynamical holes

A matrix model to describe dynamical loops on random planar graphs is analysed. It has similarities with a model studied by Kazakov, a few years ago, and the O(n) model of Kostov and collaborators. The main difference is that all loops are coherently oriented and empty. The free energy is analytically evaluated and the continuum limit is analysed in a region of parameters where the universality of the continuum description may not be expected. Our phase diagram is analogous to Kazakovs model with two phases (surface with small holes and tearing phase) with Kazakovs scaling exponents. The critical exponents of the third phase, which occurs on the boundary between the two above phases, differ from the corresponding exponents in Kazakovs model.A matrix model to describe dynamical loops on random planar graphs is analyzed. It has similarities with a model studied by Kazakov, few years ago, and the O(n) model by Kostov and collaborators. The main difference is that all loops are coherently oriented and empty. The free energy is analytically evaluated and the two critical phases are analyzed, where the free energy exhibits the same critical behaviour of Kazakovs model, thus confirming the universality of the description in the continuum limit (surface with small holes, and the tearing phase). A third phase occurs on the boundary separating the above phase regions, and is characterized by a different singular behaviour, presumably due to the orientation of loops.

Ising models on Feynman graphs

Abstract An Ising model on a lattice made of planar Feynman graphs, which was originally introduced by Kazakov, is studied. Its exact correspondence with a matrix model is reconsidered and a new correspondence suggested. The new nonperturbative phase of the model, recently found by one of us, is further analyzed.

Large-N spontaneous magnetisation in zero dimensions

A two-parameter model of Hermitian matrices in zero-dimensional space is solved in the large-N limit. A very interesting phase diagram and a spontaneous magnetisation are exhibited.

The quantum mechanical planar propagator: From vector models to planar graphs

Abstract In the framework of a systematic investigation of planar field theory, we study the planar two-point Green function. We define and evaluate an expansion which preserves the formal properties of the theory and compute the three lowest lying poles of the propagator in a one-dimensional space-time, finding excellent agreement with known results.

On the class operator of the su(2) group

The explicit expression for the class operator of the SU (2) group, previously obtained by the integration with ordered product technique, is rederived in a more direct way.

Can one expect a Kondo effect in quasiperiodic structures

We show that electrons hopping over quasiperiodic tilings give rise to a modified Kondo effect, obeying a power-law behaviour in place of the standard logarithmic behaviour.