Infinite N phase transitions in continuum Wilson loop operators

We define smoothed Wilson loop operators on a four dimensional lattice and check numerically that they have a finite and nontrivial continuum limit. The continuum operators maintain their character as unitary matrices and undergo a phase transition at infinite N reflected by the eigenvalue distribution closing a gap in its spectrum when the defining smooth loop is dilated from a small size to a large one. If this large N phase transition belongs to a solvable universality class one might be able to calculate analytically the string tension in terms of the perturbative Lambda-parameter. This would be achieved by matching instanton results for small loops to the relevant large-N-universal function which, in turn, would be matched for large loops to an effective string theory. Similarities between our findings and known analytical results in two dimensional space-time indicate that the phase transitions we found only affect the eigenvalue distribution, but the traces of finite powers of the Wilson loop operators stay smooth under scaling.Comment: 31 pages, 9 figures, typos and references corrected, minor clarifications adde

Scaled limit and rate of convergence for the largest eigenvalue from the generalized Cauchy random matrix ensemble

In this paper, we are interested in the asymptotic properties for the largest eigenvalue of the Hermitian random matrix ensemble, called the Generalized Cauchy ensemble $GCy$, whose eigenvalues PDF is given by $$\textrm{const}\cdot\prod_{1\leq j<k\leq N}(x_j-x_k)^2\prod_{j=1}^N (1+ix_j)^{-s-N}(1-ix_j)^{-\bar{s}-N}dx_j,$$where $s$ is a complex number such that $\Re(s)>-1/2$ and where $N$ is the size of the matrix ensemble. Using results by Borodin and Olshanski \cite{Borodin-Olshanski}, we first prove that for this ensemble, the largest eigenvalue divided by $N$ converges in law to some probability distribution for all $s$ such that $\Re(s)>-1/2$. Using results by Forrester and Witte \cite{Forrester-Witte2} on the distribution of the largest eigenvalue for fixed $N$, we also express the limiting probability distribution in terms of some non-linear second order differential equation. Eventually, we show that the convergence of the probability distribution function of the re-scaled largest eigenvalue to the limiting one is at least of order $(1/N)$.Comment: Minor changes in this version. Added references. To appear in Journal of Statistical Physic

Cluster simulations of loop models on two-dimensional lattices

Applied Science