Chaotic ray dynamics in an optical cavity with a beam splitter

We investigate the ray dynamics in an optical cavity when a ray splitting mechanism is present. The cavity is a conventional two-mirror stable resonator and the ray splitting is achieved by inserting an optical beam splitter perpendicular to the cavity axis. Using Hamiltonian optics, we show that such a simple device presents a surprisingly rich chaotic ray dynamics.Comment: 6 pages, 4 figure

Large n limit of Gaussian random matrices with external source, Part III: Double scaling limit

We consider the double scaling limit in the random matrix ensemble with an external source \frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM defined on $n\times n$ Hermitian matrices, where $A$ is a diagonal matrix with two eigenvalues $\pm a$ of equal multiplicities. The value $a=1$ is critical since the eigenvalues of $M$ accumulate as $n \to \infty$ on two intervals for $a > 1$ and on one interval for $0 < a < 1$. These two cases were treated in Parts I and II, where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles. For the critical case $a=1$ new limiting behavior occurs which is described in terms of Pearcey integrals, as shown by Br\'ezin and Hikami, and Tracy and Widom. We establish this result by applying the Deift/Zhou steepest descent method to a $3 \times 3$-matrix valued Riemann-Hilbert problem which involves the construction of a local parametrix out of Pearcey integrals. We resolve the main technical issue of matching the local Pearcey parametrix with a global outside parametrix by modifying an underlying Riemann surface.Comment: 36 pages, 9 figure

Output functions and fractal dimensions in dynamical systems

We present a novel method for the calculation of the fractal dimension of boundaries in dynamical systems, which is in many cases many orders of magnitude more efficient than the uncertainty method. We call it the Output Function Evaluation (OFE) method. The OFE method is based on an efficient scheme for computing output functions, such as the escape time, on a one-dimensional portion of the phase space. We show analytically that the OFE method is much more efficient than the uncertainty method for boundaries with $D<0.5$, where $D$ is the dimension of the intersection of the boundary with a one-dimensional manifold. We apply the OFE method to a scattering system, and compare it to the uncertainty method. We use the OFE method to study the behavior of the fractal dimension as the system's dynamics undergoes a topological transition.Comment: Uses REVTEX; to be published in Phys. Rev. Let

Discretization Dependence of Criticality in Model Fluids: a Hard-core Electrolyte

Grand canonical simulations at various levels, $\zeta=5$-20, of fine- lattice discretization are reported for the near-critical 1:1 hard-core electrolyte or RPM. With the aid of finite-size scaling analyses it is shown convincingly that, contrary to recent suggestions, the universal critical behavior is independent of $\zeta$ (\grtsim 4); thus the continuum $(\zeta\to\infty)$ RPM exhibits Ising-type (as against classical, SAW, XY, etc.) criticality. A general consideration of lattice discretization provides effective extrapolation of the {\em intrinsically} erratic $\zeta$-dependence, yielding (\Tc^ {\ast},\rhoc^{\ast})\simeq (0.0493_{3},0.075) for the $\zeta=\infty$ RPM.Comment: 4 pages including 4 figure

Exact solution of the six-vertex model with domain wall boundary condition. Critical line between ferroelectric and disordered phases

This is a continuation of the papers [4] of Bleher and Fokin and [5] of Bleher and Liechty, in which the large $n$ asymptotics is obtained for the partition function $Z_n$ of the six-vertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the present paper we obtain the large $n$ asymptotics of $Z_n$ on the critical line between these two phases.Comment: 22 pages, 6 figures, to appear in the Journal of Statistical Physic

Spectral statistics for quantized skew translations on the torus

We study the spectral statistics for quantized skew translations on the torus, which are ergodic but not mixing for irrational parameters. It is shown explicitly that in this case the level--spacing distribution and other common spectral statistics, like the number variance, do not exist in the semiclassical limit.Comment: 7 pages. One figure, include

Unique positive solution for an alternative discrete Painlevé I equation

We show that the alternative discrete Painleve I equation has a unique solution which remains positive for all n >0. Furthermore, we identify this positive solution in terms of a special solution of the second Painleve equation involving the Airy function Ai(t). The special-function solutions of the second Painleve equation involving only the Airy function Ai(t) therefore have the property that they remain positive for all n>0 and all t>0, which is a new characterization of these special solutions of the second Painlevé equation and the alternative discrete Painlevé I equation

Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes

Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers \cite{Adler:2009a, Daems:2007} and sample covariance matrices \cite{Baik:2005}. We consider the case when the $n\times n$ external source matrix has two distinct real eigenvalues: $a$ with multiplicity $r$ and zero with multiplicity $n-r$. The source is small in the sense that $r$ is finite or $r=\mathcal O(n^\gamma)$, for $0< \gamma<1$. For a Gaussian potential, P\'ech\'e \cite{Peche:2006} showed that for $|a|$ sufficiently small (the subcritical regime) the external source has no leading-order effect on the eigenvalues, while for $|a|$ sufficiently large (the supercritical regime) $r$ eigenvalues exit the bulk of the spectrum and behave as the eigenvalues of $r\times r$ Gaussian unitary ensemble (GUE). We establish the universality of these results for a general class of analytic potentials in the supercritical and subcritical regimes.Comment: 41 pages, 4 figure

The Julia sets and complex singularities in hierarchical Ising models

We study the analytical continuation in the complex plane of free energy of the Ising model on diamond-like hierarchical lattices. It is known that the singularities of free energy of this model lie on the Julia set of some rational endomorphism $f$ related to the action of the Migdal-Kadanoff renorm-group. We study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin of $f$. We prove that for almost all (with respect to the harmonic measure) geodesics the complex critical exponent is common, and compute it