Convex Replica Simmetry Breaking From Positivity and Thermodynamic Limit

Consider a correlated Gaussian random energy model built by successively adding one particle (spin) into the system and imposing the positivity of the associated covariance matrix. We show that the validity of a recently isolated condition ensuring the existence of the thermodynamic limit forces the covariance matrix to exhibit the Parisi replica symmetry breaking scheme with a convexity condition on the matrix elements.Comment: 11 page

Monotonicity and Thermodynamic Limit for Short Range Disordered Models

If the variance of a short range Gaussian random potential grows like the volume its quenched thermodynamic limit is reached monotonically.Comment: 2 references adde

Geometric approach to the Hamilton-Jacobi equation and global parametrices for the Schr\"odinger propagator

We construct a family of Fourier Integral Operators, defined for arbitrary large times, representing a global parametrix for the Schr\"odinger propagator when the potential is quadratic at infinity. This construction is based on the geometric approach to the corresponding Hamilton-Jacobi equation and thus sidesteps the problem of the caustics generated by the classical flow. Moreover, a detailed study of the real phase function allows us to recover a WKB semiclassical approximation which necessarily involves the multivaluedness of the graph of the Hamiltonian flow past the caustics

Ergodic Properties of Infinite Harmonic Crystals: an Analytic Approach

We give through pseudodifferential operator calculus a proof that the quantum dynamics of a class of infinite harmonic crystals becomes ergodic and mixing with respect to the quantum Gibbs measure if the classical infinite dynamics is respectively ergodic and mixing with respect to the classical infinite Gibbs measure. The classical ergodicity and mixing properties are recovered as $\hbar\to 0$, and the infinitely many particles limits of the quantum Gibbs averages are proved to be the averages over a classical infinite Gibbs measure of the symbols generating the quantum observables under Weyl quantization.Comment: 30 pages, plain LaTe

Localization in infinite billiards: a comparison between quantum and classical ergodicity

Consider the non-compact billiard in the first quandrant bounded by the positive $x$-semiaxis, the positive $y$-semiaxis and the graph of $f(x) = (x+1)^{-\alpha}$, $\alpha \in (1,2]$. Although the Schnirelman Theorem holds, the quantum average of the position $x$ is finite on any eigenstate, while classical ergodicity entails that the classical time average of $x$ is unbounded.Comment: 9 page

Absolute Continuity of the Floquet Spectrum for a Nonlinearly Forced Harmonic Oscillator

We prove that the Floquet spectrum of a class of time-periodic Schroedinger equations under a a mildly nonlinear resonant forcing is purely absolutely continuous.Comment: 8 page

A uniform quantum version of the Cherry theorem

Consider in $L^2(\R^2)$ the operator family $H(\epsilon):=P_0(\hbar,\omega)+\epsilon F_0$. $P_0$ is the quantum harmonic oscillator with diophantine frequency vector \om, $F_0$ a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space, and \ep\in\C. Then there exist \ep^\ast >0 independent of $\hbar$ and an open set \Omega\subset\C^2\setminus\R^2 such that if |\ep|<\ep^\ast and \om\in\Om the quantum normal form near $P_0$ converges uniformly with respect to $\hbar$. This yields an exact quantization formula for the eigenvalues, and for $\hbar=0$ the classical Cherry theorem on convergence of Birkhoff's normal form for complex frequencies is recovered.Comment: 17 page

PT Symmetric Schr\"odinger Operators: Reality of the Perturbed Eigenvalues

We prove the reality of the perturbed eigenvalues of some PT symmetric Hamiltonians of physical interest by means of stability methods. In particular we study 2-dimensional generalized harmonic oscillators with polynomial perturbation and the one-dimensional $x^2(ix)^{\epsilon}$ for $-1<\epsilon<0$

HSkip+: A Self-Stabilizing Overlay Network for Nodes with Heterogeneous Bandwidths

In this paper we present and analyze HSkip+, a self-stabilizing overlay network for nodes with arbitrary heterogeneous bandwidths. HSkip+ has the same topology as the Skip+ graph proposed by Jacob et al. [PODC 2009] but its self-stabilization mechanism significantly outperforms the self-stabilization mechanism proposed for Skip+. Also, the nodes are now ordered according to their bandwidths and not according to their identifiers. Various other solutions have already been proposed for overlay networks with heterogeneous bandwidths, but they are not self-stabilizing. In addition to HSkip+ being self-stabilizing, its performance is on par with the best previous bounds on the time and work for joining or leaving a network of peers of logarithmic diameter and degree and arbitrary bandwidths. Also, the dilation and congestion for routing messages is on par with the best previous bounds for such networks, so that HSkip+ combines the advantages of both worlds. Our theoretical investigations are backed by simulations demonstrating that HSkip+ is indeed performing much better than Skip+ and working correctly under high churn rates.Comment: This is a long version of a paper published by IEEE in the Proceedings of the 14-th IEEE International Conference on Peer-to-Peer Computin