Related papers: On quantum ergodicity for linear maps of the torus

In this expository note, we study several families of periodic graphs which satisfy a sufficient condition for the ergodicity of the associated continuous-time quantum walk. For these graphs, we compute the limiting distribution of the walk…

We propose a version of the Quantum Ergodicity theorem on large regular graphs of fixed valency. This is a property of delocalization of "most" eigenfunctions. We consider expander graphs with few short cycles (for instance random large…

We consider some analogs of the quantum unique ergodicity conjecture for geodesics, horocycles, or ``shrinking'' families of sets. In particular, we prove the analog of the QUE conjecture for Eisenstein series restricted to the infinite…

We consider eigenvalues of a quantized cat map (i.e. hyperbolic symplectic integer matrix), cut off in phase space to include a fixed point as its only periodic orbit on the torus. We prove a simple formula for the eigenvalues on both the…

The quantization of the continous cat maps on the torus has led to rather pathological quantum objects [6]. The non-generic behaviour of this model has led some to conclude that the Correspondence Principle fails in this case [2]. In this…

We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of "quantum chaotic dynamics". It is shown that under quite general conditions their roots…

In this paper we investigate the distribution of the set of values of a linear map at integer points on a quadratic surface. In particular we show that this set is dense in the range of the linear map subject to certain algebraic conditions…

We prove that the stationarity and the ergodicity of a quantum source are preserved by any trace-preserving completely positive linear map of the tensor product form ${\cal E}^{\otimes m}$, where a copy of ${\cal E}$ acts locally on each…

We prove that the Hecke--Maass eigenforms for a compact arithmetic triangle group have a growing number of nodal domains as the eigenvalue tends to $+\infty$. More generally the same is proved for eigenfunctions on negatively curved…

Quantum ergodicity asserts that almost all infinite sequences of eigenstates of a quantized ergodic system are equidistributed in the phase space. On the other hand, there are might exist exceptional sequences which converge to different…

W. Luo and P. Sarnak have proved the quantum unique ergodicity property for Eisenstein series on $\rm{PSL}(2,\mathbb{Z}) \backslash H$. We extend their result to Eisenstein series on $\rm{PSL}(2,O) \backslash H^n$, where $O$ is the ring of…

We define a class of dynamical maps on the quasi-local algebra of a quantum spin system, which are quantum analogues of probabilistic cellular automata. We develop criteria for such a system to be ergodic, i.e., to possess a unique…

We build an explicit $C^1$ isometric embedding $f_{\infty}:\mathbb{H}^2\to\mathbb{E}^3$ of the hyperbolic plane whose image is relatively compact. Its limit set is a closed curve of Hausdorff dimension 1. Given an initial embedding $f_0$,…

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$.…

We prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of $\PGL_n(\R)$. After attaching to each periodic…

Non-Hermitian random matrices provide a useful framework for understanding universal characteristics of dissipative quantum chaotic systems with loss or gain. We consider a model of two such system represented by two independent $N\times N$…

Using the Bargmann-Husimi representation of quantum mechanics on a torus phase space, we study analytically eigenstates of quantized cat maps. The linearity of these maps implies a close relationship between classically invariant…

In this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski.

We study the eigenvector mass distribution of an $N\times N$ Wigner matrix on a set of coordinates $I$ satisfying $| I | \ge c N$ for some constant $c >0$. For eigenvectors corresponding to eigenvalues at the spectral edge, we show that the…

We examine quantum normal typicality and ergodicity properties for quantum systems whose dynamics are generated by Hamiltonians which have residual degeneracy in their spectrum and resonance in their energy gaps. Such systems can be…