This thesis is concerned with developing the tools of differential geometry and semiclassical analysis needed to understand the the quantum ergodicity theorem of Schnirelman (1974), Zelditch (1987), and de Verdière (1985) and the quantum unique ergodicity conjecture of Rudnick and Sarnak (1994). The former states that, on any Riemannian manifold with negative curvature or ergodic geodesic flow,...

Weconsider basic ergodicity properties of adaptiveMarkov chainMonteCarlo algorithms under minimal assumptions, using coupling constructions. We prove convergence in distribution and a weak law of large numbers. We also give counterexamples to demonstrate that the assumptions we make are not redundant.

We study the semiclassical behaviour of eigenfunctions of quantum systems with ergodic classical limit. By the quantum ergodicity theorem almost all of these eigenfunctions become equidistributed in a weak sense. We give a simple derivation of an upper bound of order |ln ~|−1 on the rate of quantum ergodicity if the classical system is ergodic with a certain rate. In addition we obtain a simila...

Scar states are special many-body eigenstates that weakly violate the eigenstate thermalization hypothesis (ETH). Using explicit formalism of Lanczos algorithm, usually known as forward scattering approximation in this context, we compute Krylov state (spread) complexity typical generated by time evolution PXP Hamiltonian, hosting such states. We show for N\'eel revives an approximate sense, wh...

We establish characterization results for the ergodicity of symmetric α–stable (SαS) and α–Fréchet max–stable stationary random fields. We first show that the result of Samorodnitsky [35] remains valid in the multiparameter setting, i.e., a stationary SαS (0 < α < 2) random field is ergodic (or equivalently, weakly mixing) if and only if it is generated by a null group action. The similarity of...

We will discuss the recurrence on the state space of the adaptive MCMC algorithm using some examples. We present the ergodicity properties of adaptive MCMC algorithms under the minimal recurrent assumptions, and show the Weak Law of Large Numbers under the same conditions. We will analyze the relationship between the recurrence on the product space of state space and parameter space and the erg...

We study two properties of nonsingular and infinite measure-preserving ergodic systems: weak double ergodicity, ergodicity with isometric coefficients. show that there exist transformations are coefficients but not weakly doubly ergodic; hence these notions equivalent for measure. also give type III? examples such systems, 0<??1. prove under certain hypotheses, systems mixing along the way we a...