This survey is based on a series of talks I gave at the conference “Dynamical systems and diophantine approximation” at l’Instut Henri Poincaré in June 2003. I will present asymptotic results (transitivity, ergodicity, weak-mixing) for billiards based on the approximation technique developed by Katok and Zemlyakov. I will also present approximation techniques which allow to prove the abundance ...

We consider denumerable state nonhomogeneous Markov decision processes and extend results from both denumerable state homogeneous and finite state nonhomogeneous problems. We show that, under weak ergodicity, accumulation points of finite horizon optima (termed algorithmic optima) are average cost optimal. We also establish the existence of solution horizons. Finally, an algorithm is presented ...

Quantum many-body scars (QMBSs) are a paradigm of weak ergodicity breaking with direct technological applications. The authors show that QMBSs ubiquitous in quantum link formulations gauge theories, opening the door to large venue models where can be explored. They demonstrate not an artifact these formulations, rather, they may inherent feature ideal theory itself.

This survey is based on a series of talks I gave at the conference " Dynamical systems and Diophantine approximation " at l'Instut Henri Poincaré in June 2003. I will present asymptotic results (transitivity, ergodicity, weak-mixing) for billiards based on the approximation technique developed by Katok and Zemlyakov. I will also present approximation techniques which allow to prove the abundanc...

A discrete Gibbsian line ensemble $\mathfrak{L} = (L_1,\dots,L_N)$ consists of $N$ independent random walks on the integers conditioned not to cross one another, i.e., $L_1 \geq \cdots L_N$. In this paper we provide sufficient conditions for convergence a sequence suitably scaled ensembles $f^N (f_1^N,\dots,f_N^N)$ as number curves tends infinity. Assuming log-concavity and KMT-type coupling wa...

Due to their intrinsic link with nonlinear Fokker-Planck equations and many other applications, distribution dependent stochastic differential (DDSDEs) have been intensively investigated. In this paper, we summarize some recent progresses in the study of DDSDEs, which include correspondence weak solutions equations, well-posedness, regularity estimates, exponential ergodicity, long time large d...