In the setting of generic $\beta$-ensembles, we use loop equation hierarchy to prove a local law with optimal error up constant, valid on any scale including microscopic. This has following consequences. (i) The rigidity ordered particles is order $(\log N)/N$ in bulk spectrum. (ii) Fluctuations satisfy central limit theorem covariance corresponding logarithmically correlated field; particular ...

We consider the diffusive limit of a typical pure-jump Markovian control problem as intensity driving Poisson process tends to infinity. show that convergence speed is provided by Hölder exponent Hessian problem, and explain how correction terms can be constructed. This provides an alternative efficient method for numerical approximation optimal in situations with very high jumps. illustrate th...

Abstract We consider a general linear program in standard form whose right-hand side constraint vector is subject to random perturbations. For the corresponding program, we characterize under assumptions fluctuations of empirical optimal solutions around their population quantities after standardization by distributional limit theorem. Our approach geometric nature and further relies on duality...

Load balancing plays a critical role in efficiently dispatching jobs parallel-server systems such as cloud networks and data centers. A fundamental challenge the design of load algorithms is to achieve an optimal trade-off between delay performance implementation overhead (e.g. communication or memory usage). This has primarily been studied so far from angle amount required asymptotically perfo...