Asymptotic optimality of isoperimetric constants with respect to $L^{2}(π)$-spectral gaps

$L^{2}$-spectral gaps, weak-reversible and very weak-reversible Markov chains

The theory of $L^2$-spectral gaps for reversible Markov chains has been studied by many authors. In this paper we consider positive recurrent general state space Markov chains with stationary transition probabilities. Replacing the assumption of reversibility by a less strong one, we still obtain a simple necessary and sufficient condition for the spectral gap property of the associated Markov operator in terms of isoperimetric constant. Moreover, we define a new sequence of isoperimetric constants which provides a necessary and sufficient condition for the existence of a spectral gap in a ver…

On optimality gaps in the Halfin--Whitt regime

We consider optimal control of a multi-class queue in the Halfin--Whitt regime, and revisit the notion of asymptotic optimality and the associated optimality gaps. The existing results in the literature for such systems provide asymptotically optimal controls with optimality gaps of $o(\sqrt{n})$ where $n$ is the system size, for example, the number of servers. We construct a sequence of asymptotically optimal controls where the optimality gap grows logarithmically with the system size. Our analysis relies on a sequence of Brownian control problems, whose refined structure helps us achieve the…

$L^{2}$-spectral gaps for time discrete reversible Markov chains

In this paper we study the spectral properties of Markov-operator on $L^{2}$-spaces. Lawler and Sokal (Trans. Amer. Math. Soc., 1988, 309, pp. 557-580) used isoperimetric constants for discrete and continuous time Markov chains to obtain a spectral gap at 1. For time discrete Markov chains this does not exclude periodic behavior. We define a new constant measuring the distance from periodicity and give necessary and sufficient conditions for the existence of a global spectral gap in terms of this constant.

Optimality gaps in asymptotic dimensioning of many-server systems

The Quality-and-Efficiency-Driven (QED) regime provides a basis for solving asymptotic dimensioning problems that trade off revenue, costs and service quality. We derive bounds for the optimality gaps that capture the differences between the true optimum and the asymptotic optimum based on the QED approximations. Our bounds generalize earlier results for classical many-server systems. We also apply our bounds to a many-server system with threshold control.

Metropolis马尔可夫链的显式收敛界：等周、谱隙与轮廓

Explicit convergence bounds for Metropolis Markov chains: isoperimetry, spectral gaps and profiles

推导了$R^d$上随机游走Metropolis算法谱隙的首个显式界，改进等周轮廓不等式获得电导轮廓界，从而得到一大类模型的${\rm L}^2$混合时间显式界，并为预条件Crank--Nicolson链证明了维度无关的界。

Two Examples of COM Bounds using Spectral Gaps: Length of the LIS in a Random Permutation and Lipschitz Functions of 1d Markov Chains

We consider two examples for a well-known method for obtaining concentration of measure (COM) bounds for a given observable in a given measure. The method is to consider an auxiliary Markov chain for which the invariant distribution is the measure of interest. Then one obtains COM bounds involving two quantities. The first is the spectral gap of the Markov transition matrix. The second is an appropriate Lipschitz constant for the observable of interest with respect to 1 step of the Markov chain. We consider two examples of the basic method. The first is to obtain rough COM bounds for the lengt…