Eigenvalue analysis of an irreversible random walk with skew detailed balance conditions

An irreversible Markov-chain Monte Carlo (MCMC) algorithm with skew detailed balance conditions originally proposed by Turitsyn et al. is extended to general discrete systems on the basis of the Metropolis-Hastings scheme. To evaluate the efficiency of our proposed method, the relaxation dynamics of the slowest mode and the asymptotic variance are studied analytically in a random walk on one dimension. It is found that the performance in irreversible MCMC methods violating the detailed balance condition is improved by appropriately choosing parameters in the algorithm.

Figure 1

Graphical representation of the transition matrix. Each arrow represents a transition flow from a state to another one.

Figure 2

Schematic picture of the extended transition graph. Solid arrows represent transition flows within the same subspace and dashed ones are lifting flows.

Figure 3

Schematic picture of a part of the transition graph on the state space $I$ in a reversible Markov chain.

Figure 4

Schematic picture of a part of the transition graph on the extended state space in an irreversible Markov chain.

Figure 5

Maps of the second-largest eigenvalue of the extended transition matrix $\tilde{\mathsf{T}}$ in absolute value. The left and right panels represent cases (a) $\delta ={\delta }^{\prime }$ and (b) ${\delta }^{\prime }=1$, respectively. In each panel, the shaded portion represents the parameter region $(\delta ,\gamma )$ where the relaxation time is reduced by imposing the skew detailed balance condition. The thick dashed line represents the minimum of the relaxation time of the irreversible random walk for a fixed $\delta$ and the star symbol provides the lowest value of the relaxation time in the parameter region. Note that ${\tilde{\lambda }}_2^{+}$ is a complex number below the thick dashed line.

Figure 6

The parameter region $(\delta ,\gamma )$ where the asymptotic variance is reduced by imposing the skew detailed balance condition (shaded portion). The left and right panels represent cases (a) $\delta ={\delta }^{\prime }$ and (b) ${\delta }^{\prime }=1$, respectively. The star symbol indicates the lowest value in the parameter region.