Relaxation time of a Markov chain

The relaxation time \(1/\mathrm{gap}\) of a finite irreducible discrete-time Markov chain, the reciprocal of its spectral gap.

Usage

relaxationTime(object)

# S4 method for class 'markovchain'
relaxationTime(object)

Arguments

object: A markovchain object representing a finite, irreducible discrete-time Markov chain.

Value

A positive numeric scalar, in number of steps: Inf for a periodic chain (spectral gap zero), 1 for the trivial one-state chain.

Details

Following Levin and Peres (2017, Section 12.2) the relaxation time is \(t_{rel} = 1/\gamma\), with \(\gamma = 1 - \mathrm{SLEM}\) the spectral gap of spectralGap. It is the quantity PyDTMC calls relaxation_rate, even if it is a time, not a rate. The two are the same number.

PyDTMC's mixing_rate is \(-1/\log(\mathrm{SLEM})\), i.e. the timescale of the slowest non-trivial mode, which is already returned as the first element ("tau2") of impliedTimescales. It is therefore not repeated as a separate function.

References

Levin, D. A. and Peres, Y. (2017). Markov Chains and Mixing Times, 2nd edition. American Mathematical Society.

Examples

statesNames <- c("a", "b")
mc <- new("markovchain",
  states = statesNames,
  transitionMatrix = matrix(c(0.7, 0.3, 0.1, 0.9),
    byrow = TRUE, nrow = 2,
    dimnames = list(statesNames, statesNames)))
relaxationTime(mc)
#> [1] 2.5