relaxationTime: Relaxation time of a Markov chain

In markovchain: Easy Handling Discrete Time Markov Chains

Relaxation time of a Markov chain

Description

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 signature 'markovchain'
relaxationTime(object)

Arguments

object

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

Value

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

References

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

See Also

spectralGap, slem, impliedTimescales, mixingTime

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)

markovchain documentation built on Oct. 10, 2026, 9:07 a.m.