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Home/Math Visualization/Markov Chain Mixing

Markov Chain Mixing

Three-state Markov chain with transition matrix, stationary distribution, total variation distance, detailed-balance cue, and spectral-gap proxy.

Transition matrix

0.72
0.08
0
20

Measured values

Stationary π0.31, 0.39, 0.31
TV distance0.0000
Spectral gap proxy0.280

Self-loops slow mixing; directional bias can break detailed balance while preserving a stationary distribution.

Live graphs

About this model

A Markov chain evolves a probability vector by repeated multiplication with a transition matrix. The simulator shows a three-state chain, current probabilities, convergence toward the stationary distribution, total variation distance, a detailed-balance cue, and a spectral-gap proxy for mixing speed.

Who it's for: Probability, stochastic processes, Markov chain Monte Carlo, statistical physics, data science, and algorithms courses.

Key terms

  • Markov chain
  • Transition matrix
  • Stationary distribution
  • Mixing time
  • Detailed balance
  • Spectral gap

How it works

Markov chain mixing visualizer with transition matrix, stationary distribution, total variation distance, detailed balance, and spectral gap.

Key equations

p_{t+1}=p_t P, πP=π
Mixing speed is controlled by eigenvalues; larger spectral gap means faster convergence

Frequently asked questions

What is a stationary distribution?
It is a distribution pi such that pi P = pi. If the chain is ergodic, repeated transitions converge to this distribution from any starting state.
Why does the spectral gap matter?
The second-largest eigenvalue controls how fast transients decay. A larger gap generally means faster mixing.