Consider a system that is always in one of
states, numbered 1 through
. Every time a clock ticks, the system updates itself according to an
matrix of transition probabilities, the
entry of which gives the probability that the system moves from state
to state
at any clock tick. A Markov chain is a system like this, in which the next state depends only on the current state and not on previous states. Powers of the transition matrix approach a matrix with constant columns as the power increases. The number to which the entries in the
column converge is the asymptotic fraction of time the system spends in state
.
The image on the upper left shows the states of the chain with the current state colored red, where what is "current" is determined by the time slider. The histogram tracks the number of visits to each state over the number of time steps determined by the time slider. The transition probabilities can be changed using the new transition matrix slider. For small chains, powers of the transition matrix are shown at the bottom.
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