Let me help reconstruct and analyze this article that appears to be discussing a Markov chain problem.
The problem appears to be analyzing a Markov chain with the following properties:
- There are 12 states (labeled 0 to 11)
- State 0 is an absorbing state (terminal state)
- From state 1, there are two possible transitions:
- To state 0 with probability $$\frac{1}{2}$$
- To state 2 with probability $$\frac{1}{2}$$
- For states 2 through 10, each state can transition:
- To the previous state with probability $$\frac{1}{2}$$
- To the next state with probability $$\frac{1}{2}$$
- State 11 is also an absorbing state
Let me create a visual representation of this Markov chain:
The article presents two Mathematica implementations to solve this Markov chain:
- The first implementation uses a full transition matrix representation
- The second uses a sparse matrix representation which is more efficient
Both implementations calculate the stationary distribution of the Markov chain, particularly focusing on the probability of ending in state 11 (the final absorbing state).
The result obtained is , which means that in the long run, the probability of the process ending in state 11 (versus state 0) is approximately 0.0909 or about 9.09%.
This can be interpreted as: if you start from state 1, you have about a 9.09% chance of eventually reaching state 11, while the remaining 90.91% probability corresponds to eventually reaching state 0.
The problem is essentially analyzing a random walk with two absorbing states (0 and 11), where the process starts at state 1 and can move one step at a time in either direction with equal probability.
