A particle is placed at point A in the figure below. It travels to each of its adjacent points with equal probability. What is the probability that it has traveled to all the points after infinite time?
I am not sure how to go about this problem. First of all, most of the random walk questions I have encountered till now are about returning to the starting point. I am not able to understand how to tackle the problem of traveling to all the points. Secondly, this seems a pretty random graph. I could try using Markov chains, which is something I am not that comfortable with. I am looking for a direct probabilistic approach. Any hints would be appreciated.
Thanks.
The graph is connected. Since the particle has a nonzero probability of moving to any of the neighbours of its current position, it also has a nonzero probability of reaching any position from any current position. Thus the probability that the particle has hit all the points in the infinite limit is $1$ (almost sure).