Question

2. At the right is a section of an infinite grid--enough has been diagrammed to make the pattern clear. Some nodes are white and some are grey. Two nodes are neighbours" if they are connected by an edge. In each time unit, a counter occupying a node moves to each of its three neighbours with probability 1/3. (a) Suppose a counter is on a grey node. Calculate, by tracking the counter, the probability that it is on a grey node after 3 moves. (b) Find a way of keeping track of the change in the number of counters on white and grey nodes over time. We are looking here for a dynamical system. (c) Find a steady state distribution of counters on white and grey nodes-that is, proportions that remain constant over time. (d) Use the eigenvalues and eigenvectors of the system of (b) to get equations for the state at any time n and then set n- 3 and check your answer to (a)

Answer 1

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