Question: (a) Random walk on a finite graph. Let E be a finite set and a symmetric relation on E. Here, E is interpreted as

(a) Random walk on a finite graph. Let E be a finite set and a symmetric relation on E. Here, E is interpreted as the vertex set of a graph, and the relation x y means that x and y are connected by an (undirected) edge. Suppose that each vertex is connected by an edge to at least one other vertex or to itself. Let d.x/ D j¹y 2 E W x yºj be the degree of the vertex x 2 E, and set ….x; y/ D 1=d.x/ if y x, and ….x; y/ D 0 otherwise. The Markov chain with transition matrix … is called the random walk on the graph .E;/. Under which conditions on the graph .E;/ is … irreducible? Find a reversible distribution for ….

(b) Random walk of a knight. Consider a knight on an (otherwise empty) chess board, which chooses each possible move with equal probability. It starts (i) in a corner, (ii) in one of the 16 squares in the middle of the board. How many moves does it need on average to get back to its starting point?

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