Consider the hypercube $Q_4$. I would like to know how to compute the number of steps of a random walk in this graph such that the probability to be at a vertex is a given number $x$. I think I just have to invert the relation of the transition matrix... The probability to be at vertex $i$ should be $p_t(i)=(D^{-1}A)^tp_0$ where $D$ is the diagonal matrix with elements the degrees, $(D)_{ii}=d_i$, $A$ is the adjacency matrix of the graph and $p_0$ the initial probability distribution. I think I have to find the parameter $t$ such that one entry of the matrix is exactly that given number. The problem is that I can't use just a logarithm because of the power of the matrix. Can someone help me?
Thanks in advance.