Consider an array of \(L\) binary values all initially set as False. We want to flip them all to True by doing a \(T\)-block flip, where \(T\) consecutive values are flipped to be True (already True values stay True). What is the expected number of \(T\)-block flips until the entire array is True (assuming the position of each \(T\)-block flip is chosen randomly with equal probability for each site)?
Example: \(L = 10\), \(T = 4\): \(FFFFFFFFFF\). Flip \(1\) at position \(3\): \(FFTTTTFFFF\). Flip \(2\) at position \(5\): \(FFTTTTTTFF\), and so on.
Upper bound: If \(T = 1\), it is the coupon’s collectors problem, which has the solution \(E(X) = LH_L \approx L \ln(L) + \gamma + \frac{1}{2L}\)
Lower bound: If \(T = L\), then we simply need one \(T\)-block flip at position 1, which has \(E(X) = L\)