Random Walking in Combville The (infinite) city Combville consists of an infinite main road, along which, at regular intervals of size 1, are infinite side walks (see Fig XXXXXXXXXXA walker performs a...

Random Walking in Combville The (infinite) city Combville consists of an infinite main road, along which, at regular intervals of size 1, are infinite side walks (see Fig. 7.3). A walker performs a random walk in the city, where at every time step if they are on one of the junctions on the main road, they take a unit step with equal probability to each of the three possible directions, and otherwise they take a step up or down along the side street with equal probability (e.g., a possible route is (0 , 0) ? (1 , 0) ? (1 , 1) ? (1 , 2) ? (1 , 1) ? (1 , 0) ? (2 , 0) . . . ). (a) If a walker just entered a side street, what is the probability distribution function for returning to the main street? How long would it take them to return on average? b) If the walker started on the main road at x = 0, how would their mean squared position along the main road scale with time?