First encounters on combs

We consider two random walkers embedded in a finite, two-dimension comb and we study the mean first-encounter time (MFET) evidencing (mainly numerically) different scalings with the linear size of the underlying network according to the initial position of the walkers. If one of the two players is not allowed to move, then the first-encounter problem can be recast into a first-passage problem (MFPT) for which we also obtain exact results for different initial configurations. By comparing MFET and MFPT, we are able to figure out possible search strategies and, in particular, we show that letting one player be fixed can be convenient to speed up the search as long as we can finely control the initial setting, while, for a random setting, on average, letting one player rest would slow down the search.