Growth models and quasi-random walks

Randomness is an important concept in mathematics and in real life. From finance to machine learning, from biology to climate change, probability and random processes play an important role. The current project has at its core processes that evolve in time according to quasi-random rules. Imagine a person that walks around in a very big city and, after each step he will choose the direction of the next step according to the outcome of a die toss. The die may be fair or not, or may even show the same number on all faces, so that we have no randomness involved. The rules for the steps of the walker are, in a simplified way of saying, encoded in the symbols that appear on the faces of the die. The person keeps rolling the die and keeps following the rules forever, if we suppose that he has infinite amount of time. We aim at understanding mathematically such processes. We are interested if, by following the above rules, the person will return to the starting point infinitely many times or not. If he does not return, where is he going to escape? Can we reconstruct his escape trajectory with high probability? These questions depend on the rules and on the dies we are choosing, but also on the geometry of the city where the person is walking. Let us look at two possibilities: the first one where there are many parallel streets and many ways one can reach a given corner or a shop, and the second one where at each corner there are many bifurcation streets, but only one shortest way to return to the same corner. The behaviour of the random walker is different in these two cases, and one main important issue of the project is to understand to each extend the geometry of the city the person is moving, influences the behaviour of the process. We are also interested in understanding how does the part of the city, that the walker visited, look like geometrically. If we let the person walk for one million steps, how probable is that he has visited all the corners and streets that are not further away than five kilometers distance from the starting corner? Or are there any streets that he has not seen yet, if we let the number of steps go to infinity? Finally, we dont want to have only one walker in the city, so we will send many persons move quasi-randomly in our city and we would like to understand if these persons will ever meet or not and how many steps do they have to take in average until meeting. How fast will these walkers cover the whole city, without letting any unvisited corner? These are all fundamental problems in the theory of random processes that we deal with within the current project, and they all have big range of applications, for instance in models of virus spread in a non homogeneous population.