PhD: Starving random walks First part: We develop a framework to determine the complete statistical behavior of a fundamental quantity in the theory of random walks, namely, the probability that n1, n2, n3, . . . distinct sites are visited at times t1, t2, t3, ... . From this multiple-time distribution, we show that the visitation statistics of 1d random walks are temporally correlated and we quantify the non-Markovian nature of the process. We exploit these ideas to derive unexpected results for the two-time trapping problem and also to determine the visitation statistics of two important stochastic processes, the run-and-tumble particle and the biased random walk (Arxiv link, published in Physical Review E ).
Exploration process of a one dimensional line by a random walker.
Second part: In this work, we introduce a fundamental quantity, the elapsed time τn between visits to the nth and the (n+1)st distinct sites, from which the full dynamics about the visitation statistics can be obtained. Despite the geometrical complexity of the territory explored by a random walk (typically aspherical, as well as containing holes and islands at all scales), we find that the distribution of the τn can be accounted for by simple analytical expressions. Processes as varied as regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, fall into the same universality classes for the temporal history of distinct sites visited. We confirm our theoretical predictions by Monte Carlo and exact enumeration methods. We also determine additional basic exploration observables, such as the perimeter of the visited domain or the number of islands of unvisited sites enclosed within this domain, thereby illustrating the generality of our approach. Because of their fundamental character and their universality, these inter-visit times represent a promising tool to unravel many more aspects of the exploration dynamics of random walks (Arxiv link, published in Nature Communications).
On the left, territory visited by the random walker in 2d: how long before finding food/a new site? On the right, representation of different exploration processes.
Third part: We introduce range-controlled random walks with hopping rates depending on the range N, that is, the total number of previously visited sites. We analyze a one-parameter class of models with a hopping rate Na, and determine the large time behavior of the average range, as well as its complete distribution in two limit cases. We find that the behavior drastically changes depending on whether the exponent a is smaller, equal, or larger than the critical value, ad, depending only on the spatial dimension d. When a>ad, the forager covers the infinite lattice in a finite time. The critical exponent is a1=2 and ad=1 when d≥2. We also consider the case of two foragers who compete for food, with hopping rates depending on the number of sites each visited before the other. Surprising behaviors occur in one dimension where a single walker dominates and finds most of the sites when a>1, while for a smaller than 1, the walkers evenly explore the line. We compute the gain of efficiency in visiting sites by adding one walker. (Arxiv link).
Illustration of a walker slowing down by visiting new sites.