Anomalous persistence exponents for normal yet aging diffusion

Record ages of non-Markovian scale-invariant random walks

How long is needed for an observable to exceed its previous highest value and establish a new record? This time, known as the age of a record plays a crucial role in quantifying record statistics. Until now, general methods for determining record age statistics have been limited to observations of either independent random variables or successive positions of a Markovian (memoryless) random walk. Here we develop a theoretical framework to determine record age statistics in the presence of memory effects for continuous non-smooth processes that are asymptotically scale-invariant. Our theoretical predictions are confirmed by numerical simulations and experimental realisations of diverse representative non-Markovian random walk models and real time series with memory effects, in fields as diverse as genomics, climatology, hydrology, geology and computer science. Our results reveal the crucial role of the number of records already achieved in time series and change our view on analysing record statistics.

Range-Controlled Random Walks

We introduce range-controlled random walks with hopping rates depending on the range N, that is, the total number of previously distinct visited sites. We analyze a one-parameter class of models with a hopping rate N^{a} 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, a_{d}, depending only on the spatial dimension d. When a>a_{d}, the forager covers the infinite lattice in a finite time. The critical exponent is a_{1}=2 and a_{d}=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 1D where a single walker dominates and finds most of the sites when a>1, while for a<1, the walkers evenly explore the line. We compute the gain of efficiency in visiting sites by adding one walker.

Universal exploration dynamics of random walks

The territory explored by a random walk is a key property that may be quantified by the number of distinct sites that the random walk visits up to a given time. We introduce a more fundamental quantity, the time τ_n required by a random walk to find a site that it never visited previously when the walk has already visited n distinct sites, which encompasses the full dynamics about the visitation statistics. To study it, we develop a theoretical approach that relies on a mapping with a trapping problem, in which the spatial distribution of traps is continuously updated by the random walk itself. Despite the geometrical complexity of the territory explored by a random walk, the distribution of the τ_n 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.

Joint statistics of space and time exploration of one-dimensional random walks

The statistics of first-passage times of random walks to target sites has proved to play a key role in determining the kinetics of space exploration in various contexts. In parallel, the number of distinct sites visited by a random walker and related observables has been introduced to characterize the geometry of space exploration. Here, we address the question of the joint distribution of the first-passage time to a target and the number of distinct sites visited when the target is reached, which fully quantifies the coupling between the kinetics and geometry of search trajectories. Focusing on one-dimensional systems, we present a general method and derive explicit expressions of this joint distribution for several representative examples of Markovian search processes. In addition, we obtain a general scaling form, which holds also for non-Markovian processes and captures the general dependence of the joint distribution on its space and time variables. We argue that the joint distribution has important applications to various problems, such as a conditional form of the Rosenstock trapping model, and the persistence properties of self-interacting random walks.

Cell migration guided by long-lived spatial memory

Living cells actively migrate in their environment to perform key biological functions-from unicellular organisms looking for food to single cells such as fibroblasts, leukocytes or cancer cells that can shape, patrol or invade tissues. Cell migration results from complex intracellular processes that enable cell self-propulsion, and has been shown to also integrate various chemical or physical extracellular signals. While it is established that cells can modify their environment by depositing biochemical signals or mechanically remodelling the extracellular matrix, the impact of such self-induced environmental perturbations on cell trajectories at various scales remains unexplored. Here, we show that cells can retrieve their path: by confining motile cells on 1D and 2D micropatterned surfaces, we demonstrate that they leave long-lived physicochemical footprints along their way, which determine their future path. On this basis, we argue that cell trajectories belong to the general class of self-interacting random walks, and show that self-interactions can rule large scale exploration by inducing long-lived ageing, subdiffusion and anomalous first-passage statistics. Altogether, our joint experimental and theoretical approach points to a generic coupling between motile cells and their environment, which endows cells with a spatial memory of their path and can dramatically change their space exploration.