Complete visitation statistics of one-dimensional random walks

Full-record statistics of one-dimensional random walks

We develop a comprehensive framework for analyzing full-record statistics, covering record counts M(t_{1}),M(t_{2}),..., their corresponding attainment times T_{M(t_{1})},T_{M(t_{2})},..., and the intervals until the next record. From this multiple-time distribution, we derive general expressions for various observables related to record dynamics, including the conditional number of records given the number observed at a previous time and the conditional time required to reach the current record given the occurrence time of the previous one. Our formalism is exemplified by a variety of stochastic processes, including biased nearest-neighbor random walks, asymmetric run-and-tumble dynamics, and random walks with stochastic resetting.

From Maximum of Inter-Visit Times to Starving Random Walks

Very recently, a fundamental observable has been introduced and analyzed to quantify the exploration of random walks: the time τ_{k} required for a random walk to find a site that it never visited previously, when the walk has already visited k distinct sites. Here, we tackle the natural issue of the statistics of M_{n}, the longest duration out of τ_{0},…,τ_{n-1}. This problem belongs to the active field of extreme value statistics, with the difficulty that the random variables τ_{k} are both correlated and nonidentically distributed. Beyond this fundamental aspect, we show that the asymptotic determination of the statistics of M_{n} finds explicit applications in foraging theory and allows us to solve the open d-dimensional starving random walk problem, in which each site of a lattice initially contains one food unit, consumed upon visit by the random walker, which can travel S steps without food before starving. Processes of diverse nature, including regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, share common properties within the same universality classes.

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.