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We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time n, whose moves to the right or to the left are prescribed by the rise-and-descent sequence associated with a given random permutation. We determine exactly the probability of finding the trajectory of such a permutation-generated random walk at site X at time n, obtain the probability measure of different excursions and define the asymptotic distribution of the number of “U-turns" of the trajectories - permutation “peaks" and “through". In the second part, we focus on some statistical properties of surfaces obtained by randomly placing natural numbers 1,2,3, ...,L on sites of a 1d or 2d lattices containing L sites. We calculate the distribution function of the number of local “peaks" - sites the number at which is larger than the numbers appearing at nearest-neighboring sites - and discuss surprising collective behavior emerging in this model.

The first-passage time, defined as the time a random walker takes to reach a target point in a confining domain, is a key quantity in the theory of stochastic processes. Its importance comes from its crucial role in quantifying the efficiency of processes as varied as diffusion-limited reactions, target search processes or the spread of diseases. Most methods of determining the properties of first-passage time in confined domains have been limited to Markovian (memoryless) processes. However, as soon as the random walker interacts with its environment, memory effects cannot be neglected: that is, the future motion of the random walker does not depend only on its current position, but also on its past trajectory. Examples of non-Markovian dynamics include single-file diffusion in narrow channels, or the motion of a tracer particle either attached to a polymeric chain or diffusing in simple or complex fluids such as nematics, dense soft colloids or viscoelastic solutions. Here we introduce an analytical approach to calculate, in the limit of a large confining volume, the mean first-passage time of a Gaussian non-Markovian random walker to a target. The non-Markovian features of the dynamics are encompassed by determining the statistical properties of the fictitious trajectory that the random walker would follow after the first-passage event takes place, which are shown to govern the first-passage time kinetics. This analysis is applicable to a broad range of stochastic processes, which may be correlated at long times. Our theoretical predictions are confirmed by numerical simulations for several examples of non-Markovian processes, including the case of fractional Brownian motion in one and higher dimensions. These results reveal, on the basis of Gaussian processes, the importance of memory effects in first-passage statistics of non-Markovian random walkers in confinement.

We consider a model of surface-mediated diffusion with alternating phases of bulk and surface diffusion for two geometries: the disk and rectangles. We develop a spectral approach to derive an exact formula for the mean exit time of a particle through a hole on the boundary. The spectral representation of the mean exit time through the eigenvalues of an appropriate self-adjoint operator is particularly well-suited to investigate the asymptotic behavior in the limit of large desorption rate $$\lambda $$ λ . For a point-like target, we show that the mean exit time diverges as $$\sqrt{\lambda }$$ λ . For extended targets, we establish the asymptotic approach to a finite limit. In both cases, the mean exit time is shown to asymptotically increase as $$\lambda $$ λ tends to infinity. That implies that the pure bulk diffusion is never an optimal search strategy. We also investigate the influence of rectangle elongation onto the mean exit time, in particular, the dependence of the critical ratio of bulk and surface diffusion coefficients on the rectangle aspect ratio. We show that the intermittent search strategy can significantly outperform pure surface diffusion for elongated rectangles.

In prokaryotes and eukaryotes, genes are transcribed stochastically according to various temporal patterns that range from simple first-order kinetics to marked bursts, resulting in temporal and cell-to-cell variations of mRNA and protein levels. Here, we consider the effect of the transport of regulatory molecules on the noise in gene expression by taking into account explicitly the dynamics of a finite number of transcription factors confined in the cell. We calculate analytically time-dependent correlation functions of mRNA levels for a wide range of transport mechanisms and find that in the limit of small-transcription-factor copy number, the results differ significantly from standard approaches, which ignore confinement. It is shown how such dynamical quantities, which can now be obtained experimentally, can be used to identify the underlying mechanisms of transcription. Of particular importance, it is demonstrated that the geometry of transcription-factor trajectories in the cellular environment plays a key role in transcription kinetics, and can intrinsically generate the observed various transcription patterns ranging from simple first-order kinetics to bursts.

It is widely recognized that the cleaving rate of a restriction enzyme on target DNA sequences is several orders-of-magnitude faster than the maximal one calculated from the diffusion-limited theory. It was therefore commonly assumed that the target site interaction of a restriction enzyme with DNA has to occur via two steps: one-dimensional diffusion along a DNA segment, and long-range jumps coming from association-dissociation events. We propose here a stochastic model for this reaction which comprises a series of one-dimensional diffusions of a restriction enzyme on nonspecific DNA sequences interrupted by three-dimensional excursions in the solution until the target sequence is reached. This model provides an optimal finding strategy which explains the fast association rate. Modeling the excursions by uncorrelated random jumps, we recover the expression of the mean time required for target site association to occur given by Berg et al. in 1981, and we explicitly give several physical quantities describing the stochastic pathway of the enzyme. For competitive target sites we calculate two quantities: processivity and preference. By comparing these theoretical expressions to recent experimental data obtained for EcoRV-DNA interaction, we quantify: 1), the mean residence time per binding event of EcoRV on DNA for a representative one-dimensional diffusion coefficient; 2), the average lengths of DNA scanned during the one-dimensional diffusion (during one binding event and during the overall process); and 3), the mean time and the mean number of visits needed to go from one target site to the other. Further, we evaluate the dynamics of DNA cleavage with regard to the probability for the restriction enzyme to perform another one-dimensional diffusion on the same DNA substrate following a three-dimensional excursion.

An abstract of the article "Intermittent Search Strategies," by O. Bénichou, C. Loverdo, M. Moreau and R. Voituriez is presented.