Average Number Of Distinct Sites Research Articles

Number of distinct sites visited by a resetting random walker

We investigate the number V p (n) of distinct sites visited by an n-step resetting random walker on a d-dimensional hypercubic lattice with resetting probability p. In the case p = 0, we recover the well-known result that the average number of distinct sites grows for large n as ⟨V 0(n)⟩ ∼ n d/2 for d < 2 and as ⟨V 0(n)⟩ ∼ n for d > 2. For p > 0, we show that ⟨V p (n)⟩ grows extremely slowly as . We observe that the recurrence-transience transition at d = 2 for standard random walks (without resetting) disappears in the presence of resetting. In the limit p → 0, we compute the exact crossover scaling function between the two regimes. In the one-dimensional case, we derive analytically the full distribution of V p (n) in the limit of large n. Moreover, for a one-dimensional random walker, we introduce a new observable, which we call imbalance, that measures how much the visited region is symmetric around the starting position. We analytically compute the full distribution of the imbalance both for p = 0 and for p > 0. Our theoretical results are verified by extensive numerical simulations.

The average number of distinct sites visited by a random walker on random graphs

We study the linear large-n behaviour of the average number of distinct sites S(n) visited by a random walker after n steps on a large random graph. An expression for the graph topology-dependent prefactor B in S(n) = Bn is proposed. We use generating function techniques to relate this prefactor to the graph adjacency matrix and then devise message-passing equations to calculate its value. Numerical simulations are performed to evaluate the agreement between the message passing predictions and random walk simulations on random graphs. Scaling with system size and average graph connectivity are also analysed.

Quantifying exciton hopping in disordered media with quenching sites: Application to arrays of quantum dots

We present an analytical method for quantifying exciton hopping in an energetically disordered system with quenching sites. The method is subsequently used to provide a quantitative understanding of exciton hopping in a quantum dot (QD) array. Several statistical quantities that characterize the dynamics (survival probability, average number of distinct sites visited, average hopping distance, and average hopping rate in the initial stage) are obtained experimentally by measuring time-resolved fluorescence intensities at various temperatures. The time evolution of these quantities suggests in a quantitative way that at low temperature an exciton tends to be trapped at a local low-energy site, while at room temperature, exciton hopping occurs repeatedly, leading to a large hopping distance. This method will serve to facilitate highly efficient optoelectronic devices using QDs such as photovoltaic cells and light-emitting diodes, since exciton hopping is considered to strongly influence their operational parameters. The presence of a dark QD (quenching site) that exhibits fast decay is also quantified.

Exploration and Trapping of Mortal Random Walkers

Exploration and trapping properties of random walkers that may evanesce at any time as they walk have seen very little treatment in the literature, and yet a finite lifetime is a frequent occurrence, and its effects on a number of random walk properties may be profound. For instance, whereas the average number of distinct sites visited by an immortal walker grows with time without bound, that of a mortal walker may, depending on dimensionality and rate of evanescence, remain finite or keep growing with the passage of time. This number can in turn be used to calculate other classic quantities such as the survival probability of a target surrounded by diffusing traps. If the traps are immortal, the survival probability will vanish with increasing time. However, if the traps are evanescent, the target may be spared a certain death. We analytically calculate a number of basic and broadly used quantities for evanescent random walkers.

Revisiting random walks in fractal media: On the occurrence of time discrete scale invariance

This paper addresses the kinetic behavior of random walks in fractal media. We perform extensive numerical simulations of both single and annihilating random walkers on several Sierpinski carpets, in order to study the time behavior of three observables: the average number of distinct sites visited by a single walker, the mean-square displacement from the origin, and the density of annihilating random walkers. We found that the time behavior of those observables is given by a power law modulated by soft logarithmic-periodic oscillations. We conjecture that logarithmic-periodic oscillations are a manifestation of a time domain discrete scale iNvariance (DSI) that occurs as a consequence of the spatial DSI of the substrate. Our conjecture implies that the logarithmic periods of oscillations in space and time domains are linked by a dynamic exponent z, through z=log(tau)/log(b(1)), where tau and b(1) are the fundamental scaling ratios of the DSI symmetry in the time and space domains, respectively. We use this relationship in order to compute z for different observables and fractals. Furthermore, we check the values obtained with independent measurements provided by the power-law behavior of the mean-square displacement with time [R(2)(t) proportional variant t(2/z)]. The very good agreement obtained between both computations of the z exponent gives strong support to the idea of an intimate interplay between spatial and time symmetry properties that we expect will have a quite general scope. We expect that the application of the outlined concepts in the field of dynamic processes in fractal media will stimulate further research.

Target decay on irregular networks

We investigate the survival probability of immobile targets, which get annihilated by randomwalkers (RWs) at first encounter. We focus on scale-free networks (SFNs) and on smallworld networks (SWNs) as examples of irregular lattices. On SFNs we consider degreedistributions (DDs) with long time tails. Interestingly, it turns out that the survival probabilityand the quality of its description through the average number of distinct sites visited,Sn, depend on the details of the DD: SFNs which are more ramified have survivalprobabilities which are more regular, whereas SFNs with long chain-like segmentsdisplay decay laws similar to those of SWNs, where a description only in terms ofSn is rather poor.

Scaling properties of random walks on small-world networks.

Using both numerical simulations and scaling arguments, we study the behavior of a random walker on a one-dimensional small-world network. For the properties we study, we find that the random walk obeys a characteristic scaling form. These properties include the average number of distinct sites visited by the random walker, the mean-square displacement of the walker, and the distribution of first-return times. The scaling form has three characteristic time regimes. At short times, the walker does not see the small-world shortcuts and effectively probes an ordinary Euclidean network in d dimensions. At intermediate times, the properties of the walker shows scaling behavior characteristic of an infinite small-world network. Finally, at long times, the finite size of the network becomes important, and many of the properties of the walker saturate. We propose general analytical forms for the scaling properties in all three regimes, and show that these analytical forms are consistent with our numerical simulations.

Anomalous behavior of the target decay on small world networks

We investigate on small world networks (SWN’s) the survival probability of immobile targets, which get annihilated by random walkers at first encounter. On SWN’s we find (distinct from regular lattices, Cayley trees, and regular ultrametric spaces) that in general the survival probability cannot be directly related to the average number of distinct sites visited. We underline this finding with arguments related to the structural disorder of SWN’s and through the derivation of a lower bound for the targets’ decay.

Number of distinct sites visited by a random walker trapped by an absorbing boundary.

The number of distinct sites visited by a lattice random walker is a subject of continuing interest in both mathematics and physics. All previous investigations have used the assumption that the lattice is unbounded. An assessment of the amount of tissue interrogated by a photon in reflectance measurements for diagnostic purposes suggests analyzing properties of the average number of distinct sites visited by a random walker trapped by an absorbing plane at time t. We show that for sufficiently large t this number is the same as the average number of distinct sites visited for this time when the surface is not present. A more complete analysis is possible for a random walk on a line terminated by an absorbing point.

Scaling exponents for random walks on Sierpinski carpets and number of distinct sites visited: a new algorithm for infinite fractal lattices

The scaling exponent for the mean square distance covered in a random walk (dw) and the average number of distinct sites visited (dn) are determined for a family of Sierpinski carpet patterns. We suggest a new random walk algorithm to generate walks on an effectively infinite deterministic fractal lattice. The algorithm is applied to several Sierpinski carpet patterns with the same Hausdorff dimension. We show that the systems have a quite different scaling exponent dw and, further, that the generally accepted result dn = ds does not hold for all of these, where ds is the spectral dimension.

Number of distinct sites visited by Lévy flights injected into a d-dimensional lattice

We study the average number of distinct sites $〈{S}_{{N}_{0}}(t)〉$ visited by L\'evy flights injected in the center of a lattice: ${N}_{0}$ new particles appear in the center of the lattice at each time step. L\'evy flights are particles which have the probability $p(\mathcal{l})=A{\mathcal{l}}^{\ensuremath{-}(1+\ensuremath{\alpha})},0&lt;\ensuremath{\alpha}&lt;2$ of making an $\mathcal{l}$-length jump. We show analytically that the asymptotic form of $〈{S}_{{N}_{0}}(t)〉$ is related to that of the case of constant initial number $N$ of particles. We find that different ranges of $\ensuremath{\alpha}$ correspond to different limits, $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ and $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$, in the behavior of the number of sites visited by constant number of particles. The results obtained analytically are in good agreement with Monte Carlo simulations. We also discuss possible results for $\ensuremath{\alpha}&gt;~2$.

Dimensional crossovers and anomalous scaling of single and reacting random walkers in baguettelike lattices: Monte Carlo simulations of the number of distinct sites visited and of bimolecular A+A and A+B reactions.

We performed Monte Carlo simulations on baguettelike lattices of random-walk-based bimolecular $A+A$ and $A+B$ reactions, and of the number of distinct sites visited. The emphasis is on the crossover times, from high (two- or three-) dimensional behavior to one-dimensional behavior, and their scaling laws with respect to tube width. We find that these dimensional crossovers deviate significantly from a mean square displacement law and are specific to both tube dimensionality (2 or 3) and reaction type (e.g., $A+A$ or $A+B$), instead of an expected power of 2, the exponents range between 1 and 4. Thus, the global information propagation is either faster or slower than single particle diffusion. The fractional densities of the $A+B$ reactions at the dimensional crossover are compared to the fractional densities at the segregation crossover in nonconfined media. The time evolutions of the $A+A$ reactions approximately mimic those of the average number of distinct sites visited. All asymptotic time behaviors exhibit one-dimensional character.

Random walks, trapping and reactions in quasi-one dimensional lattices

We performed Monte Carlo simulations of diffusion processes on baguette-like lattices, which have very small width and height, but long length, in effect providing quasi-one dimensional systems. This is done by investigating random walk properties of single particles, and also the well known model bimolecular reactions A + A and A + B. We monitor the number of distinct sites visited, as a function of time, and also the survival probability in the presence of static traps. For the reaction systems we monitor, as usual, the decay of the particle density. The expected one-dimensional behavior is recovered, in the long time limit, for all cases studied. Our interest here is in the crossover time, from 3- (or 2-)dimensionaI behavior (early time) to one-dimensional (long time). We find that this crossover time scales with respect to the baguette's short dimension. However, this scaling deviates significantly from a mean square displacement law, and it is specific to both tube dimensionality and reaction type (e.g. A + A or A + B). Specifically, instead of an expected power of 2, the exponents range between 1 and 4. The densities of the A + B reactions at the dimensional crossover are compared to the densities at the segregation crossover in regular lattices at all three dimesnions. As expected, the time evolution of the A + A reaction parallels the behavior of the average number of distinct sites visited.

Dimensional recurrence formulas for Green functions of cubic lattices

I apply the theory of random walks to prove dimensional recurrence formulas for return probabilities and Green functions of cubic lattices. I use these formulas to compute the average number of distinct sites visited by the random walker on an “almost one-dimensional” lattice.

The number of distinct sites visited by a tracer particle

The authors calculate the average number of distinct sites visited by a tracer particle in a three- or higher-dimensional lattice gas with vacancy density rho , in the limit of low density and large times, i.e., rho 0 and rho t>>1.

The average number of distinct sites visited by a one-dimensional random walker and its application to isotope exchange in polypeptides

The average number of distinct sites visited by a random walker moving with arbitrary transition probability on a one-dimensional lattice is calculated. Asymptotic forms of this quantity for both asymmetric and symmetric random walks are determined, and an exact solution for the latter case is also given for any number of steps. The average number of sites visited is then analyzed for intermediate numbers of steps by introducing an exponent. This approach is applied to explain the results of isotope exchange experiments in polypeptides, and applications of asymmetric random walks to other biological problems are briefly discussed.

Random walks with variable step length on regular lattices

We study random walks on regular lattices, where the probability of steps of length r is proportional to r −6. For the diamond and the three cubic lattices we determine numerically the average number of distinct sites S( n) visited in an n-step walk. Applied to walks restricted to nearest-neighbor steps, our method agrees with the analytical expressions.

The Number of Distinct Sites Visited in a Random Walk on a Lattice

A general formalism is developed from which the average number of distinct sites visited in n steps by a random walker on a lattice can be calculated. The asymptotic value of this number for large n is shown to be (8n/π)12 for a one-dimensional lattice and cn for lattices of three or more dimensions. The constant c is evaluated exactly, with the help of Watson's integrals, for the simple cubic, body-centered cubic, and face-centered cubic lattices. An analogy is drawn with an electrical network in which unit resistors replace all near-neighbor bonds in a lattice, and the resistance of such a network on each of the three cubic lattices is evaluated.